text
stringlengths 1.79k
370k
| summary
stringlengths 34
23.5k
|
|---|---|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent years frustrated quantum spin systems on regular two - dimensional ( 2d ) lattices have aroused a great deal of research interest . @xcite in particular the interplay of magnetic frustration and quantum fluctuations has been seen to be a very effective route to destabilize or destroy magnetic order and thereby to create new quantum phases . such 2d magnetic systems can thus in turn develop a diverse array of phases with widely different ordering properties , such as antiferromagnets with quasiclassical nel ordering , quantum `` spirals '' , valence - bond crystals / solids , phases with nematic ordering , and spin liquids . other factors that influence the ground - state ( gs ) phase structures are the nature of the underlying crystallographic lattice , the number and nature of the bonds on this lattice , and the spin quantum numbers of the atoms localized to the sites on the lattice . the theoretical investigation of these models has proceeded hand in hand with the discovery and experimental investigation of ever more quasi-2d magnetic materials with novel properties . one of the most intensively studied of all of the frustrated 2d models is the spin-@xmath0 @xmath1@xmath2 heisenberg antiferromagnet ( haf ) on the square lattice with nearest - neighbor ( nn ) bonds ( of strength @xmath19 ) competing with next - nearest - neighbor ( nnn ) bonds ( of strength @xmath20 ) . this quantum system has two different quasiclassical phases with collinear magnetic long - range order ( lro ) at small ( @xmath21 ) and large ( @xmath22 ) values of the frustration strength parameter @xmath23 , separated by an intermediate quantum paramagnetic phase with no magnetic lro in the regime @xmath24 . interest in this model has been greatly stimulated recently by its experimental realization in such layered magnetic materials as li@xmath25vosio@xmath26,@xcite li@xmath25vogeo@xmath26,@xcite and vomoo@xmath26.@xcite the syntheses of such layered quasi-2d materials has stirred up a great deal of renewed interest in the model ( and see also , e.g. , refs . [ ] ) . amongst several methods that have been very successfully applied to the @xmath1@xmath2 model has been the coupled cluster method ( ccm),@xcite which has also been applied to many similar strongly - interacting and highly frustrated spin - lattice models with comparable success . other frustrated 2d models that have similarly engendered great recent interest include the spin-@xmath0 hafs on the triangular@xcite and kagome lattices.@xcite there has been a large amount of recent experimental investigation of the properties of quasi-2d magnetic materials with a ferromagnetic ( fm ) nn coupling ( @xmath27 ) and an antiferromagnetic ( afm ) nnn coupling ( @xmath28 ) . examples include pb@xmath29vo(po@xmath30)@xmath29,@xcite ( cucl)lanb@xmath29o@xmath31,@xcite srznvo(po@xmath30)@xmath29,@xcite bacdvo(po@xmath30)@xmath29,@xcite pbznvo(po@xmath30)@xmath29,@xcite and ( cubr)lanb@xmath25o@xmath32.@xcite these experimental studies have also served to reignite interest in the theoretical investigation of the gs and thermodynamic properties of the fm @xmath1@xmath2 model , i.e. , the model with fm nn exchange ( @xmath27 ) and frustrating afm nnn exchange ( @xmath5 ) . @xcite interestingly , arguments for the existence of a spin - nematic phase between two quasiclassical magnetically - ordered phases were presented.@xcite on the other hand , the existence of such a non - classical magnetically - disordered phase was also questioned in ref . other systems that have grown in importance in the last few years are various spin-@xmath0 magnetic models defined on the 2d honeycomb lattice . several such systems have been both theoretically and experimentally studied @xcite intensively , partly because of their special properties and partly due to the recent syntheses of various quasi-2d honeycomb - lattice materials . one reason for the theoretical interest in such models on the 2d honeycomb lattice is that a spin - liquid phase has been found for the exactly solvable kitaev model,@xcite in which the spin-@xmath0 particles reside on just such a lattice . furthermore , the honeycomb lattice is obviously germane to the very active research field of graphene , where the relevant physics may well be described by hubbard - like models on this lattice.@xcite interestingly , meng _ et al._@xcite found that for the hubbard model on the honeycomb lattice with moderate values of the coulomb repulsion parameter @xmath33 , strong quantum fluctuations lead to an insulating spin - liquid phase between the non - magnetic metallic phase and the afm mott insulator phase . from the experimental side recent observations on the spin-@xmath34 honeycomb - lattice haf bi@xmath35mn@xmath26o@xmath36(no@xmath35 ) demonstrate a spin - liquid - like behavior at temperatures much lower than the curie - weiss temperature.@xcite we have recently studied@xcite the afm @xmath1@xmath2@xmath3 honeycomb model for the case where the spin quantum number @xmath37 of each of the spins on every lattice site is @xmath38 , and with afm nearest - neighbor exchange bonds ( @xmath39 ) in the presence of frustration caused by afm nnn bonds ( @xmath5 ) and with next - next - nearest neighbor ( nnnn ) bonds of strength @xmath3 also present , for the special case where @xmath7 . we found@xcite that the scenario of deconfined criticality may hold for this model ( and see also ref . to date there exist only limited studies of the corresponding fm @xmath1@xmath2@xmath3 model ( namely where @xmath27 ) . in this paper , we further the investigation into the fm @xmath1@xmath2@xmath3 honeycomb model with fm nn bonds ( of strength @xmath27 ) in the presence of frustrating afm nnn bonds ( of strength @xmath5 ) and nnnn bonds ( of strength @xmath40 ) . once again we consider only the interesting special case where @xmath7 . we focus our attention in the present study particularly on the detection and characterization of the gs phases of the quantum model . bearing in mind the controversial discussion of the corresponding @xmath1@xmath2 square - lattice model with fm nn exchange bonds ( @xmath27 ) , the question naturally arises as to whether any indications for a non - classical magnetically - disordered phase might now be found for the honeycomb model . to determine the relevant gs phases and their properties we calculate the gs energy , the spin - spin correlation function , and the magnetic order parameter for the stripe - ordered state discussed below that is present as a gs phase in the corresponding classical version ( equivalent to the @xmath41 limit ) of the model . in view of its proven past ability to give results of high accuracy for a wide variety of highly frustrated 2d spin - lattice models , we again use the coupled cluster method ( ccm ) as our main computational tool in this paper . additionally , we use the exact diagonalization ( ed ) method for a large finite lattice of @xmath42 spins as a validity check of our ccm results . since at the classical level the model now under consideration also exhibits some similarities with the corresponding model with afm nn bonds ( @xmath39 ) , we compare our results for the quantum model of the fm case ( @xmath43 ) with those of the corresponding afm case ( @xmath44 ) . the rest of the paper is organized as follows . after describing the model in sec . [ model_section ] , we apply the ccm to investigate its gs properties . the ccm itself is very briefly described in sec . [ ccm ] , before presenting and discussing our ccm and ed results in sec . [ results ] . we conclude in sec . [ conclusions ] with a summary of the main results . the hamiltonian of the spin-@xmath0 @xmath1@xmath2@xmath3 heisenberg model on the honeycomb lattice , which we studied recently@xcite for the afm case ( @xmath39 ) is defined as @xmath45 where @xmath46 runs over all lattice sites on the lattice , and where @xmath47 runs over all nn sites connected to site @xmath46 by @xmath1 bonds , @xmath48 runs over all nnn sites connected to site @xmath46 by @xmath2 bonds , and @xmath49 runs over all nnnn sites connected to site @xmath46 by @xmath3 bonds , but counting each bond once and once only in the three sums . each site @xmath46 of the lattice carries a spin-@xmath0 particle with spin operator @xmath50 . we note that precisely the same model has also been studied recently on the square lattice , both in the case where all the bonds are afm in nature,@xcite and in the fm case where @xmath27 and @xmath51.@xcite the aim of the present work is now to study further the spin-@xmath0 @xmath1@xmath2@xmath3 fm model ( namely the above model in the case @xmath27 ) on the honeycomb lattice.@xcite the lattice and the exchange bonds are illustrated in fig . [ model](a ) . the classical gs phase diagram for the @xmath1@xmath2@xmath3 afm model ( with @xmath39 ) on the honeycomb lattice model displays collinear nel and striped phases , both afm in nature , as well as a spiral phase . these phases meet in a triple point at @xmath52 ( and for more details see , e.g. , ref . [ ] ) . for the remainder of this paper we again focus on the case where @xmath53 , but now where the nn exchange bond is fm in nature ( @xmath27 ) . the gs energies of the only two corresponding classical collinear states are then given by @xmath54 for the fm state and collinear striped afm state shown in figs . [ model](a ) and ( b ) respectively . if these were the only gs phases in this @xmath27 regime we would thus have a classical transition between the fm state and the striped afm state at @xmath55 ( @xmath10 for @xmath4 ) and a classical energy per site at this point of @xmath56 for the @xmath38 system with @xmath4 . for the corresponding afm model with @xmath39 such a striped afm state also exists as stated above , but the classical transition between the afm nel state and the striped afm state is at @xmath57 . the reason why the corresponding phase transition in the fm model occurs at a smaller value of the frustration parameter , @xmath58 than the value @xmath59 for the afm model is due to the @xmath3(@xmath60 ) nnnn exchange bonds that act to frustrate the fully polarized fm state , whereas they reinforce the afm nel state . by contrast , the @xmath2(@xmath60 ) nnn exchange bonds act to frustrate the @xmath61 bonds for both the fm state of the fm model and the nel state of the afm model . we note that the classical fm state is also an eigenstate of the hamiltonian , with energy eigenvalue equal to the energy of its classical fm counterpart . we note , however , that in fact the classical @xmath1@xmath2@xmath3 heisenberg model on the honeycomb lattice with @xmath27 , @xmath5 , and @xmath6 also has a spiral phase that intervenes in a very narrow strip between the fm phase and the collinear striped afm phase . ( in refs . [ ] this is referred to as phase v. ) the region in the @xmath62-@xmath63 plane ( where @xmath64 and @xmath65 ) in which it is the stable gs phase in the case @xmath27 is bounded by the three curves ( i ) @xmath66 , @xmath67 , ( ii ) @xmath68 , @xmath69 , and ( iii ) @xmath70 $ ] , @xmath71 . the point @xmath72 is a classical tetracritical point at which the spiral phase v meets the fm phase , the striped collinear afm phase , and the afm nel phase ( and see fig . 3 of ref . [ ] for further details ) . thus , in our case , where @xmath7 and @xmath27 , the classical spiral phase v exists in the narrow region @xmath73 . naturally this includes the point @xmath58 discussed above at which the fm and striped collinear afm phases would meet in the absence of the spiral phase v as a stable gs phase intervening between them . in all of our results below for the fm @xmath1@xmath2@xmath3 honeycomb system we henceforth set @xmath43 with no loss of generality , since this simply sets the overall scale of the hamiltonian , and we consider the case where @xmath53 , such that both the nnn and nnnn bonds act to frustrate the ferromagnetism the ccm ( see , e.g. , refs . [ ] and references cited therein ) that we use here is one of the most powerful and most versatile modern techniques in quantum many - body theory . it has been used to study various quantum magnets ( see , e.g. , refs . [ ] and references cited therein ) very successfully . the method is particularly suitable for investigating frustrated systems , due to the fact that some of the main alternative methods are restricted by certain problems that arise in such cases . for instance , quantum monte carlo ( qmc ) techniques suffer from the infamous and well - known `` sign problem '' for such systems . the exact ed method is also usually restricted by available computational power to relatively small finite - sized lattices . nevertheless it can often be used , as here , to provide a handy tool to check and validate the results of other numerical or approximate methods . we briefly describe here some of the important features of the ccm as applied to spin - lattice problems ( and see , e.g. , refs . [ ] and references cited therein for further details ) . the starting point for any ccm calculation is to select a normalized state @xmath74 as a reference or model state against which to incorporate in a systematic and potentially exact fashion the correlations present in the exact ground state . we often use a relevant classical ground state as the model state for spin systems for the sake of convenience , but other appropriate states may certainly also be used . in order to treat each site equivalently , a mathematical rotation of the local axes of the spins is conveniently performed in such a way that all spins in the reference state align along the same direction , say the negative @xmath75-axis . clearly , such rotations leave unchanged the su(2 ) commutation relations between components of the spin operators . the exact ket and bra gs energy eigenstates , @xmath76 and @xmath77 , of the many - body system are then parametrized in the ccm form as : 0.2 cm @xmath78 @xmath79 where @xmath80 are the schrdinger gs ket and bra equations respectively . the multiconfigurational creation operators @xmath81 are defined so that @xmath82 , and where we have defined @xmath83 . they are required to form a complete set of mutually commuting many - body creation operators in the hilbert space , defined with respect to @xmath74 as a cyclic vector . clearly the states are normalized such that @xmath84 . for spin - lattice systems they take the form of multi - spin raising operators and are written as products of single - spin raising operators , @xmath85 , where @xmath86 . the gs energy is calculated in terms of the correlation coefficients @xmath87 as @xmath88 ; and the average on - site magnetization @xmath89 in the rotated spin coordinates is calculated equivalently in terms of the coefficients @xmath90 as @xmath91 , which now plays the role of the order parameter . finally , the complete set of unknown ket- and bra - state correlation coefficients @xmath92 is calculated by requiring the expectation value @xmath93 to be a minimum with respect to all parameters @xmath94 . this readily leads to the coupled set of nonlinear equations for the ket - state creation correlation operators @xmath87 , @xmath95 , and to the coupled set of linear equations , @xmath96 , which can then be solved for the bra - state destruction correlation operators @xmath97 . when all many - body configurations @xmath98 are included in the expansions of the correlation expansions operators @xmath99 and @xmath100 , the ccm formalism is exact . however , it is necessary of course in practice to use approximation schemes to truncate the sets of configurations @xmath98 contained in the expansions of eqs . ( [ eq : ket_eq ] ) and ( [ eq : bra_eq ] ) for the ccm correlation operators . for systems defined on a regular periodic spatial lattice as here , it is convenient to use the well - established lsub@xmath101 approximation scheme in which all possible multi - spin - flip correlations over different locales on the ( here , honeycomb ) lattice defined by @xmath101 or few contiguous lattice sites are retained . clusters are defined to be contiguous in this sense if every site in the cluster is adjacent ( as a nearest neighbor ) to at least one other site in the cluster . this is the scheme we use for all our results presented below . the number @xmath102 of independent fundamental clusters ( i.e. , those that are inequivalent under the symmetries of the hamiltonian and of the model state ) increases rapidly with the truncation index @xmath101 , as shown in table [ table_fundconfig ] .number of fundamental lsub@xmath101 configurations ( @xmath102 ) for the collinear striped afm state of the spin-@xmath0 @xmath1@xmath2@xmath3 honeycomb model . [ cols="^,^",options="header " , ] for the present spin-@xmath0 @xmath1@xmath2@xmath3 model on the honeycomb lattice , where we use the natural lattice geometry itself to define the notion of adjacency inherent in the definition of the lsub@xmath101 scheme . we see , for example , that the number @xmath102 of such fundamental clusters ( and hence the number of simultaneous nonlinear equations we need to solve for the retained correlation coefficients @xmath87 ) for the striped model state is 250891 at the highest lsub12 level of approximation that we utilize here . the corresponding numbers , @xmath103 , of fundamental configurations are appreciably higher at a given lsub@xmath101 level when the spiral phase v is used as the ccm model state , due to the considerably reduced symmetry . it is necessary to use massive parallelization and supercomputing resources in order to perform the ccm calculations at such high level of approximation.@xcite thus , for example , to obtain a single data point ( i.e. , for a given value of @xmath7 ) for the striped model state at the lsub12 level typically requires about 0.5 h computing time using 1000 processors simultaneously . we present ccm results below based on the striped collinear afm state as model state , at various lsub@xmath101 levels of approximation with @xmath104 , and also in the corresponding @xmath105 extrapolated limits ( lsub@xmath106 ) based on the well - tested extrapolation schemes described below and in more detail elsewhere.@xcite we have also performed extrapolations for the data set with @xmath107 . both sets of results agree well with one another , which gives added credence to our results . note that we do not use the lsub@xmath101 approximation scheme for values @xmath108 of the truncation index , since these low - order approximations will not capture the natural hexagonal structure of the lattice . we remark that , as always , the ccm exactly obeys the goldstone linked - cluster theorem at every lsub@xmath101 level of approximation . hence we work from the outset in the limit @xmath109 , where @xmath110 is the number of sites on the honeycomb lattice , and extensive quantities such as the gs energy are always guaranteed to be linearly proportional to @xmath110 in this limit . we clearly do not need to perform any finite - size scaling of our results , as all ccm approximations are automatically performed from the outset in the infinite - lattice limit , @xmath111 , as discussed above . it is , however , necessary to extrapolate to the exact @xmath105 limit in the lsub@xmath101 truncation index @xmath101 , in which limit the complete ( infinite ) hilbert space is reached . for the gs energy per spin , @xmath112 , a well - tested and very accurate extrapolation ansatz ( and see , e.g. , refs . , [ ] ) is @xmath113 while for the magnetic order parameter , @xmath89 , different schemes have been employed for different situations . for models showing no or only relatively small amounts of frustration , a well - tested and accurate rule ( and see , e.g. , refs . [ ] ) is @xmath114 for highly frustrated systems , particularly those showing a gs order - disorder transition , a more appropriate extrapolation rule with fixed exponents that has been found to give good results ( and see , e.g. , refs . [ ] ) is @xmath115 we give illustrations here of the use of each of these schemes , wherever and whenever possible . we now present and discuss our ccm results . in order to have an independent check on the accuracy and consistency of our ccm results , we have also performed additional computations of selected gs properties of the present models using the ed technique that is a well - established and successful tool for studying frustrated quantum spin systems ( and see , e.g. , refs . [ ] ) . in fig . [ e](a ) we show the ccm results for the gs energy per spin , @xmath112 , in various lsub@xmath101 approximations based on the striped state as ccm model state , as well as the exact gs energy for a finite lattice of size @xmath42 . we also show separately , in fig . [ e](b ) , the extrapolated ( @xmath105 ) results obtained from eq . ( [ e_extrapo ] ) using the data set @xmath104 . comparison is made with the results for the corresponding afm version of the model with @xmath12 . the ccm lsub@xmath101 data displayed in fig . [ e](a ) show that the gs energy results converge extremely rapidly as the truncation index @xmath101 is increased , such that the difference between the lsub12 results and the extrapolated ( @xmath105 ) results obtained from eq . ( [ e_extrapo ] ) is very small indeed . we note too that , just as in the corresponding afm case of the model with @xmath12 , the various ccm lsub@xmath101 solutions based on the striped model state now also terminate at some lower termination point @xmath116 as @xmath2 is decreased . such terminations of ccm solutions are very common and have been very well documented.@xcite in all such cases a termination point always arises due to the solution of the ccm equations becoming complex at this point , beyond which there exist two branches of entirely unphysical complex conjugate solutions.@xcite in the region where the solution reflecting the true physical solution is real there actually also exists another ( unstable ) real solution . however , only the shown branch of these two solutions reflects the true ( stable ) physical ground state , whereas the other branch does not . the physical branch is usually easily identified in practice as the one which becomes exact in some known ( e.g. , perturbative ) limit . this physical branch then meets the corresponding unphysical branch at some termination point with infinite slope , beyond which no real solutions exist . the lsub@xmath101 termination points are themselves also reflections of the quantum phase transitions in the real system and may be used to estimate the position of the phase boundary,@xcite although we do not do so for this critical point in the fm model , since we have more accurate criteria that we now discuss . we note first from fig . [ e](a ) that the lsub@xmath101 termination points using the striped state as the ccm model state for the present fm version of the model with @xmath4 , lie very close indeed to the points where the curves cross ( or nearly cross ) the corresponding curve for the fm state given by eq . ( [ h_classical ] ) . this gives us our first evidence that either there is no intermediate phase between the quantum striped phase and the fm phase for the case @xmath4 , or , if one exists , it can occur only over a very narrow regime indeed . this situation may be contrasted with that of the afm version of the model ( @xmath12),@xcite where the lsub@xmath101 results for the gs energy using the striped model state terminate before they meet the corresponding results using the nel state as model state ( which themselves also terminated at some upper termination points that were lower in value than the lower termination points for the striped state ) . in the latter case there is an intermediate plaquette valence - bond crystal ( pvbc ) phase . at the classical level the difference in the values of the gs energy per spin of the collinear striped states between the two @xmath38 cases ( i.e. , for positive and negative values of @xmath1 with @xmath117 ) is 0.25 , independent of @xmath2 and @xmath3 . the quantum versions follow this pattern for larger values of @xmath7 , as seen from fig . [ e ] , but the constancy in the difference breaks down at around @xmath118 , where the afm case ( @xmath12 ) exhibits a critical point marking a transition to the pvbc phase , which then in turn undergoes a further phase transition to the nel phase at another lower critical value . the corresponding best available ccm estimates for those two critical values for the afm case of the model with @xmath119 are @xmath14 and @xmath17 respectively.@xcite in the present fm case of the model with @xmath120 we see no evidence ( apart from the seeming termination of the solutions to the equations for the lsub@xmath121 approximation based on the striped state as ccm model state very slightly before the gs energy crossing point with the fm state ) of any similar intermediate state between the fm state and the collinear striped afm state . if any such intermediate state exists at all , however , it must be confined to a very narrow region indeed around @xmath122 , probably confined to @xmath18 . we return to a more detailed discussion of this region later . for the moment we note only that it is much reduced from the region @xmath123 in which the corresponding classical version of the model has the spiral phase v as its stable gs phase . for the present fm case with @xmath4 the ccm lsub@xmath101 gs energy curves using the striped model state cross the corresponding gs energy curve for the fm state from eq . ( [ h_classical ] ) for @xmath107 at corresponding critical values @xmath124 ( where @xmath125 ) , @xmath126 ( where @xmath127 ) , and @xmath128 ( where @xmath129 ) . the corresponding lsub@xmath121 result for the gs energy using the striped state as the ccm model state appears to terminate just before meeting the gs energy curve for the fm phase . however we note that for such very high - order ccm calculations it becomes very computationally expensive to determine the termination point with high accuracy . if we use the extrapolated lsub@xmath106 results for the gs energy for the striped phase by making use of eq . ( [ e_extrapo ] ) and employing the whole data set @xmath104 , we thus need to perform a further very small extrapolation of the ccm results to lower values of @xmath2 to find the presumed crossing point of the energies of the striped and fm phases , in the scenario in which these two phases meet at a first - order transition with no intermediate phase ( that would itself be confined to the very narrow intervening region @xmath18 , as discussed above ) . as expected , simple power - law expansions give very accurate fits , and give crossing points very close to those above . putting all the energy data together , our best estimate for the critical point of the first - order phase transition from the collinear striped phase to the fm phase ( in the scenario where this transition occurs directly , with no intermediate phase confined to the narrow region @xmath18 ) for the spin-@xmath0 @xmath1@xmath2@xmath3 heisenberg ferromagnet ( with @xmath4 ) on the honeycomb lattice , and with @xmath53 , is @xmath130 , at which point the gs energy per spin is @xmath131 . we see from fig . [ e](a ) that the agreement between the ed ( @xmath42 ) and the ccm energies is very satisfactory . moreover , due to the finite - size scaling of the gs energy , @xmath132 with @xmath133 ( and see , e.g. refs . [ ] and [ ] ) , the difference between the ccm and the ed gs energies would become even smaller if finite lattices of larger size could be considered . the ed turnover point in the energy curve that marks the termination of the fm phase occurs at a value of about 0.1003 for the @xmath42 lattice used , and for the same reasons as above this value will increase as @xmath110 is increased . thus , in summary , while the ccm estimates for the gs energy per spin for the spin-@xmath0 @xmath1@xmath2@xmath3 heisenberg model on the extended infinite honeycomb lattice are much more accurate than the ed results , the latter do serve as an independent check on the former . the hypothetical phase transition ( i.e. , when the existence of the intervening spiral phase v is momentarily ignored ) from fm order to collinear striped afm order for the classical version of the fm model with @xmath4 occurs at a value @xmath134 , compared with the corresponding value @xmath135 found here . thus quantum fluctuations act to stabilize the collinear afm order , at the expense of the fm order , to higher values of frustration than in the classical case . it is interesting to note that a similar situation was found in the fm version ( @xmath136 ) of the spin-@xmath0 @xmath1@xmath2 model on the square lattice,@xcite where a quantum critical point exists at a value @xmath137 for a similar transition from fm order to collinear striped order , compared with a corresponding classical value of @xmath138 . it is well known , from many cases studied , that quantum fluctuations almost always favor collinear states over noncollinear ones ( e.g. , spiral or canted states ) . what is interesting in both the present case and the spin-@xmath0 @xmath1@xmath2 model on the square lattice cited above , is that quantum fluctuations seem also to favor one collinear state ( namely the collinear striped afm state in these two cases ) where the quantum fluctuations are present , over another collinear state ( namely the fm state in these two cases ) where quantum fluctuations are absent . it is intriguing to wonder whether these are examples of a more general rule . we present results in fig . [ m ] for the ccm collinear stripe order parameter @xmath89 , as defined in sec . figure [ m](a ) shows lsub@xmath101 results with @xmath104 , while fig . [ m](b ) shows the corresponding extrapolated ccm lsub@xmath106 ( @xmath139 ) results using both eqs . ( [ m_extrapo_standard ] ) and ( [ m_extrapo_frustrated ] ) . we note firstly that the ccm lsub@xmath101 order parameter results depend on the approximation level @xmath101 much more strongly than those for the gs energy . it is clear that the order parameter behaves similarly for large values of @xmath140 for both the fm model ( @xmath27 ) and the afm model ( @xmath39 ) . however , once again there are considerable differences in the behavior of @xmath89 between the two models for values of the frustration parameter @xmath13 . the extrapolated ccm results for @xmath89 for the afm model in fig . [ m](b ) clearly show the breakdown of the quasiclassical collinear magnetic lro near the critical value of @xmath118 , i.e. , significantly above the classical transition point @xmath141 ( and see , e.g. , refs . [ ] ) . indeed , the ccm estimate for the critical value of the frustration parameter in the afm case for the disappearance of collinear striped order is @xmath14 from the point at which @xmath89 becomes zero , using the extrapolation scheme of eq . ( [ m_extrapo_frustrated]).@xcite by contrast , the order parameter for the fm model stays almost constant over the whole parameter region shown in fig . we do not observe any indication of the breakdown of the collinear striped magnetic lro until @xmath122 for the fm model , which is below the hypothetical classical transition point @xmath142 , as we observed previously in the results for the gs energy . lastly , we present results for various spin - spin correlation functions for the fm as well as for the afm model in fig . [ corrfunctn ] . figure [ corrfunctn](a ) shows the ccm lsub10 results and fig . [ corrfunctn](b ) shows the corresponding ed results . once again we note that for large values of the frustration parameter @xmath140 the corresponding spin - spin correlations functions for both the fm ( @xmath4 ) and afm ( @xmath12 ) models agree remarkably well with one another for both the ccm and ed calculations . furthermore , for the fm model the agreement of the ccm correlation functions with the ed data is excellent . for the afm model the agreement between the ccm and ed results is again excellent for values of @xmath140 above the transition point at which the afm collinear striped order disappears , namely @xmath143 , but around and below this value there are noticeable differences . in particular , the very steep change in the correlation functions at @xmath144 present in the ed ( @xmath42 ) data for the afm model is not observed in the ccm lsub10 data . instead the ccm data show a smoother change in that region . however , we have argued@xcite that for @xmath145 no striped magnetic lro order exists . indeed we argued that no magnetic lro order exists at all for the afm model in the regime @xmath146 , where instead we have a pvbc state . hence , it is not surprising that the ccm solution in a finite order of lsub@xmath101 approximation based on the collinear stripe reference state does not provide such accurate results for the correlation functions inside this magnetically disordered phase . to conclude , we return to examine more closely the very narrow region @xmath18 for which our ccm results based on the striped afm state as model state could not exclude the possibility of an intervening phase between the striped afm and the fm phases . in fig . [ corrfunctn_zoom ] shown in fig . [ corrfunctn](b ) for the spin-@xmath0 @xmath1@xmath2@xmath3 model on the honeycomb lattice for the fm case with @xmath4 and @xmath7 using the ed method on a lattice of size @xmath42.,width=226 ] we show a more detailed view of the ed results for the same spin - spin correlation functions shown in fig . [ corrfunctn](b ) in this narrow region just above the fm transition point . the ed data does definitely indicate the existence of a phase in precisely the region @xmath18 . it is difficult from this data to say with any certainty whether or not the state is the quantum - mechanical remnant of the classical spiral phase v that exists in the classical regime @xmath123 . furthermore , without ed calculations on larger lattices , for which the computational cost would be prohibitive , it is also not possible to say whether these results over such a narrow region are an artefact of the finite lattice size . our results are summarized in sec . [ conclusions ] . in this paper we have presented results on the gs properties of the spin-@xmath0 @xmath1@xmath2@xmath3 heisenberg model with fm nn ( @xmath4 ) exchange bonds in the presence of frustrating afm nnn ( @xmath5 ) and nnnn ( @xmath6 ) exchange bonds of equal strength ( @xmath7 ) on the honeycomb lattice , using both the ccm and lanczos ed . by comparison with previous studies for the afm ( @xmath12 ) version of the model,@xcite we find similar behavior for both models for values @xmath147 , but for values of @xmath148 the models differ markedly . the results of the present paper for the fm version of the model and that of the previous paper @xcite for the afm version may conveniently be combined and summarised in the phase diagram shown in fig . [ phase - diagram ] . @xmath1@xmath2@xmath3 honeycomb model in the @xmath61-@xmath140 plane , for the case @xmath7 . the continuous transition between the afm nel and pvbc phases at @xmath149 is shown by a broken line , while the first - order transition between the pvbc and afm striped phases at @xmath150 is shown by a solid line . our results indicate that the transition between the striped afm and fm phases is either a first - order one at @xmath151 or occurs via an intermediate phase , probably with noncollinear spiral order , which exists in the region @xmath152 . the region between the fm and afm nel phases with @xmath153 has not been investigated by us.,width=415 ] we note that , by contrast with the corresponding model with afm nn exchange ( @xmath12 ) we do not find indications for a non - classical magnetically - disordered phase for the model with fm nn exchange ( @xmath4 ) . if such a phase exists at all it must be confined to a very small range of the frustration parameter around @xmath18 . however , any such phase is much more likely to be a quasiclassical remnant of the spiral phase v that exists in the corresponding classical model ( with @xmath4 ) in the parameter regime @xmath123 . as expected , quantum fluctuations then usually favor a collinear phase over a noncollinear one , and the extent of any spiral phase is smaller in the quantum spin-@xmath0 case than in the classical ( @xmath154 ) case . in one scenario the results presented here for the case @xmath4 indicate a direct first - order transition between the two magnetically ordered phases , namely the fm ground state at small values of the frustration parameter @xmath140 and the striped collinear afm ground state at larger values of @xmath140 . our best estimate of the phase transition point is then @xmath130 . although in this scenario a quasiclassical gs phase ( viz . , the collinear striped afm state ) exists in the whole parameter space down to the fm gs phase , the frustration might still have a strong effect on the low - temperature thermodynamics near the transition point at @xmath130.@xcite for values @xmath155 the fm multiplet becomes a low - lying excitation , and this might lead to an additional low - temperature peak in the specific heat @xmath156.@xcite we note that indications for such an additional low - temperature peak in @xmath156 were also found on the fm side near such a transition@xcite ( i.e. , at @xmath157 ) in other frustrated spin models . in an alternative scenario our results also indicate the possibility of an intervening phase between the collinear fm and striped afm phases . any such phase , however , is limited to lie within the very narrow range @xmath18 , as shown in fig . [ phase - diagram ] . in principle we could more accurately establish the existence of such a phase as a quasiclassical remnant of the classical spiral phase v , and thence also more accurately establish its phase boundaries , by performing another comparable set of ccm calculations to those performed here with the striped afm state as model state , but using instead the spiral state v as model state . such calculations would be much more onerous and computationally expensive , however , since on the one hand the number @xmath103 of fundamental ccm configurations at a given lsub@xmath101 level is greater for the spiral model state than for the striped model state and , furthermore , the ccm results would need to be optimized at a given lsub@xmath101 level with respect to the spiral pitch angle parameters by minimizing the corresponding result for the energy per spin separately for each set of values for the bond strength parameters . we note finally that we have not yet investigated the present model in the case where @xmath153 . for the fm version of the model when @xmath27 also , the fm phase is then obviously the stable ground state . conversely , when @xmath39 and frustration occurs , there is a direct first - order transition in the classical version of the model between the fm and nel afm states at a value @xmath158 . following the discussion in sec . [ results ] we might expect that quantum fluctuations could again act either ( a ) to retain the direct transition but to stabilize the collinear afm order in preference to the fm order , thus pushing the phase boundary to a somewhat lower value , @xmath159 , for the spin-@xmath0 case ; or ( b ) to permit an intervening state with no classical counterpart . indeed , very preliminary ccm calculations indicate that scenario ( a ) is realized and that this corresponding critical point may be pushed to a value @xmath160 . we hope to report in more detail on this region and to give a more accurate value of this phase boundary in a future paper . as discussed briefly in sec . [ intro ] , it has been proposed @xcite that the competition between fm heisenberg interactions between nn pairs of spins and afm interactions between other spins in frustrated spin-@xmath0 systems on the square lattice could lead to gapless spin - liquid states with multipolar order ( e.g. , spin - nematic states ) adjacent to the fm state . similar states have also been proposed to arise in frustrated multiple cyclic spin - exchange models on the triangular lattice with fm nn pairwise interactions , @xcite either in the presence of a magnetic field ( where octupolar order occurs ) or in its absence ( where quadratic or nematic ordering occurs in a state bordering the fm state ) . in the case of the frustrated honeycomb - lattice ferromagnet considered here we have found no evidence for such states . however , the multipolar - ordering phenomenon in the zero - field case considered here is evidently rather fragile , and in the square - lattice case for the spin-@xmath0 fm version of the @xmath61@xmath140 model ( i.e. , with @xmath27 ) even their existence has been questioned in recent rather accurate work @xcite that also employed both high - order ccm and ed techniques . no evidence was found for such states either in a very recent schwinger boson study on the square lattice,@xcite using the same fm version of the spin-@xmath0 @xmath1@xmath2@xmath3 heisenberg model that we studied here on ther honeycomb lattice . nevertheless , the history of the study of quantum magnets has shown us that the detection of phases with novel quantum ordering , such as nematic states of various kinds , is extremely subtle . in particular , the present honeycomb - lattice model surely warrants further investigation before the absence of nematic states in the fm case discussed here is considered definite . finally we mention that frustrated ferromagnets are also interesting with respect to multi - magnon bound states appearing in high magnetic fields ( and see , e.g. , refs . the present model also warrants further investigation when the coupling to an external magnetic field is included . we thank the university of minnesota supercomputing institute for digital simulation and advanced computation for the grant of supercomputing facilities , on which we relied for the numerical calculations reported here . one of the authors ( djjf ) acknowledges and thanks the european science foundation for financial support under the research network program `` highly frustrated magnetism '' ( short visit grant number 3858 ) . r. shindou , s. yunoki , and t. momoi , phys . rev . b * 84 * , 134414 ( 2011 ) . f. wang , phys . b * 82 * , 024419 ( 2010 ) . s. okumura , h. kawamuro , t. okubo , and y. motome , j. phys . jpn . * 79 * , 114705 ( 2010 )
|
we study the ground - state ( gs ) properties of the frustrated spin-@xmath0 @xmath1@xmath2@xmath3 heisenberg model on the two - dimensional honeycomb lattice with ferromagnetic nearest - neighbor ( @xmath4 ) exchange and frustrating antiferromagnetic next - nearest - neighbor ( @xmath5 ) and next - next - nearest - neighbor ( @xmath6 ) exchanges , for the case @xmath7 .
we use the coupled - cluster method implemented to high orders of approximation , complemented by the lanczos exact diagonalization of a large finite lattice with 32 sites , in order to calculate the gs energy , magnetic order parameter , and spin - spin correlation functions . in one scenario
we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic ( `` striped '' ) collinear order at @xmath8 , which is below the corresponding hypothetical transition point at @xmath9 ( @xmath10 ) for the classical version of the model , in which we momentarily ignore the intervening noncollinear spiral phase in the region @xmath11 .
hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model .
we compare results for the present ferromagnetic case ( with @xmath4 ) to previous results for the corresponding antiferromagnetic case ( with @xmath12 ) .
the magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter @xmath2 . however , there are considerable differences in the behavior of the order parameters for the two models for @xmath13 . for example
, the quasiclassical collinear magnetic long - range order for the antiferromagnetic model ( with @xmath12 ) breaks down at @xmath14 , whereas the `` equivalent '' point for the ferromagnetic model ( with @xmath4 ) occurs at @xmath15 .
unlike in the antiferromagnetic model ( with @xmath12 ) , where a plaquette valence - bond crystal phase intrudes between the two corresponding quasiclassical antiferrmagnetic phases ( with nel and striped order ) for @xmath16 , with @xmath17 , we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order .
instead the evidence for the ferromagnetic model ( with @xmath4 ) points to one of two scenarios : either there is a direct first - order transition between the two magnetically ordered phases , as mentioned above ; or there exists an intervening phase between them in the very narrow range @xmath18 , which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range @xmath11 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it is generally accepted that percolation is an essential aspect of gelation or vulcanization it is doubtful that even in a highly entangled melt of long polymers a nonzero value of the static shear modulus could exist in the absence of an infinite connected network . however , percolation has usually been studied in rather special limits . site and bond percolation of a single species on regular lattices are very well characterized and off - lattice percolation seems to present no new features @xcite , at least insofar as critical behavior is concerned . more closely related to real gels are the so - called correlated percolation models where the distribution of crosslinks is drawn from a boltzmann distribution appropriate for a nearest neighbor lattice gas @xcite . except at special points in the phase diagram these models are also in the universality class of the simple percolation problem . in our previous work on transport properties near the gel point @xcite , we have also used a simple one - species percolation process to produce the incipient gel . we found that the shear viscosity diverges as the percolation concentration @xmath2 is approached according to @xmath5 with @xmath6 . this value of the exponent @xmath4 is in excellent agreement with a prediction of de gennes based on a superconductor - normal conductor analogy @xcite and with recent analytical work on a rouse model @xcite . it is also reasonably close to some experimental results for @xmath4 @xcite but quite different from that produced by another set of experiments @xmath7 @xcite . thus it seems reasonable to ask if different versions of the crosslinking process might produce significantly different cluster size distributions from percolation and , consequently , different rheological properties . gelation often occurs in the presence of a solvent and over some period of time rather than instantaneously , as in the usual percolation models . to simulate this feature , we have considered a two - species model consisting of a fraction @xmath8 of @xmath0-functional particles that are eligible to bond irreversibly to others of the same kind . the remaining particles are inert and function as a background liquid through which the gel particles and clusters diffuse . crosslinking occurs in stages : the equations of motion of all the particles are integrated forward for a fixed number of time steps between crosslinking attempts and this process is continued until the desired number of crosslinks is attained . at a critical concentration of crosslinks , @xmath2 , ( in the thermodynamic limit ) the largest cluster percolates and an amorphous solid forms . for this process one can calculate the usual static or geometrical quantities used to characterize percolating systems , _ e.g. _ , the fraction of particles on the ` infinite cluster ' , @xmath9 , the mean mass of finite clusters , @xmath10 , the fraction of samples percolating @xmath11 , the cluster size distribution @xmath12 where @xmath13 is the mass of a cluster and the radius of gyration @xmath14 where @xmath15 is the fractal dimension of the clusters . for simple percolation processes , @xmath16 , @xmath17 and these two exponents determine the others through scaling relations @xcite . here we find , at least for small @xmath8 , that the cluster size distribution , even at @xmath2 , is not well described by a simple power law . however , the other static quantities listed above do display power law behavior near @xmath2 and a standard finite - size scaling analysis provides a very good collapse of our data . moreover , the hyperscaling relation @xmath18 , where @xmath19 is the dimensionality and @xmath20 the correlation length exponent , is satisfied . this suggests that this percolation transition is fundamentally describable in terms of a fixed point with two ( at least ) relevant scaling fields . as the percolation point is approached from below , the shear viscosity diverges according to @xmath3 . in contrast to our previous work on a model without solvent , we find values of @xmath4 in the range @xmath21 as compared with @xmath6 . these results suggest that the critical behavior of transport coefficients of systems close to the gel point is nonuniversal . the structure of this article is as follows . in section [ sec : model ] we describe the present model and simulation procedures in more detail . the geometric properties of the system are discussed in section [ sec : percolation ] and the data on the shear viscosity are presented in section [ sec : visco ] . we conclude with a brief summary and discussion in section [ sec : discu ] . we consider a system of @xmath22 particles in three dimensions , all of which interact with each other through the soft - sphere potential @xmath23 for @xmath24 @xcite with @xmath25 and @xmath26 . we simulated systems at a volume fraction @xmath27 which is well below the liquid - solid coexistence density . in the absence of any other interactions , this system would be a simple three - dimensional liquid . we initially place the particles on a simple cubic lattice that fills the computational box . we then randomly select @xmath28 particles to be the gel forming component . after equilibration of the system with brownian dynamics , with periodic boundary conditions , for 10000 time steps we begin the crosslinking process . at this point , the calculation proceeds via _ conservative _ molecular dynamics ( md ) so as to allow hydrodynamic modes to develop . here we use a time step @xmath29 . in the smallest system , crosslinking is carried out one bond at a time . a single gel particle is randomly selected and all other gel particles within a distance of @xmath30 are identified . one of the particles in this list is randomly selected and bonded irreversibly to the central particle through the tethering potential @xmath31 with @xmath32 and @xmath33 . each gel particle is allowed to bond to no more than six others and bonding between any pair of particles occurs at most once . the configuration of the entire system is then updated for 100 time steps and the entire bonding process is repeated until @xmath34 crosslinks have been added . the parameter @xmath1 is analogous to the occupation probability in a bond percolation process on the simple cubic lattice . in larger systems , the number of crosslinks added in the bonding steps is scaled by the system size in order to keep the crosslinking rate per gel particle constant . the parameters in the potentials and the total volume fraction @xmath35 are the same as in our previous work @xcite . the differences are that in this earlier work all particles were considered to be gel particles and that the crosslinking was done instantaneously , at @xmath36 , when the particles were on the vertices of a cubic lattice and thus all structural properties were those of percolation in three dimensions . the present model is similar in some ways to a model discussed by gimel _ _ @xcite and hasmy and jullien @xcite who studied percolation in the context of diffusion - limited cluster - cluster aggregation using monte carlo methods . their model differs from ours in that it is a lattice model , in the details of the crosslinking process , in the lack of solvent and in the nature of the cluster dynamics . in monte carlo simulations , one is forced to arbitrarily choose the mass - dependent diffusion constant @xmath37 whereas in our molecular dynamics calculations it is determined by the existing structure and the interparticle forces . in the regime that is of interest here , _ i.e. _ , high enough gel density that percolation is possible , these authors find the critical behavior of ordinary percolation . in a separate set of runs , we calculate the stress - stress autocorrelation function and , through the appropriate green - kubo formula , the shear viscosity . equilibration and crosslinking are carried out as described above and the calculation of the viscosity is again done with conservative md . the adjustable parameters in our calculations are the gel fraction @xmath8 , the crosslink density @xmath1 and the system size . here we report results for @xmath38 , @xmath39 and @xmath40 . calculations for other values of @xmath8 are in progress and will be reported in a future publication @xcite . we parametrize the size of our system in terms of the dimensionless length @xmath41 where @xmath22 is the total number of gel and solvent particles . because the crosslinking process is itself quite time consuming , we are able only to simulate systems up to size @xmath42 ( 32768 particles ) and this makes our estimates of critical exponents rather imprecise . a second factor contributing to the uncertainty in critical exponents is that we need to determine the critical crosslink density @xmath2 for each value of @xmath8 whereas for lattice percolation this number is known to high accuracy . we next discuss the static ( geometric ) properties of our model . the critical concentration @xmath2 at which percolation occurs in the thermodynamic limit @xmath43 is accurately estimated from the intersection of curves @xmath44 as function of @xmath1 for different values of @xmath45 . here @xmath44 is the fraction of samples percolating in a system of size @xmath45 at crosslink concentration @xmath1 . for the two cases of interest here , @xmath46 and @xmath38 , we find @xmath47 and @xmath48 . once @xmath2 has been determined , the correlation length exponent @xmath20 can be estimated from the collapse of the data for the function @xmath0 when plotted as function of @xmath49 . we show this collapse of the data for @xmath38 and @xmath39 in figure ( [ fig1 ] ) . for @xmath46 , the best collapse of the data for @xmath50 is obtained for @xmath51 which should be compared to the three - dimensional percolation result @xmath52 . for @xmath38 , finite - size effects are more pronounced and the data for @xmath53 have been excluded . for this case , the best collapse of the data is obtained for @xmath54 . this method of estimating a critical exponent is not very accurate but the three - dimensional percolation value @xmath52 provides a significantly worse collapse of the data . we next discuss the mean size of finite clusters because this data provides an unbiased estimate of the ratio @xmath55 . in the thermodynamic limit , @xmath10 with @xmath56 for @xmath19 percolation . for finite @xmath45 , @xmath57 is peaked near @xmath2 with a peak height that grows as @xmath58 . therefore , rescaling the peak heights to the same value for different @xmath45 provides an estimate of @xmath55 that is not affected by errors in either @xmath2 or @xmath20 . of course , the overall collapse of the data to a universal curve depends on accurate determination of these two quantities but the peak height does not . in figures ( [ fig2 ] ) and ( [ fig3 ] ) we show the function @xmath59 plotted as function of @xmath60 for the previously determined values of @xmath2 and @xmath20 . the collapse to a universal curve is quite respectable for both @xmath46 and @xmath61 for @xmath62 and @xmath63 , respectively . as above , the data for @xmath53 have been excluded for @xmath38 . we note that in the case of three - dimensional percolation the ratio @xmath64 . use of this value of @xmath55 in figure ( [ fig2 ] ) would result in a 40% difference between the peak heights for @xmath42 and @xmath53 . in the scaling theory of percolation @xcite , the ratio @xmath65 , where @xmath19 is the dimensionality and @xmath66 is the exponent characterising the cluster size distribution at @xmath67 . if we enforce this scaling relation , we obtain @xmath68 for both @xmath38 and @xmath46 . using @xmath69 , we find @xmath70 and @xmath71 . using @xmath72 for the fractal dimension results in the prediction @xmath73 and @xmath74 for the fractal dimensions of the clusters . as well , the hyperscaling relation @xmath75 yields @xmath76 and @xmath77 for @xmath46 and @xmath61 respectively . the accuracy of these scaling predictions is tested in figures ( [ fig4 ] ) to ( [ fig7 ] ) . in figure ( [ fig4 ] ) we show the number of clusters @xmath78 of mass @xmath13 at @xmath79 for @xmath38 and 0.3 for @xmath42 and @xmath80 . for the case @xmath81 , we have only counted the uncrosslinked gel particles . in neither case is the data well described by a simple power law , in contrast to percolation on a lattice or in the absence of solvent where the exponent @xmath16 is already obtained for @xmath82 . a fit to a power law over the range @xmath83 yields @xmath84 for @xmath38 and @xmath85 for @xmath46 . the straight lines in figure ( [ fig4 ] ) are the best fits to the form @xmath86 over the range @xmath83 and while the fit is not perfect , the data are not inconsistent with this behavior in the limit of large @xmath13 . in figure ( [ fig5 ] ) we show the square of the radius of gyration @xmath87 as function of @xmath13 for a system of size @xmath42 together with curves @xmath88 with @xmath74 and @xmath73 as determined above . the data again show considerable curvature but the fit to the assumed functional form is reasonable over the range @xmath89 . finally , in figures ( [ fig6 ] ) and ( [ fig7 ] ) we display the scaled form of @xmath90 , the probability that a gel particle is part of the percolating cluster using the predicted exponent ratios @xmath76 for @xmath46 and @xmath91 for @xmath38 . these two figures present the least impressive collapse of data to a universal curve , especially at the larger values of @xmath92 . one can improve the collapse by different choice of @xmath93 and @xmath20 but at the expense of violating hyperscaling . we also note that the data for the two largest values of @xmath45 are reasonably close to each other over the entire range of @xmath94 . we have also carried out a limited number of simulations for @xmath95 and @xmath96 with the crosslinking process described above . in both cases , the critical exponents and the cluster size distributions are entirely consistent with ordinary three - dimensional percolation . this suggests that either there is a critical gel fraction @xcite below which the geometric properties of the clusters are described by continuously varying exponents or that the apparent variation of the exponents with @xmath8 described above is a finite - size artefact . only simulations of larger systems can resolve this issue . we have calculated the shear viscosity for systems up to size @xmath97 as function of the crosslink density @xmath1 for @xmath46 and for @xmath98 for @xmath38 . systems are equilibrated as a liquid , crosslinked as described above and then evolved by constant energy md for 40000 or 80000 time steps , depending on the crosslink density . here we have typically used 500 to 2000 different realizations of the crosslinks at each @xmath1 . we calculate , as in @xcite , the stress - stress autocorrelation function @xmath99 where @xmath100 are elements of the stress tensor . here the sum is over both gel and solvent particles and @xmath101 is the derivative of the pair potential between particles @xmath102 and @xmath103 . the analysis of the stress - stress correlation function has been described in @xcite and is done in the same way here . as @xmath104 , @xmath105 decays extremely slowly and is fitted , at long times , to a stretched exponential . the static shear viscosity is then obtained from the appropriate green - kubo formula @xcite , @xmath106 the results , for @xmath46 are shown in finite - size scaled form in fig . ( [ fig8 ] ) where @xmath107 is plotted as function of the scaled concentration @xmath94 @xcite . in contrast to our previous result for @xmath108 and instantaneous crosslinking where we found @xmath6 , we find that @xmath109 provides an excellent collapse of the data with @xmath51 . we note that , outside the critical region , consistency of the finite - size scaling ansatz requires the scaled viscosity to vary as @xmath110 and it is clear that the data are consistent with this behavior . we have also calculated the shear viscosity for @xmath38 for @xmath98 . the raw data are displayed in figure ( [ fig9 ] ) as function of @xmath111 together with the corresponding results for @xmath46 . fitting to a power law outside the critical region produces an exponent @xmath112 suggesting , as in the case of the static properties , a variation of critical exponents with @xmath8 and an absence of universality . in this article we have proposed and investigated a new model for gelation which incorporates a solvent on a microscopic level . for relatively small concentrations of gel , the geometric properties of the system close to the gel point seem to depend continuously on this gel fraction and are , at least for the system sizes investigated , markedly different from three - dimensional percolation . in particular , the fractal dimension of the clusters seems to be smaller than those of percolation clusters and this more spidery morphology may be responsible for the slower divergence of the shear viscosity as the gel point is approached . the change in the exponents controlling the geometric properties is rather small and further study of larger systems is certainly necessary to confirm this result . however , the exponent @xmath4 that characterizes the divergence of the shear viscosity at the gel point is reduced by almost a factor of 2 from its value in the absence of solvent and it is unlikely that this can be attributed to finite - size effects . in light of this result , it seems implausible that a single universality class describes the behavior of transport coefficients and , presumably , the moduli of the amorphous phase near the gel point . the considerable dispersion found in experimental values of the critical exponents @xcite is another indicator that this may be the case . in future work we intend to explore this new model in greater detail . it will be interesting to investigate if the exponent @xmath4 and the static exponents are tunable by varying the concentration of the solvent and the solubility of the solute . we also intend to study diffusion constants as function of cluster size and to investigate the existence of long time tails . finally , one of the original motivations for this model is the existence of a body of experimental work that has yielded values in the range 1.1 1.3 for the viscosity exponent @xmath4 . clearly , we have moved further from this range of values compared to our previous results . if the cluster size distribution and cluster geometry is the determining factor in the critical behavior of the transport coefficients then this indicates that models that produce more compact rather than more tenuous clusters than those arising from percolation may be appropriate . one of us ( bj ) thanks the physics department at simon fraser university for its hospitality during a sabbatical visit . we thank ralph colby , paul goldbart and sune jespersen for helpful discussions . this research is supported by the nserc of canada . see for example d. stauffer and a. aharony , _ introduction to percolation theory _ , 2nd edition , ( taylor and francis , london , 1994 ) . see d. stauffer , a. coniglio and m.adam , adv . sci . * 44 * , 103 ( 1982 ) for a review . d. vernon , m. plischke and b. jos , phys . rev e * 64 * , 03105 ( 2001 ) . p.g . de gennes , j. phys ( paris ) , * 40 * , l197 ( 1979 ) . k. broderix , h. lwe , p. mller and a. zippelius , europhys . lett . , * 48 * , 421 ( 1999 ) ; phys . e * 63 * , 011510 ( 2001 ) . m. adam , m. delsanti , d. durand , g. hild and j.p . munch , pure appl . , * 53 * , 1489 ( 1981 ) ; m. adam , m. delsanti and d. durand , macromolecules , * 18 * , 2285 ( 1985 ) ; d. durand , m. delsanti and j.m . luck , europhys . * 3 * , 297 ( 1987 ) . lusignan , t.h . mourey , j.c . wilson and r.h . colby , phys . e * 52 * , 6271 ( 1995 ) ; j.e . martin and j. wilcoxon , phys . lett . , * 61 * , 373 ( 1988 ) ; d. adolf and j.e . martin , macromolecules , * 23 * , 3700 ( 1990 ) ; j.e . martin , j. wilcoxon and j. odinek , phys . rev . a * 43 * , 858 ( 1991 ) ; j.e . martin , d. adolf and j. wilcoxon , phys . lett . , * 61 * , 2620 ( 1988 ) . powles and d.m . heyes , mol . phys . * 98 * , 917 ( 2000 ) . these authors have studied the properties of systems with a pair potential of the form @xmath113 , for @xmath114 , including our case @xmath115 . gimel , d. durand and t. nicolai , phys . b * 51 * , 11348 ( 1995 ) . a. hasmy and r. jullien , phys . e * 53 * , 1789 ( 1996 ) . m. plischke , s. jespersen and b. jos , unpublished . a critical volume fraction appears in the model of gimel _ et al . _ @xcite , albeit in a rather trivial way : when the volume fraction is increased above 0.31 , the system percolates instantaneously ( in the thermodynamic limit ) and the features associated with cluster - cluster aggregation disappear . allen and d.j . tildesley , _ computer simulation of liquids _ ( oxford university press , new york , 1987 ) , chap . 2 ; j.p . hansen and i.r . macdonald , _ theory of simple liquids _ ( academic press , new york , 1986 ) . a fit of the raw data for @xmath98 and @xmath116 to the form @xmath117 produces the estimates @xmath118 and @xmath119 , in good agreement with the percolation theory estimate of @xmath2 and with the finite - size scaling analysis reported above . for a review , see m. adam and d. lairez in _ the physical properties of polymeric gels _ , j.p . cohen addad , ed . ( john wiley and sons , ltd . , new york , 1996 )
|
we study a two - component model for gelation consisting of @xmath0-functional monomers ( the gel ) and inert particles ( the solvent ) .
after equilibration as a simple liquid , the gel particles are gradually crosslinked to each other until the desired number of crosslinks has been attained . at a critical crosslink density the largest gel cluster percolates and an amorphous solid forms .
this percolation process is different from ordinary lattice or continuum percolation of a single species in the sense that the critical exponents are new .
as the crosslink density @xmath1 approaches its critical value @xmath2 , the shear viscosity diverges : @xmath3 with @xmath4 a nonuniversal concentration - dependent exponent .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the goal of this paper is two - fold ; the first goal is to present an algorithm for developing reduced - order models of the input - output dynamics of _ unstable _ high - dimensional linear state - space systems ( such as linearized navier - stokes equations with actuation and sensing ) , while the second goal is to demonstrate the algorithm by developing estimation - based controllers to stabilize unstable steady states of a two dimensional low - reynolds - number flow past a flat plate at a large angle of attack . development of feedback control strategies based on linearized navier - stokes equations is attractive due to the ready availability of a large class of control techniques , and there has been substantial progress in this direction in the past decade , reviewed in detail by @xcite . however , many of these techniques are limited to relatively small dimensional systems @xmath0 , while the numerical discretization of fluid flows invariably result in huge dimensional systems , typically @xmath1 . thus , model reduction has played an important role in making these tools further accessible to fluid flows . extensive research effort in model reduction has focused on the method of proper orthogonal decomposition ( pod ) and galerkin projection , developed first by @xcite . the main disadvantage of this technique is that , although the pod modes capture the energetically important structures of the flow , the reduced - order models obtained by the subsequent galerkin projection of the governing equations onto these modes often do not faithfully represent the dynamics . various modifications to improve this method have been proposed and used for flow control ; refer to the introduction of @xcite for a review of these techniques . the pod / galerkin methods have been applied for flow control in various contexts , such as bluff - body wake suppression ( @xcite , @xcite , @xcite , @xcite ) , noise reduction in cavity flow ( @xcite , @xcite ) , and drag reduction in turbulent boundary layers ( @xcite , @xcite ) . another model - reduction technique , based on projection onto the global ( stable or unstable ) eigenmodes of the flow linearized about steady states , has been used by @xcite and @xcite in the context of spatially developing flows such as separated boundary layers . in this paper , we focus on an approximate balanced truncation method developed by @xcite as an approximation to the original method of @xcite . this technique captures the dynamically important modes of the system , and the non - approximate version provides rigorous bounds for the resulting reduced - order models . the method , sometimes called balanced pod , was used to obtain models of the linearized channel flow by @xcite and the blasius boundary layer by @xcite , and was shown to accurately capture the control actuation and also to outperform the pod / galerkin models . the balanced truncation method of @xcite is applicable only to systems linearized about _ stable _ steady states . an extension to _ unstable _ linear systems was proposed by @xcite , by introducing frequency - domain definitions of controllability and observability gramians . reduced - order models were obtained by first decoupling the dynamics on the stable and unstable subspaces , and then truncating the relatively uncontrollable and unobservable modes on each of the two subspaces . in this paper , we present an approximation algorithm for balanced truncation of linear unstable systems , which results in models that are equivalent to those of @xcite on the _ stable _ subspace . the dynamics on the unstable subspace is treated _ exactly _ by a projection onto the global eigenmodes , as in @xcite . as a proof - of - concept study , the modeling procedure is applied to the problem of two dimensional low - reynolds - number flow past a flat plate at a large angle of attack . we develop reduced - order models and design controllers that stabilize the unstable steady states of this flow . our motivation for the choice of this problem comes from our interest in regulating vortices in separated flows behind low aspect - ratio wings , which is of importance in design of micro air vehicles ( mavs ) . recently , design of mavs has been inspired from experimental observations in insect and bird flights of a stabilizing leading edge vortex ( see @xcite and @xcite ) , which remains attached throughout the wing stroke and provides enhanced lift . so , it could be beneficial to design controllers that can manipulate the wake of mavs to enhance lift and achieve better maneuverability in presence of wind gusts . recent studies in this direction , using open - loop control of the flow past low - aspect - ratio wings using steady or periodic blowing , were performed computationally by @xcite and experimentally by @xcite . these studies explored different forcing amplitudes and frequencies , locations and directions . however , the design of feedback controllers remains a challenge , due to the large dimensionality of the problem and complex flow physics . we present computational tools , that we hope can at least pave a direction and provide techniques towards addressing some of these challenges . we consider the 2-d flow past a flat plate , actuated by a localized body force close to the leading edge , with two near - wake velocity sensors . we design a reduced - order compensator and show that it is able to suppress vortex shedding at high angles of attack . many previous studies have focused on the control of flow past a cylinder , which is qualitatively similar to the flow past a flat plate at large angle of attack , with the natural flow in both the cases being periodic vortex shedding . for the cylinder , the flow undergoes a transition from steady state to periodic shedding with increasing reynolds number , while a similar transition occurs in the flat plate with increasing angle of attack . there has been considerable research effort on suppression of this shedding in cylinder and other bluff body wakes , using passive and active , open - loop and feedback control , as reviewed by @xcite . among those , some techniques are based on reduced - order models ; for instance , @xcite developed models using artificial neural networks and a pod basis , @xcite modified the pod / galerkin method to account for actuation by means of cylinder - rotation , while @xcite developed a double pod method to account for changes in the wake structure during transients . some earlier efforts in the control of a _ flat - plate _ wake include those by @xcite , @xcite who used vortex - based methods to model the flow past a vertical plate ( angle of attack = @xmath2 ) ; vortex - based models form their own class of modeling techniques reviewed recently by @xcite . lagrangian coherent structures were used by @xcite to enhance mixing in flow past a bluff body with the trailing surface similar to the vertical flat plate . one of the few efforts towards control of flat plate at an angle of incidence was by @xcite , who used a passive leading - edge suction control along with a potential flow vortex model . @xcite also used reduced - order vortex models for drag reduction on an elongated d - shaped bluff body . this paper is organized as follows : in section [ sec : model_red ] , we first briefly describe the balanced truncation method for unstable systems as developed by @xcite , and the approximate balanced truncation procedure called balanced pod of @xcite for large dimensional stable systems . then , we present an algorithm for approximate balanced truncation of large dimensional _ unstable _ systems , assuming that the dimension of unstable subspace is small and the corresponding global eigenmodes can be computed . in section [ sec : numerics_ibfs ] , we briefly describe the numerical technique of @xcite using a fast immersed boundary method , and present the linearized and adjoint formulations of this numerical method . in section [ section : model_results ] , we present numerical results , using the model example of two - dimensional flow past a flat plate at a large angle of attack and a low reynolds number @xmath3 . first , we perform a steady - state analysis and compute the branch of steady states in the entire range of angles of attack , @xmath4 . we also compute the left and right global eigenmodes of the flow linearized about the unstable steady state at @xmath5 . we present reduced - order models of the linearized dynamics and use linear optimal control techniques to design controllers that stabilize this steady state . full - state feedback and more practical near - wake velocity - measurement based feedback controllers are derived , implemented in the nonlinear equations , and shown to suppress vortex shedding . the paper concludes with a brief discussion in section [ sec : discussion ] . we briefly describe a model reduction procedure using the balanced truncation method for unstable systems developed by @xcite . consider the state - space system @xmath6 where @xmath7 is the state , @xmath8 is the input , and @xmath9 is the output of the system ; the dot over @xmath10 represents differentiation with respect to time . the eigenvalues of @xmath11 are assumed to be anywhere on the complex plane , except on the imaginary axis . the standard balanced truncation procedure developed by @xcite , valid only for stable systems , starts with defining the controllability and observability gramians of the system ( [ ss ] ) as follows : @xmath12 where asterisks are used to denote adjoint operators . a co - ordinate transformation is then obtained such that the gramians ( [ gramians_stable ] ) of the transformed system are equal and diagonal . the diagonal entries of the transformed gramians , called hankel singular values ( hsvs ) , decrease monotonically and are directly related to the controllability and observability of the corresponding states . a reduced - order model is obtained by truncating the states with relatively small hsvs , that is , the states which are almost uncontrollable and unobservable . for unstable systems , the integrals in ( [ gramians_stable ] ) are unbounded and hence the gramians are ill - defined . a modified technique was proposed by @xcite based on the following frequency - domain definitions of the gramians : @xmath13 by using parseval s theorem , it can be shown that for stable systems , the frequency - domain definitions ( [ gram_freq ] ) are equivalent to the time - domain definitions ( [ gramians_stable ] ) . the model - reduction procedure of @xcite begins by first transforming the system ( [ ss ] ) to coordinates in which the stable and unstable dynamics are decoupled . that is , let @xmath14 be a transformation such that if @xmath15 , the system ( [ ss ] ) transforms to @xmath16 here , @xmath17 and @xmath18 are such that all their eigenvalues are in the right- and left - half complex planes respectively , while @xmath19 and @xmath20 are the corresponding states . next , denote the controllability and observability gramians corresponding to the set @xmath21 describing the stable dynamics by @xmath22 and @xmath23 respectively . similarly , denote the gramians corresponding to the set @xmath24 by @xmath25 and @xmath26 . the gramians of the original system are then related to those corresponding to the subsystems by : @xmath27 a system is said to be balanced if its gramians defined by ( [ gramians_general ] ) are equal and diagonal , in which case the diagonal entries are called the _ generalized _ hankel singular values . a reduced - order model is obtained by truncating the states with small generalized hsvs . for systems of large dimension @xmath28 such as those encountered here , the gramians ( [ gramians_general ] ) are huge matrices which can not be easily computed or stored . a computationally tractable procedure was introduced by @xcite for obtaining an approximate balancing transformation . we first briefly describe this method , valid only for stable systems , and then present an extension to unstable systems . the procedure consists of computing the impulse responses of the system ( [ ss ] ) and stacking the resulting snapshots of the state @xmath10 as columns of a matrix @xmath29 . it also requires state - snapshots of the impulse responses of the adjoint system @xmath30 which are stacked as columns of a matrix @xmath31 . then , the gramians ( [ gramians_stable ] ) can be approximated as @xmath32 the approximate gramians ( [ gramians_approx ] ) are not actually computed due to the large storage cost , but the leading columns ( or modes ) of the transformation that balances these gramians are computed using a cost - efficient algorithm . it involves computing the singular value decomposition of @xmath33 where @xmath34 is a diagonal matrix of the most significant hsvs greater than a cut - off which is a modeling parameter , while @xmath35 is a diagonal matrix of smaller and zero hsvs . note that @xmath36 is a small matrix , where @xmath37 and @xmath38 are the number of snapshots of the impulse responses of systems ( [ ss ] ) and ( [ ss_adj ] ) respectively . for fluid systems that we are interested in , the typical number of snapshots is @xmath39 , thus resulting in a reasonable computational cost , and typically @xmath40 . the leading columns and rows of the balancing transformation and its inverse are obtained using : @xmath41 where @xmath42 , and the two sets of modes are bi - orthogonal ; that is , @xmath43 . the reduced - order model of ( [ ss ] ) is then obtained by expressing @xmath44 , @xmath45 , and using the bi - orthogonality of @xmath46 and @xmath47 : @xmath48 when the number of outputs of the system ( rows of @xmath49 ) is large , the algorithm described in section [ sec : approx_baltrunc_stable ] can become intractable . the reason for this is that it involves one simulation of the adjoint system ( [ ss_adj ] ) for each component of @xmath50 , the dimension of which is the same as the number of outputs . this number is often large in fluids systems where a good model needs to capture the response of the entire system ( @xmath51 ) to a given input . to overcome this problem , @xcite proposed a technique called _ output projection _ , which involves projecting the output @xmath52 of ( [ ss ] ) onto a small number of energetically important modes obtained using proper orthogonal decomposition ( pod ) . let the columns of @xmath53 consist of the first @xmath54 pod modes of the dataset consisting of outputs obtained from an impulse response of ( [ ss ] ) . then , for the purpose of obtaining a reduced - order model , the output of ( [ ss ] ) is approximated by @xmath55 where @xmath56 is an orthogonal projection of the output onto the first @xmath54 pod modes . the resulting output - projected system is optimally close ( in the @xmath57sense ) to the original system , for an output of fixed rank @xmath54 . with this approximation , only @xmath54 adjoint simulations are required to approximate the observability gramian ; refer to @xcite for details . the number of pod modes @xmath54 for output projection is a design parameter and is typically chosen to capture more than @xmath58 of the output energy . in the rest of this paper , the models resulting from this approximation of the output are referred to as _ @xmath59mode output projected models . _ the approximate balancing procedure described in the previous section , which is essentially a snapshot - based method , does not extend to unstable systems since the impulse responses of ( [ ss ] ) and ( [ ss_adj ] ) are unbounded . we could consider applying the algorithm to the two sub - systems given in ( [ decouple_ss ] ) , but the transformation @xmath14 that decouples ( [ ss ] ) itself is not available . however , when the dimension of the unstable sub - system is small , we show that it is not necessary to compute the entire transformation @xmath14 and it is still possible to obtain an approximate balancing transformation . here , we present an algorithm for computing such a transformation and also show that it essentially results in a method that is a slight variant of the technique of @xcite presented in section [ sec : unstable_baltrunc ] . the idea behind the algorithm is to project the original system ( [ ss ] ) onto the stable subspace of @xmath11 . then , one obtains a reduced - order model of the projected system using the snapshot - based procedure described in section [ sec : approx_baltrunc_stable ] . the dynamics projected onto the unstable subspace can be treated exactly on account of its low dimensionality . we assume that the number of unstable eigenvalues @xmath60 is @xmath61 and can be computed numerically , say using the computational package arpack developed by @xcite . we further assume that the bases for the right and the left unstable eigenspaces @xmath62 can be computed . for the algorithm , we need the following projection operator onto the stable subspace of @xmath11 : @xmath63 where @xmath64 and @xmath65 have been scaled such that @xmath66 . we use the operator @xmath67 to obtain the dynamics of ( [ ss ] ) restricted to the stable subspace of @xmath11 as follows : @xmath68 where @xmath69 the adjoint of ( [ ss_stable ] ) is the same as the dynamics of ( [ ss_adj ] ) restricted to the stable subspace of @xmath70 using @xmath71 , and is given by @xmath72 where @xmath73 . we compute the state - impulse responses of ( [ ss_stable ] ) and ( [ ss_adj_stable ] ) and stack the resulting snapshots @xmath74 and @xmath75 in matrices @xmath76 and @xmath77 respectively . as in ( [ svd ] ) , we compute the singular valued decomposition of @xmath78 and use the expressions ( [ bal_transf ] ) to obtain the balancing modes @xmath79 and the adjoint modes @xmath80 , where again @xmath81 . the reduced - order modes are obtained by expressing the state @xmath10 as @xmath82 where @xmath83 and @xmath84 . substituting ( [ modal_exp ] ) in ( [ ss ] ) and pre - multiplying by @xmath85 and @xmath86 , we obtain @xmath87 now , since range@xmath88 span@xmath89 , we can write @xmath90 for some @xmath91 , and using the properties of eigenvectors , we have @xmath92 . similarly , it can be shown that @xmath93 . thus , the cross terms in ( [ roma ] ) are zero and the reduced - order model is @xmath94 the procedure described so far to obtain the reduced - order model ( [ rom ] ) is related to the procedure of @xcite described in section [ sec : unstable_baltrunc ] . it can be shown that the transformation that balances the gramians defined by ( [ gramians_general ] ) results in a system in which the unstable and stable dynamics are decoupled . furthermore , the resulting stable dynamics are the same as those given by the equations describing the @xmath95-dynamics of ( [ rom ] ) . that is , balancing the stable part of the gramians @xmath96 and @xmath97 defined in ( [ gramians_general ] ) ( balancing @xmath22 and @xmath23 ) is the same as balancing the gramians of the stable subsystem ( [ ss_stable ] ) ; a proof is outlined in appendix [ sec : app : baltrunc ] . in our algorithm , the unstable dynamics are not balanced , while they are by @xcite . a disadvantage of zhou s approach is that an unstable mode might be truncated resulting in a model which does not capture the instability , which is undesirable for control purposes . for systems with a large number of outputs , the number of adjoint simulations ( [ ss_adj_stable ] ) can become intractable ; however , the output projection of section [ sec : outproj ] can readily be extended to unstable systems . instead of projecting the entire output @xmath52 onto pod modes , we first express the state @xmath98 , where @xmath99 and @xmath100 are projections on the unstable and stable subspaces of @xmath11 respectively . we similarly express the output as @xmath101 . we then project the component @xmath102 onto a small number of pod modes , of the data set consisting of the outputs from an impulse response of ( [ ss_stable ] ) . if the pod modes are represented as columns of the matrix @xmath103 , the output of ( [ ss ] ) is approximated by @xmath104 x = cx_u + \theta_s \theta_s^\ast c x_s . \label{yapprox_unstable}\end{aligned}\ ] ] finally , with the state @xmath10 expressed by the modal expansion ( [ modal_exp ] ) , the output of the reduced - order model ( [ rom ] ) is given by @xmath105 the steps involved in obtaining reduced - order models of ( [ ss ] ) , for the case where the output is the entire state ( @xmath51 ) , can now be summarized as follows : 1 . [ primal_step ] project the original system ( [ ss ] ) onto the subspace spanned by the stable eigenvectors of @xmath11 in the direction of the unstable eigenvectors of @xmath11 to obtain ( [ ss_stable ] ) . compute the ( state ) response to an impulse on each input of ( [ ss_stable ] ) and stack the snapshots in a matrix @xmath76 . 2 . assemble the resulting snapshots , and compute the pod modes @xmath106 of the resulting data - set . these pod modes are stacked as columns of @xmath107 . 3 . choose the number of pod modes one wants to use to describe the output of ( [ ss_stable ] ) . for instance , if @xmath108 error is acceptable , and the first @xmath54 pod modes capture @xmath58 of the energy , then the output is the velocity field projected onto the first @xmath54 modes . thus , the output is represented as @xmath109 . [ adjoint_step ] project the adjoint system ( [ ss_adj ] ) onto the subspace spanned by the stable eigenvectors of @xmath70 in the direction of the unstable eigenvectors of @xmath70 to obtain ( [ ss_adj_stable ] ) . compute the ( state ) response of ( [ ss_adj_stable ] ) , starting with each pod mode @xmath106 as the initial condition ( one simulation for each of the first @xmath54 modes ) . stack the snapshots in a matrix @xmath77 5 . compute the singular value decomposition of @xmath110 , where @xmath111 , and @xmath112 rank(@xmath113 ) . define balancing modes @xmath114 and the corresponding adjoint modes @xmath115 as columns of the matrices @xmath79 and @xmath80 , where @xmath116 7 . obtain the reduced - order model using ( [ rom ] ) , which can be written as @xmath117 alternatively , the outputs can be considered to be simply the coefficients of the unstable modes @xmath118 and the coefficients of the pod modes @xmath107 of the stable subspace . with this choice , the output can be represented as @xmath119 finally , if the initial state @xmath120 is known , the initial condition of ( [ rom_compact ] ) can be obtained using @xmath121 in the remainder of this paper , we demonstrate the algorithm developed in this section by obtaining reduced order models of the 2-d uniform flow past a flat plate , and develop controllers based on these models to stabilize the unstable steady states that exist at high angles of attack . the numerical scheme used is a fast immersed boundary method developed by @xcite , and is briefly described here . the method is then used to develop the linearized and adjoint formulations . consider the following form of the incompressible navier - stokes equations , based on the continuous analog of the immersed boundary formulation introduced by @xcite : @xmath122 where @xmath123 , @xmath124 and @xmath125 are the appropriately non - dimensionalized fluid velocity , pressure and surface force respectively . the force @xmath125 acts as a lagrange multiplier that imposes the no - slip boundary condition on the lagrangian points @xmath126 , which arise from the discretization of a body moving with velocity @xmath127 . we consider the body to be a stationary flat plate at an angle of attack @xmath128 ; that is , here @xmath129 . the reynolds number is defined as @xmath130 where @xmath131 is the free - stream velocity , @xmath132 is the chord - length and @xmath133 is the kinematic viscosity . equations ( [ navier1]-[navier3 ] ) are discretized in space using a second - order finite - volume scheme on a staggered grid , and a discrete curl operation @xmath134 is introduced to eliminate the pressure and obtain a semi - discrete formulation in terms of the circulation @xmath135 : @xmath136 the incompressibility condition ( [ navier2 ] ) is implicitly satisfied by an appropriate construction of @xmath49 . the discrete laplacian is represented by @xmath137 , using the identity @xmath138 ; the constant @xmath139 , where @xmath140 is the uniform grid spacing . the operator @xmath141 smears the dirac delta function of ( [ navier1 ] ) over a few grid points . the nonlinear term @xmath142 is the spatial discretization of @xmath143 , where @xmath144 is the discrete velocity flux , in turn related to the discrete stream function @xmath145 and circulation @xmath135 as : @xmath146 a uniform grid and a choice of simple boundary conditions result in a _ fast _ algorithm . with a uniform grid , the discrete poisson equation ( [ stream ] ) is solved by means of the efficient discrete sine transform . the boundary conditions specified are dirichlet and neumann for the velocity components normal and tangential to the domain boundaries , which for the flow past a flat plate imply a uniform - flow in the far - field . these boundary conditions are however valid for only a sufficiently large domain , and are imposed by employing a computationally efficient multi - domain approach . the domain is considered to be embedded in a series of domains , each twice - as - large as the preceding , with a uniform but a _ coarser _ grid having the same number of grid points . the poisson equation , with zero boundary conditions , is solved on the largest domain and the stream function is interpolated on the boundary of the smaller domain , which are in turn used to solve the poisson equation on the smaller domain . for the flow past a flat plate considered here , the typical size of the largest domain is around 40 chord lengths in each direction . finally , the time - integration is performed using the implicit crank - nicolson scheme for the viscous terms and the second - order accurate , adams - bashforth scheme for the convective terms . for deriving reduced - order models useful for control design , we first linearize equation ( [ semidisc ] ) about a pre - computed steady state ( @xmath147 , @xmath148 ) . the linearized equations are the same as equations ( [ semidisc],[constraint ] ) with the nonlinear term @xmath142 replaced by its linearization about the steady state , and is denoted by @xmath149 where the flux @xmath144 is related to @xmath135 by ( [ stream ] ) . thus , the linearized equations are : @xmath150 the boundary conditions for the linearized equations are @xmath151 on the outer boundary of the largest computational domain . in order to derive the reduced - order models using the procedure described earlier , we need to perform adjoint simulations . in order to derive the adjoint equations , we define the following inner - product on the state - space : @xmath152 that is , the inner - product defined on the state - space is the standard @xmath153-inner product weighted with the inverse - laplacian operator . this choice is convenient as it results in the adjoint equations which differ from the linearized equations only in the nonlinear term . a derivation is outlined in appendix [ sec : app : adjoint ] and the resulting equations are : @xmath154 where the variables @xmath155 , @xmath126 and @xmath156 are the duals of the discrete circulation @xmath135 , stream function @xmath145 and body force @xmath157 , respectively , and @xmath158 is the dual of flux @xmath144 . the adjoint of the linearized nonlinear term is @xmath159 , which can be shown to be a spatial discretization of @xmath160 . since equation ( [ adj1 ] ) differs from ( [ linear ] ) only in the last term on the right hand side , the numerical integrator for the adjoint equations can be obtained by a small modification to the linearized equations solver . the nature of the multi - domain scheme used to approximate the boundary conditions of the smallest computational domain , results in a multi - domain discrete laplacian which is not exactly self - adjoint to numerical precision . as a result , the adjoint formulation given by ( [ adj1],[adj2 ] ) which also uses the same multi - domain approach , is not precise and results in small , rather insignificant , errors in the computation of the reduced - order models . . vorticity contours are plotted , with negative contours shown by dashed lines . the velocity - sensor locations are marked by solid circles.,height=192 ] we apply the model reduction techniques developed in the previous sections to the uniform flow past a flat plate in two spatial dimensions , at a low reynolds number , @xmath3 . we obtain reduced - order models of a system actuated by means of a localized body force near the leading edge of the flat plate ; the vorticity contours of the flow field obtained on an impulsive input to the actuator are shown in fig . [ fig : actuator_sensor ] . using these reduced - order models , we develop feedback controllers that stabilize the unstable steady state at high angles of attack . we first assume full - state feedback , but use output projection described in section [ sec : outproj ] to considerably decrease the number of outputs in order to make the model computation tractable . later , we relax the full - state feedback assumption , and develop a more practical observer - based controller which uses a few velocity measurements in the near - wake of the flat plate ( shown in fig . [ fig : actuator_sensor ] ) to reconstruct the entire flow . the grid size used is @xmath161 , with the smallest computational domain given by @xmath162\times[-2.5,2.5]$ ] , where lengths are non - dimensionalized by the chord of the flat plate , with its center located at the origin . we use 5 domains in the multiple - grid scheme , resulting in an effective computational domain @xmath163 times larger the size of the smallest domain ; thus the largest domain is given by @xmath164\times[-40,40]$ ] . the timestep used for all simulations is @xmath165 . since our approach is to obtain reduced - order models of the flow linearized about a given steady state , we first need to compute these steady states . the model - reduction of unstable systems involves projecting the dynamics onto a stable subspace , for which we also need to compute the right and left eigenvectors of the linearized dynamics . this section concerns this steady - state analysis , using a `` timestepper - based '' approach as outlined in @xcite and @xcite . a simple way of computing stable steady states is by simply evolving the time - accurate simulation to stationarity . however , unstable steady states can not be found in this manner , and stable steady states near a bifurcation point could take very long to converge . instead , we use a timestepper - based approach which involves writing a computational wrapper around the original computational routine to compute the steady states using a newton iteration . if the numerical timestepper advances a circulation field @xmath166 at a timestep @xmath167 to a circulation field @xmath168 after @xmath14 timesteps , the steady state is given by the field @xmath147 that satisfies @xmath169 the steady states are given by zeros of @xmath170 , which could , in principle , be solved for using newton s method . however , the standard newton s method involves computing and inverting jacobian matrices at each iteration , which is computationally infeasible due to the large dimension of fluid systems . instead of computing the jacobian , we use a krylov - space based iterative solver called generalized minimal residual method ( gmres ) developed by @xcite to compute the newton update ( see @xcite and @xcite for a description of the method ) . this method requires computation of only jacobian - vector products @xmath171 , which can be approximated using finite differences as @xmath172/\epsilon$ ] , for @xmath173 . so , the jacobian - vector products can also be computed by invoking the appropriately - initialized timestepper . a nice feature of gmres is relatively fast convergence to the steady state when the eigenvalues of the jacobian @xmath174 occur in clusters ; see @xcite and @xcite for details . for systems with multiple time - scales , such as navier - stokes , most of the eigenvalues of the continuous jacobian lie in the far - left - half of the complex plane . thus , the corresponding eigenvalues of the discrete jacobian @xmath175 , for a sufficiently large value of @xmath14 , cluster near the origin . and at @xmath3 , showing a transition from a stable equilibrium to periodic vortex shedding at @xmath176 . shown are the force coefficients corresponding to the stable ( @xmath177 ) and unstable ( @xmath178 ) steady states , and the maximum and minimum ( @xmath179 ) , and the mean ( @xmath180 ) values during periodic vortex shedding . also shown are the vorticity contours ( negative values in dashed lines ) of steady states at @xmath181 and the flow fields corresponding to the maximum and minimum force coefficients at @xmath182 . , title="fig:",height=134 ] and at @xmath3 , showing a transition from a stable equilibrium to periodic vortex shedding at @xmath176 . shown are the force coefficients corresponding to the stable ( @xmath177 ) and unstable ( @xmath178 ) steady states , and the maximum and minimum ( @xmath179 ) , and the mean ( @xmath180 ) values during periodic vortex shedding . also shown are the vorticity contours ( negative values in dashed lines ) of steady states at @xmath181 and the flow fields corresponding to the maximum and minimum force coefficients at @xmath182 . , title="fig:",height=134 ] and at @xmath3 , showing a transition from a stable equilibrium to periodic vortex shedding at @xmath176 . shown are the force coefficients corresponding to the stable ( @xmath177 ) and unstable ( @xmath178 ) steady states , and the maximum and minimum ( @xmath179 ) , and the mean ( @xmath180 ) values during periodic vortex shedding . also shown are the vorticity contours ( negative values in dashed lines ) of steady states at @xmath181 and the flow fields corresponding to the maximum and minimum force coefficients at @xmath182 . , title="fig:",height=182 ] \(a ) ( b ) + . ( b ) @xmath183 vs. time , with the steady state as an initial condition.,title="fig:",height=144 ] . ( b ) @xmath183 vs. time , with the steady state as an initial condition.,title="fig:",height=144 ] the procedure described above is used to compute the branch of steady states for the angles of attack @xmath184 ; the parameter @xmath14 in ( [ newton ] ) is fixed to 50 timesteps . the lift and drag coefficients , @xmath183 and @xmath185 , and their ratio @xmath186 with changing @xmath128 are plotted in fig . [ fig : bifurc ] . as with flow past bluff bodies with increasing reynolds number ( for example , see @xcite ) , the flow undergoes a hopf bifurcation from a steady flow to periodic vortex shedding as the angle of attack @xmath128 is increased beyond a critical value @xmath187 , which in our computations is @xmath188 . also plotted in the figure are the maximum , minimum , and mean values of the forces during shedding for @xmath189 . we see that the ( unstable ) steady state values of the lift coefficient are smaller than the minimum for the periodic shedding till @xmath190 , after which they are slightly higher , but still smaller than the mean lift for the periodic shedding . the ( unstable ) steady state drag is much lower than the minimum value for periodic shedding . the ratio @xmath186 of the ( unstable ) steady state is close to the mean value for shedding . thus , if the large fluctuations in the forces are undesirable at high angles of attack , it would be useful to stabilize the unstable state . the steady state at @xmath5 is shown in fig . [ fig : ss](a ) , and a time history of the lift coefficient @xmath183 with this steady state as an initial condition is shown in fig . [ fig : ss](b ) . since the steady state is unstable , the numerical perturbations excite the instability , and the flow eventually transitions to periodic vortex shedding . + we also compute a basis spanning the right and left unstable eigenspaces ( @xmath64 and @xmath65 ) of the flow linearized about the unstable steady states , which are required in our model reduction procedure , for restricting dynamics onto the stable subspace . as the flow undergoes a hopf bifurcation , a complex pair of eigenvalues crosses the imaginary axis from the left half of the complex plane ; thus the dimension of the unstable subspace is two . here , we obtain the basis spanning this subspace by a numerical implementation of the power method ( see page 191 of @xcite ) . we begin the linearized simulation with a very small random noise to excite the instability . after a sufficiently long time , the stable modes decay , and the dynamics lies close to the unstable subspace . any two independent snapshots of the remaining dynamics gives a good approximation of the basis spanning the unstable eigenspace . similarly , a basis spanning the left unstable eigenspace can be computed by initializing the adjoint equations with a small random noise . a basis spanning the right and left unstable eigenspaces of the flow linearized about the steady state at @xmath5 is plotted in fig . [ fig : evec ] . these modes are qualitatively similar to the structures during periodic vortex shedding , but have different spatial wavelengths , as reported in earlier studies by @xcite and @xcite . \(a ) + , projected onto the stable subspace , and ( b , c ) the first- and fifth - most energetic pod modes of the impulse response , restricted to the stable subspace.,title="fig:",height=124 ] + ( b ) ( c ) + , projected onto the stable subspace , and ( b , c ) the first- and fifth - most energetic pod modes of the impulse response , restricted to the stable subspace.,title="fig:",height=124 ] , projected onto the stable subspace , and ( b , c ) the first- and fifth - most energetic pod modes of the impulse response , restricted to the stable subspace.,title="fig:",height=124 ] we now describe the process involved in deriving reduced - order models of the input - output response of ( [ ss ] ) , which in this example are the linearized incompressible navier - stokes equations ( [ linear ] , [ lin_constraint ] ) . the actuator used is a localized body force close to the leading edge of the flat plate , plotted in fig . [ fig : actuator_sensor ] . the models are derived using the procedure outlined in section [ sec : approx_baltrunc_unstable ] . as seen in equation ( [ rom ] ) , the output of the system is considered to be the entire velocity field , observed as a projection onto ( a ) the unstable eigenspace , and ( b ) the span of the leading pod modes of the impulse response restricted to the stable subspace . the first step in computing the reduced - order models is to project the flow field @xmath191 onto the stable subspace of ( [ linear ] , [ lin_constraint ] ) using the projection operator @xmath67 defined in equation ( [ proj_stable ] ) ; the unstable eigenvectors computed in section [ sec : steady_states ] are used to define @xmath67 numerically . the vorticity contours of the corresponding flow field @xmath192 are plotted in fig . [ fig : actuator]a . the next step is to compute the impulse response of ( [ ss_stable ] ) . instead , for practical reasons , we compute the impulse response of @xmath193 that is , at each timestep of integration , we project the state @xmath74 onto the stable subspace of @xmath11 . because the stable subspace is an invariant subspace for the linearized dynamics ( [ linear ] ) , theoretically , the impulse responses of equations ( [ ss_stable ] ) and ( [ ss_stable_numerical ] ) are exactly the same , and they are the same as that obtained by restricting the impulse response of ( [ ss ] ) to its stable subspace . however , due to the ( small ) numerical inaccuracy of the projection @xmath67 ( which is a result of the numerical inaccuracy of the unstable eigenspaces @xmath64 and @xmath65 ) , the dynamics of ( [ ss_stable ] ) is not strictly restricted to the stable subspace and , in the long term , grows without bound in the unstable direction . next , we compute the pod modes @xmath194 of the impulse response of ( @xmath195 ) , and consider the output of ( [ ss_stable_numerical ] ) to be the state @xmath74 projected onto a certain number of these pod modes . here , 200 snapshots spaced every 50 timesteps were used to compute the pod modes . the leading 4 and 10 pod modes contain @xmath196 and @xmath197 of the energy respectively and , as it has been observed in previous studies ( see @xcite and @xcite ) , these modes come in pairs in terms of their energy content , a characteristic of traveling structures ; the leading first and third pod modes are shown in fig . [ fig : actuator ] . + the next step is to compute the adjoint snapshots , with the pod modes of the impulse response ( projected onto the stable subspace of the adjoint ) as the initial conditions . as the linearized impulse response , these simulations are also restricted to the stable subspace . again , instead of computing the response of ( [ ss_adj_stable ] ) , we compute that of the following system : @xmath198 the snapshots of the impulse responses of systems ( [ ss_stable_numerical ] ) and ( [ ss_adj_stable_numerical ] ) are stacked as columns of @xmath29 and @xmath31 , and using the expressions ( [ svd ] ) and ( [ bal_transf ] ) , we obtain the balancing modes @xmath199 and the adjoint modes @xmath200 . we used 200 snapshots of the linearized simulation and 200 snapshots of each adjoint simulation , with the spacing between snapshots fixed to 50 timesteps , to compute the balancing transformation . these number of snapshots and the spacing were sufficient to accurately compute the modes ; further reduction in the spacing did not significantly change the singular values from the svd computation ( [ svd ] ) . we considered the outputs to be a projection onto 4 , 10 and 20 pod modes ( corresponding to 4 , 10 and 20 mode _ output - projections _ , as introduced in section [ sec : outproj ] ) . using these modes , we use the expressions in equation ( [ rom_compact],[output_compact2 ] ) to obtain the matrices @xmath201 defining the reduced - order model of the stable - subspace dynamics . the vorticity contours of the balancing and the adjoint modes , for a 10-mode output projected system , are plotted in fig . [ fig : modes ] . the adjoint modes provide a direction for projecting the linearized equations onto the subspace spanned by the balancing modes . since these modes are quite different from the pod and the balancing modes , the resulting models are also quite different from those obtained using the standard pod - galerkin technique wherein an orthogonal projection is used . since the models obtained using balanced truncation are known to perform better than the pod - galerkin models , as reported by @xcite , the better performance could be attributed to a better choice of projection using the adjoint modes . ) and the diagonal elements of the controllability ( @xmath178 , @xmath202 ) and observability ( @xmath203 , @xmath204 ) gramians of a 25-mode model with a 4 , 10 , and 20-mode output projection , for the unstable steady state at @xmath205.,height=230 ] since the reduced - order models of the stable - subspace dynamics are approximately balanced , the controllability and observability gramians of the @xmath95-dynamics of ( [ rom_compact ] ) , given by expressions ( [ gramians_stable ] ) , are approximately equal and diagonal . further , their diagonal values are approximately the same as the hankel singular values @xmath206 obtained by the svd ( [ svd ] ) . the diagonal values of the gramians and the singular values for different output projections are plotted in fig . [ fig : unstable_hsv ] for a 30-state reduced - order model . with increasing order of output projection , the hsvs converge to the case with full - state output , and the number of converged hsvs is roughly equal to the order of output projection , as was observed by @xcite . we see that the diagonal elements of both the gramians are very close to the hsvs for the first 20 modes . for higher modes , the observability gramians are inaccurate , which is due to a small inaccuracy of the adjoint formulation mentioned in section [ sec : lin_adjoint ] . for controller design , we use models of order @xmath207 , for which these gramians are sufficiently accurate . , @xmath202 ) are compared with predictions of models with 4 ( @xmath203 , @xmath204 ) , 10 ( @xmath178 , @xmath208 ) , and 20 ( @xmath180 , @xmath209 ) modes.,title="fig:",height=163 ] , @xmath202 ) are compared with predictions of models with 4 ( @xmath203 , @xmath204 ) , 10 ( @xmath178 , @xmath208 ) , and 20 ( @xmath180 , @xmath209 ) modes.,title="fig:",height=163 ] finally , to test the accuracy of the reduced - order models , we compare the impulse responses of system ( [ ss_stable_numerical ] ) ( that is , restricted to the stable subspace ) with that of the model ( [ rom_compact ] ) , restricting @xmath210 . in particular , we compare the outputs of the two systems , which are the projection onto the pod modes ; a representative case in fig . [ fig : outputs_stable_oproj10 ] shows the results of 4 , 10 and 20 mode models of a system approximated using a 20-mode output projection ( the outputs are projection onto the leading 10 pod modes ) . the first output , which is a projection onto the first pod mode , is well captured by all the models until @xmath211 , while the 20-mode model performs well for all time . also shown is the eleventh output , which is well captured only by the 20-mode model . as we will see later , it is important to capture the higher - order outputs for design of observers . . the state is then multiplied by the gain @xmath212 , computed based on the reduced - order model using lqr , to obtain the control input @xmath123.,height=124 ] . comparison of the outputs @xmath213 and @xmath214 of a 12-mode reduced - order model ( @xmath215 ) with the projection of data from the linearized simulation ( @xmath216).,title="fig:",height=134 ] . comparison of the outputs @xmath213 and @xmath214 of a 12-mode reduced - order model ( @xmath215 ) with the projection of data from the linearized simulation ( @xmath216).,title="fig:",height=134 ] the resulting models can now be used along with standard linear control techniques to obtain stabilizing controllers . we use linear quadratic regulator ( lqr ) to compute the gain @xmath212 so that the eigenvalues of @xmath217 ( where the matrices were defined in ( [ rom_compact ] ) ) are in the left - half of the complex plane , and the input @xmath218 minimizes the cost @xmath219 = \int_0^\infty \ , ( a^\ast q a + u^\ast r u ) \ , dt , \label{lqr_cost}\end{aligned}\ ] ] where @xmath220 and @xmath221 are positive weights computed as follows . we choose @xmath220 such that the first term in the integrand of ( [ lqr_cost ] ) represents the energy , that is , we use @xmath222 , with @xmath223 defined in ( [ output_compact ] ) . the weight @xmath221 is chosen to be a multiple of the identity @xmath224 , and typically @xmath132 is chosen to be a large number @xmath225 , to avoid excessively aggressive controllers . the control implementation steps are sketched in fig . [ fig : cartoon_fullstate_control ] ; first compute the reduced - order state @xmath226 , using the expression ( [ init_model ] ) , then the control input is given by @xmath227 . here , we derive the gain @xmath212 based on a 12-mode reduced - order model ( with 2 unstable and 10 stable modes ) , using @xmath228 , and include the same in the original linearized and nonlinear simulations . the output is approximated using a 4-mode output projection . the difference between the linear and nonlinear simulations is that , in the latter , the steady state field @xmath120 is subtracted from the state @xmath10 , before projecting onto the modes to compute the reduced - order state @xmath226 . [ fig : contlin ] compares the model predictions with the projection of data from the simulations of the linearized system ( [ linear ] , [ lin_constraint ] ) , with a control input . the initial condition used is the flow field obtained from an impulsive input to the actuator . both the states shown in the figure eventually decay to zero , which implies that the perturbations decay to zero , thus stabilizing the unstable steady state . more importantly , the model predicts the outputs accurately for the time horizon shown in the plots . vs. time @xmath229 , for full - state feedback control , with control turned on at different times in the base uncontrolled simulation . the base case with no control ( @xmath178 ) has the unstable steady state as the initial condition , and transitions to periodic vortex shedding . the control is tested for different initial conditions , corresponding to @xmath230 of the base case , and stabilizes the steady state in all the cases ( @xmath177).,title="fig:",width=432 ] we now use the same controller in the full nonlinear simulations and test the performance of the model for various perturbations of the steady state . a plot of the lift coefficient @xmath183 vs. time @xmath229 , with the control turned on at different times of the base simulation , is shown in fig . [ fig : cont_fullfback_lift ] . the initial condition for the base case ( no control ) is the unstable steady state ; eventually , small numerical errors excite the unstable modes and the flow transitions to periodic vortex shedding . in separate simulations , control is turned on at times @xmath231 corresponding to the base case . as the figure shows , the control is effective and is able to stabilize the steady state in each case , even when the flow exhibits strong vortex shedding . we remark that the latter two of these perturbations are large enough to be outside the range of validity of the linearized system , but the control is still effective , implying a large basin of attraction of the stabilized steady state . we also compare the output of the reduced - order model with the outputs of the nonlinear simulation ; the plots are shown in fig . [ fig : outputs_control_oproj4 ] . the models perform well for the initial transients , but for longer times fail to capture the actual dynamics . this is not surprising as these perturbations are outside the range of validity of the linear models . for control purposes , it appears to be sufficient to capture the initial transients ( approximately one period ) , during which the instability is suppressed to a great extent . we remark that one could possibly compute nonlinear models by projecting the full nonlinear equations onto the balancing modes , or enhance the model subspace by adding pod modes of vortex shedding and the shift modes as proposed by @xcite to account for the nonlinear terms . of the uncontrolled case plotted in fig . [ fig : cont_fullfback_lift ] . comparison of the outputs @xmath213 and @xmath232 of a 12-mode ( 2 unstable and 10 stable modes ) reduced - order model ( @xmath215 ) with the projection of data from the full nonlinear simulation ( @xmath216).,title="fig:",height=134 ] of the uncontrolled case plotted in fig . [ fig : cont_fullfback_lift ] . comparison of the outputs @xmath213 and @xmath232 of a 12-mode ( 2 unstable and 10 stable modes ) reduced - order model ( @xmath215 ) with the projection of data from the full nonlinear simulation ( @xmath216).,title="fig:",height=134 ] finally , we note that the reduced - order model ( [ rom_compact ] ) decouples the dynamics on the stable and unstable subspaces , and also , the dynamics on the unstable subspace can be computed only using the unstable eigen - bases @xmath64 and @xmath65 . thus , we could derive a control gain @xmath233 , based only on the two - dimensional unstable part of the model , such that the eigenvalues of @xmath234 are in the left half complex plane . that is , we can obtain a stabilizing controller _ without _ modeling the stable subspace dynamics . we have performed simulations to test such a controller and found that it also is capable of suppressing the periodic vortex shedding and thus results in a large basin of attraction for the stabilized steady state . the choice of weight matrices @xmath220 and @xmath221 in the lqr cost ( [ lqr_cost ] ) needs to be different to obtain a comparable performance . however , as shown in the next section , it is essential to model the stable subspace dynamics to design a practical controller based on an observer that reconstructs the entire flow field using a few sensor measurements . the full - state feedback control of section [ sec : fullstatefb ] is not practical since it is not possible to measure the entire flow field . here , we consider a more practical approach of measuring certain flow quantities at a small number of sensor locations . we assume that we can measure the velocities at the sensors shown in fig . [ fig : actuator_sensor ] , in the near - wake of the plate . we remark that , even though these sensors are not experimentally realizable , they serve as a good testing ground for our models . and the sensor measurements @xmath52 are used as inputs to the observer , which reconstructs the reduced - order state @xmath235 . this state is then multiplied by the gain @xmath212 , to obtain the control input @xmath123 . both , the controller and observer gains @xmath212 and @xmath236 are computed based on the reduced - order model using lqr and lqg respectively.,height=144 ] using the reduced - order models derived as outlined in section [ sec : models ] , we design observers that dynamically estimate the entire flow field . since the models ( [ rom_compact ] ) have a different output ( projection onto certain modes ) , we modify the output equation so that it appropriately represents the sensor measurements . we replace the output equation of ( [ rom_compact ] ) with @xmath237 where @xmath238 and @xmath145 is the number of sensor measurements . the matrix @xmath113 is sparse and extracts the values of the output of ( [ rom_compact ] ) at the sensor locations ; thus , each row of @xmath113 is filled with 0s except for the entry corresponding to a sensor measurement , which is 1 . with the output equation ( [ observer_output ] ) , we design an observer using a linear quadratic gaussian ( lqg ) estimation . this method assumes that the errors in representing the state @xmath226 and and the measurement @xmath52 ( due to the inaccuracies of the model ) are stochastic gaussian processes , and results in an estimate @xmath235 of the state @xmath226 that is optimal in the sense that it minimizes the mean of the squared error ; refer to @xcite for details . we now discuss briefly our procedure for modeling these noises ; consider the reduced - order model ( [ rom_compact ] ) , but with process noise @xmath239 and sensor noise @xmath50 which enter the dynamics as follows : @xmath240 a key source of the process ( state ) noise @xmath239 arises from model truncation , and second , from ignoring the nonlinear terms in the reduced - order model . the nonlinearity of the dynamics is important , for instance , when the model is used to suppress vortex shedding . a source of the sensor noise arises from two sources ; first , the state @xmath10 is approximated as a sum of a finite number of modes ( [ modal_exp ] ) , and second , in the output projection step , the output is considered as a projection of the ( approximated ) state @xmath10 onto a finite number of pod modes ( [ yrom_unstable_outproj ] ) . here , we approximate these two noises as gaussian processes whose variances are @xmath241 and @xmath242 gives the expected value . here , @xmath243 is the operator obtained by projecting the nonlinear navier - stokes equations ( [ navier1 ] ) onto the balancing modes @xmath46 , using the adjoint modes @xmath47 . the state @xmath244 is obtained by projecting the snapshots , obtained from a representative simulation of the full nonlinear system , onto the balancing modes . the representative simulation we used here is the base case , with no control , shown in fig . [ fig : cont_fullfback_lift ] , which includes the transient evolution from the steady state to periodic vortex shedding . the resulting estimator is of the form : @xmath245 where @xmath235 is the estimate of state @xmath226 , @xmath246 is the estimated output , and @xmath236 is the observer gain . the estimator is then used along with the full - state feedback controller designed in section [ sec : fullstatefb ] to determine the control input ; a schematic is shown in fig . [ fig : cartoon_obsv_control ] . norm ) obtained from an impulse response ( @xmath177 , @xmath202 ) of ( [ ss_stable_numerical ] ) , and the energy captured by 4 ( @xmath178 , @xmath204 ) , 10 ( @xmath179 , @xmath208 ) , and 20 ( @xmath180 , @xmath247 ) leading pod modes.,height=192 ] , @xmath202 ) , of an impulse response of ( [ ss_stable_numerical ] ) , compared with the reconstruction using 10 ( @xmath179 , @xmath208 ) and 20 ( @xmath178 , @xmath204 ) leading pod modes.,height=259 ] since the observability gramian corresponding to the pair @xmath248 is different from that for the pair @xmath249 , the model ( [ rom_compact ] ) with the output represented by ( [ observer_output ] ) , is not balanced . in principle , it is possible to construct reduced - order models of a system with the sensor measurements as the outputs , using the procedure of section [ sec : approx_baltrunc_unstable ] . since the number of outputs will be typically small , the output projection step would not be required for such models . however , if this were done , the cost function ( [ lqr_cost ] ) , based on total kinetic energy in the perturbation , would not be captured well by the model and the model would not be as effective for control design . vs. time @xmath229 , for estimator - based feedback control , with control turned on at different times in the base uncontrolled simulation . the base case ( @xmath178 ) is the same as in fig . [ fig : cont_fullfback_lift ] , and the control is tested for different initial conditions , corresponding to @xmath230 of the base case ( @xmath177 ) . in both the cases , the controller stabilizes the flow to a small neighborhood of the steady state.,title="fig:",width=432 ] , @xmath204 ) by a 22-mode observer quickly converge to the actual states ( @xmath177 , @xmath202 ) . the initial conditions used are those corresponding to @xmath250 ( top and bottom ) of the uncontrolled case shown in fig . [ fig : cont_obsvfback_lift].,title="fig:",height=134 ] , @xmath204 ) by a 22-mode observer quickly converge to the actual states ( @xmath177 , @xmath202 ) . the initial conditions used are those corresponding to @xmath250 ( top and bottom ) of the uncontrolled case shown in fig . [ fig : cont_obsvfback_lift].,title="fig:",height=134 ] + , @xmath204 ) by a 22-mode observer quickly converge to the actual states ( @xmath177 , @xmath202 ) . the initial conditions used are those corresponding to @xmath250 ( top and bottom ) of the uncontrolled case shown in fig . [ fig : cont_obsvfback_lift].,title="fig:",height=134 ] , @xmath204 ) by a 22-mode observer quickly converge to the actual states ( @xmath177 , @xmath202 ) . the initial conditions used are those corresponding to @xmath250 ( top and bottom ) of the uncontrolled case shown in fig . [ fig : cont_obsvfback_lift].,title="fig:",height=134 ] the models used in section [ sec : fullstatefb ] for full - state feedback were those of a system whose stable - subspace output was the velocity field projected onto the leading 4 pod modes . these 4 pod modes capture only about @xmath251 of the energy , but the resulting models were effective in suppressing vortex shedding . however , for observer design , this representation of the output is inadequate , as the energy content of the flow at the sensor locations is very small , while the pod modes capture the energetically dominant modes . hence , a greater number of pod modes is required to accurately represent the velocity at the sensor locations . the temporal evolution of the energy content of the flow , obtained from an impulse response of the system restricted to evolve on the stable subspace , is plotted in figure [ fig : l2norm ] . also plotted is the energy content of the same flow , but projected onto the leading 4 , 10 and 20 pod modes ; thus , a 4-mode projection leads to noticeable errors , while both 10- and 20-mode projections accurately represent the energy . the velocity field at the sensor locations , reconstructed by 10 and 20 pod modes , plotted in figure [ fig : sensor_reconst ] , shows that a 10-mode projection does not accurately represent the velocities at the sensor locations . since 20 pod modes are sufficient to represent these velocities , we derive models using a _ 20-mode output projection _ , and use the same for observer design . the models obtained using the modified output ( [ observer_output ] ) are used to design dynamic observers based on the vertical ( @xmath50- ) velocity measurements at the sensor locations . a 22-mode reduced - order model , with 2 and 20 modes describing the dynamics on the unstable and stable subspaces respectively , is used to design a kalman filter for producing an optimal estimate of the velocity field based on gaussian approximations of error terms ( [ process_noise_model ] , [ sensor_noise_model ] ) . this estimate is then used along with reduced - order model controller to determine the control input , as shown in fig . [ fig : cartoon_obsv_control ] . the results of this observer - based controller ( or compensator ) are shown in figs . [ fig : cont_obsvfback_lift ] , [ fig : states_obsvcontrol_oproj20 ] . the compensator again stabilizes the unstable operating point , and furthermore , the observer reconstructs the reduced - order model states accurately . initially , the observer has no information about the states ( the initial condition is zero ) , but it quickly converges to and follows the actual states . there is a key difference from the full - state feedback control , that the compensator does not exactly stabilize the unstable operating point but converges to its small neighborhood . the reason is that the observer design is based on the velocities at the sensor locations projected onto the leading 20 pod modes , rather than the exact velocities at these locations . these small errors enter into the observer s dynamics in the same way that sensor noise enters , resulting in small errors in the state estimate . we presented an algorithm for developing reduced - order models of the input - output dynamics of high - dimensional linear unstable systems , extending the approximate balanced truncation method developed by @xcite for stable systems . we assumed that the dimension of the unstable eigenspace is small and the corresponding global eigenmodes can be numerically computed . the modeling procedure treats the dynamics on the unstable subspace exactly and obtains a reduced - order model of the dynamics on the stable subspace . in a proof - of - concept study , the procedure was applied to control the 2-d low - reynolds - number flow past a flat plate at a large angle of attack @xmath128 , where the natural flow state is periodic vortex shedding . we first performed a continuation study at @xmath3 and computed the branch of steady states with @xmath128 varying from @xmath252 to @xmath2 ; we show that the flow undergoes a hopf bifurcation from steady state to periodic shedding at @xmath253 . we developed reduced - order models of the linearized dynamics at @xmath5 actuated by a localized body force close to the leading edge of the plate . the outputs were considered to be the entire flow field , projected onto the unstable eigenmodes and the leading pod modes of the impulse response simulation ( restricted to the stable subspace ) . we developed stabilizing controllers based on the reduced - order models to stabilize the unstable steady state and showed that the models agreed well with the actual simulations . we also included the controllers in the full nonlinear simulations , and showed that they had a large - enough basin of attraction to even suppress the vortex shedding . for such large perturbations , however , the model agreement with the full simulation was good only for short times . a natural step towards improving these models would be to project the full nonlinear equations onto the balancing modes to obtain nonlinear models . alternately , the balanced models , which accurately capture the transient dynamics , could be combined with the pod - based models using shift - modes of @xcite which accurately capture vortex shedding and some of the transient dynamics . an interesting future direction is development of algorithms to compute _ nonlinear _ balanced models , for instance based on the theoretical work of @xcite . instead of computing nonlinear models , here we pursued a step towards more practical controllers by considering an observer - based control design , in which the outputs were modified to be just two near - wake velocity measurements . the nonlinear terms in the equations , which our models do not capture , were treated as process noise , and the error in modeling the outputs was treated as sensor noise . we designed a 22-mode reduced - order observer which reconstructed the flow field accurately , and along with the controllers , suppressed vortex shedding and stabilized the flow in a small neighborhood of the unstable steady state . we remark that the actuator and sensors considered here are not practically realizable , but the methodology presented here can be extended to a more practical actuation such as blowing and suction through the plate and measurements using surface pressure sensors . furthermore , the choice of sensor locations in this study was ad hoc , and an interesting problem is of finding the optimal sensor locations , for a given actuator . the controllers present here are designed to operate at a fixed set of parameters ( such as @xmath254 and @xmath128 ) , and it would be interesting to test their performance at off - design parameter values ; the study on the performance of models of the linearized channel flow at off - design @xmath254 by @xcite shows promise in that direction . a motivation for the choice of our model problem was to develop tools towards manipulating wakes of micro - air vehicles . recently , @xcite performed a numerical study of flow past low - aspect - ratio plates , and a future direction we intend to undertake is to perform a detailed continuation study of the same flow to explore the existence and stabilization of high - lift unstable steady states in this 3-d flow . the authors would like to thank tim colonius and kunihiko taira for their tremendous help in adopting their immersed boundary solver . the authors would also like to thank ioannis g. kevrekidis , sung joon moon and liang qiao for their help with the timestepper - based steady - state analysis . this work was funded by the u. s. air force office of scientific research grant fa9550 - 05 - 1 - 0369 and this support is gratefully acknowledged . without loss of generality , a transformation @xmath14 ( and its inverse ) that decouples the stable and unstable dynamics of ( [ ss ] ) can be written as : @xmath255 where the columns of @xmath256 and @xmath257 span the unstable and stable _ right _ eigenspaces of @xmath11 , while the columns of @xmath258 and @xmath259 span the unstable and stable _ left _ eigenspaces of @xmath11 . further , these matrices are scaled such that @xmath260 and @xmath261 . the transformation ( [ transf ] ) decouples the dynamics of ( [ ss ] ) as given in ( [ decouple_ss ] ) with the various matrices defined as follows : @xmath262 using ( [ transf ] ) in ( [ gramians_general ] ) , the gramians of the original system ( [ ss ] ) are @xmath263 where , @xmath22 and @xmath23 are the gramians corresponding to the system defined by @xmath21 , while @xmath25 and @xmath26 are the gramians corresponding to the system defined by @xmath24 . let @xmath264 be the transformation that balances the gramians @xmath25 and @xmath26 , while @xmath265 be the transformation that balances @xmath22 and @xmath23 . then , it can be verified that the transformation that balances the gramians @xmath96 and @xmath97 is given by @xmath266 thus , the balancing transformation consists of two parts @xmath64 and @xmath79 which respectively balance the dynamics on the unstable and stable subspaces of @xmath11 . as per the technique of @xcite , a reduced - order model can be obtained by truncating the columns of @xmath46 that correspond to the relatively uncontrollable and unobservable states . as we will show now , the algorithm outlined in section [ sec : approx_baltrunc_unstable ] essentially computes the leading columns of @xmath79 ( and the corresponding rows of its inverse ) . we show that the controllability gramian of the stable dynamics of ( [ ss ] ) , which are defined by ( [ ss_stable ] ) , is the same as the `` stable '' part of the gramian defined in ( [ gramians_general_expand ] ) . note that using ( [ transf ] ) and the definition ( [ proj_stable ] ) , the projection operator @xmath67 can be written as @xmath267 using the definition ( [ gramians_stable ] ) , the controllability gramian of ( [ ss_stable ] ) is @xmath268 which is the same as the stable part of @xmath96 . similarly , it can be shown that the observability gramian @xmath269 of ( [ ss_stable ] ) is the same as the `` stable '' part of the observability gramian @xmath97 : @xmath270 thus , balancing the gramians @xmath269 and @xmath271 is identical to balancing the parts of the gramians @xmath96 and @xmath97 of the original system ( [ ss ] ) that are related to the dynamics on the stable subspace of @xmath11 . in this appendix , we derive the adjoint of the linearized semi - discrete equations ( [ linear],[lin_constraint ] ) . let @xmath272 be the weighting functions corresponding to @xmath273 . then , using the inner product defined in equation ( [ ip ] ) , the weak form of ( [ linear],[lin_constraint ] ) is : @xmath274 integrating by parts with respect to @xmath229 and rearranging terms , @xmath275 for linearization about stable steady states , @xmath276 , as @xmath277 , and if the adjoint equations are integrated backwards in time , @xmath278 . so , the last term on the left hand side of equation ( [ weak2 ] ) vanishes identically . if equation ( [ weak2 ] ) is to hold for all values of @xmath135 and @xmath157 , we get the following adjoint equations hold : @xmath279 where @xmath280 and @xmath158 can be thought of as the weighting functions corresponding to the streamfunction @xmath145 and the flux @xmath144 respectively . now , equations ( [ adjoint1],[adjoint2 ] ) have the same form as ( [ linear],[lin_constraint ] ) except for the nonlinear term . thus , the same time - integration scheme can be used for both , with the appropriate ( linearized ) nonlinear terms .
|
we present an estimator - based control design procedure for flow control , using reduced - order models of the governing equations , linearized about a possibly unstable steady state .
the reduced - order models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system .
the original method is valid only for stable linear systems , and in this paper , we present an extension to unstable linear systems .
the dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes , assumed to be small in number .
a snapshot - based algorithm is developed , using approximate balanced truncation , for obtaining a reduced - order model of the dynamics on the stable subspace .
the proposed algorithm is used to study feedback control of two - dimensional flow over a flat plate at a low reynolds number and at large angles of attack , where the natural flow is vortex shedding , though there also exists an unstable steady state . for control design
, we derive reduced - order models valid in the neighborhood of this unstable steady state .
the actuation is modeled as a localized body force near the leading edge of the flat plate , and the sensors are two velocity measurements in the near - wake of the plate .
a reduced - order kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements , and the resulting estimator - based control is shown to stabilize the unstable steady state . for small perturbations of the steady state , the model accurately predicts the response of the full simulation .
furthermore , the resulting controller is even able to suppress the stable periodic vortex shedding , where the nonlinear effects are strong , thus implying a large domain of attraction of the stabilized steady state .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
magnetic fields are thought to play a significant role in all stages of star formation ( e.g. , recent reviews by mouschovias & ciolek 1999 ; shu et al . 1999 ) . however , the magnetic field is the most poorly measured quantity in the star formation process . observations of the linear polarization from the thermal emission of magnetically aligned dust grains provide a relatively easy approach to explore the magnetic field morphology ( heiles et al . such observations give the field direction in the plane of the sky perpendicular to the direction of polarization ( davis & greenstein 1951 ; roberge 1996 ) . information on the magnetic field morphology is useful for testing the predictions of theoretical models and simulations . the standard star formation theory predicts certain morphological evolution of magnetic fields . theory predicts that molecular clouds will tend to be flat with their minor axes parallel to the field lines , because magnetic fields prevent collapse perpendicular to the direction of the field lines ( mouschovias 1976 ) . as contracting cores form , the field morphology achieves an hourglass " shape with a collapsing accretion disk @xmath2100 au at the pinch " and a magnetically supported envelope @xmath21000 au ( fiedler & mouschovias 1993 ; galli & shu 1993 ) . furthermore , the rotation of disks may twist field lines into the direction along the disk and form a toroidal morphology ( holland et al . numerical simulations performed by ostriker , gammie , & stone ( 1999 ) show that the field morphology is more random for larger ratios of the thermal to magnetic energy . because of the high sensitivity requirement in polarization observations , most previous observations of even the nearest star - forming regions have been made with single - dish telescopes whose large beams cover a region greater than the physical extent of protostellar cores . for example , the angular resolution of the james clerk maxwell telescope ( jcmt ) at 850 @xmath3 m is @xmath214 , which corresponds to @xmath26000 au at the distance of the orion molecular cloud . therefore , in order to test theoretical models and simulations of star formation , it is essential to obtain high - resolution observations of magnetic fields . the best available approach to acquire high resolution information is to conduct interferometric observations . pioneer interferometric polarization observations at millimeter wavelengths have been done with the owens valley radio observatory ( akeson & carlstrom 1997 ) . recently , the berkeley - illinois - maryland - association millimeter array ( bima ) has been able to successfully provide extended polarization maps with high resolution up to 2 ( rao et al . 1998 ; girart , crutcher , & rao 1999 ; lai et al . 2001 , hereafter paper i ) . ngc2024 , a massive star formation region in the orion b giant molecular cloud ( distance @xmath2 415 pc : anthony - twarog 1982 ) , is a good candidate to explore the magnetic field structure at small scales . it contains a luminous region with a north - south molecular ridge at its center corresponding to the dust lane in the optical image . mezger et al . ( 1988 , 1992 ) identify seven dust cores in the molecular ridge at 1300 and 350 @xmath3 m and assign designations using the acronym fir. they interpret these cores as isothermal protostars in the stage of free - fall contraction . however , detailed studies of fir4/5/6 show that these cores contain either a near - infrared source or strong outflows that are traditionally related to protostellar cores in a more evolved stage ( moore & chandler 1989 ; chandler & carlstrom 1996 ) . our interest here focuses on the brightest core , fir5 . with a highly collimated unipolar molecular outflow extended over @xmath25 south of the core ( richer et al . 1992 ) , fir5 appears to be the most evolved object among the fir cores in ngc2024 . continuum observations at 3 mm by wiesemeyer et al . ( 1997 ) resolve fir5 into two compact cores , and they suggest fir5 is a binary disk with an envelope . the magnetic field structure in ngc2024 has been mapped with vla oh and zeeman observations by crutcher et al . ( 1999 ) and with far - infrared dust polarization by dotson at al . both observations have beam sizes larger than the fir5 core . our observations provide new information on detailed field morphology in the fir5 core . the observations were carried out from 1999 march to 2001 february using nine bima antennas with 1-mm superconductor - insulator - superconductor ( sis ) receivers and quarter - wave plates . the digital correlator was set up to observe both continuum and the co 21 line simultaneously . the continuum was observed with a 750 mhz window centered at 226.9 ghz in the lower sideband and a 700 mhz window centered at 230.9 ghz in the upper sideband . strong co @xmath4=21 line emission was isolated in an additional 50 mhz window in the upper sideband . the primary beam was @xmath250 at 1.3 mm wavelength . data were obtained in the d , c , and b array configurations , and the projected baseline ranges were 4.520 , 568 , and 5180 kilowavelengths . the integration time in the d , c , and b array was 3.4 , 19.1 , and 11 hours , respectively . the d - array observations were made first on 1999 march with a pointing center between ngc2024 fir5 and fir6 ( @xmath5 , @xmath6 ) . because the d - array data showed significant amount of polarized flux toward fir5 near the edge of the primary beam , the follow - up c- and b - array observations were therefore made with a pointing center at fir5 ( @xmath7 , @xmath8 ) . the bima polarimeter and the calibration procedure are described in detail in paper i ( also see rao et al . 2001 , in preparation ) . the average instrumental polarization of each antenna was 5.6%0.4% for our observations . the stokes @xmath9 image of the continuum was made by mosaicing the b- , c- , and d - array data with briggs robust weighting of 0.5 ( briggs 1995 ; sault & killeen 1998 ) to obtain a smaller synthesized beam ( hpbw 2314 , pa=9 ) without losing significant amount of flux . d - array data were needed to recover the extended emission in order to obtain better measurements of the polarization percentage . the stokes @xmath10 and @xmath11 images were made with b- and c - array data only , because the precise pattern of the off - axis polarization across the primary beam is unknown . rao et al . ( 1999 ) measured the off - axis instrumental polarization at selected positions and showed that it only provides an uncertainty less than 0.5% ( comparable to the uncertainty of the leakages ) within 4/5 of the primary beam ; therefore , the off - axis polarization calibration was ignored . natural weighting was used to produce the maps of the stokes @xmath10 and @xmath11 in order to obtain the highest s / n ratio , and the resulting synthesized beam was 24@xmath1214 with pa=8 . maps of stokes @xmath10 and @xmath11 were deconvolved and binned to approximately half - beamwidth per pixel ( 1206 ) to reduce oversampling in our statistics . these maps were combined to obtain the linearly polarized intensity ( @xmath13 ) , the position angle ( @xmath14 ) , and the polarization percentage ( @xmath15 ) , along with their uncertainties as described in section 2 of paper i. when weighted with @xmath13 , the average measurement uncertainty in the position angle for our observations was 6023 . figure 1 shows the b+c - array stokes i map ( contours ) superposed on the d - array stokes i map ( grey - scale ) . the b+c - array map resolves fir5 and fir6 into several clumps with the highest resolution ever obtained ( 1612 , pa=11 ) . the clumps with peak flux stronger than 7@xmath16 ( @xmath17 = 3.3 ) are named and tabulated in table 1 . figure 2 displays the mosaiced map of ngc2024 fir5 overlaid with polarization vectors . this map contains b , c , and d array data . polarization vectors are plotted at positions where the observed linearly polarized intensity is greater than 3@xmath18 ( 1@xmath18 = 2.1 ) and the total intensity is greater than 3@xmath16 ( 1@xmath16 = 7.4 ; note that @xmath16 is dominated by incomplete deconvolution rather than thermal noise ) . under these criteria , the polarized emission extends over an area of @xmath28 beam sizes . table 2 lists the polarization measurements in ngc2024 fir5 at selected positions separated by approximately the synthesized beamsize . the distributions and the model fitting of the polarization angle are plotted in figure 3 and figure 4 respectively , and the distribution of the polarization percentage is shown in figure 5 . the continuum emission of fir5 can be separated into three main components fir5:main , fir5:ne , and fir5:sw ( figure 1 ) . fir5:main are resolved into seven clumps , and the seven clumps are assigned designations with lcgr ( table 1 ) . only the two brightest clumps ( fir5:lcgr 4 and 6 ) were previously identified at 96 ghz ( wiesemeyer et al . 1997 ; note that we have renamed these two clumps ) . wiesemeyer et al . ( 1997 ) suggested that fir5 is a binary disk ; however , our high resolution data present complex morphology in fir5 . our data also show that the continuum of fir6 consists of two compact sources . the three main components of fir5 and the two compact sources of fir6 are clearly seen in the 450 @xmath3 m continuum maps of visser et al . ( 1998 ) , suggesting that they are dust dominated sources . figure 2 shows that our detection of the polarized flux in fir5 is mainly distributed in a @xmath2104 strip along the major axis of fir5:lcgr 4 extending into fir5:lcgr 2 , which is roughly perpendicular to the disk previously proposed by wiesemeyer et al . the peak of the polarized flux occurs at a position about 18 north of the continuum peak between fir5:lcgr 3 and 4 . detections with @xmath19 were also made toward positions near fir5:lcgr 7 and in fir5:ne . the histogram of polarization angles in fir5 is shown in figure 3 . most of the polarization detected is associated with fir5:lcgr 4 . toward this source the average position angle is @xmath0606 , and the angles appear to decrease from south to north from @xmath07 to @xmath071 . compact polarization is also observed associated with fir5:ne and fir5:lcgr 7 . both these sources show a similar position angle ( average = 549 ) which is very distinct from that of the vectors in fir5:lcgr 4 . our results are consistent with recent jcmt observations at 850 @xmath3 m , which also show @xmath20 50 in fir5:ne , low polarization around fir5:lcgr 6 , and @xmath20 @xmath060 in fir5:lcgr 4 ( matthews et al . 2001 , private communication ) . since the magnetic field direction inferred from the dust polarization is perpendicular to the polarization vectors , the variation of the polarization angles around fir5:lcgr 4 suggests that the magnetic field lines are curved . the curved field lines can be successfully modeled with a set of parabolas with the same focal point , and the best model with minimum @xmath21 is presented in figure 4 . the symmetry axis of the best model is at @xmath077 , which is consistent with the position angle of the line connecting fir5 : d and f at @xmath067 . the histogram in figure 4 shows the distribution of the deviation between our measurements and the best model , which shows a gaussian - like distribution with a dispersion of @xmath22=9233 . after deconvolving the measurement uncertainty in this region ( @xmath23 ) from the observed dispersion , we obtain the intrinsic dispersion of the polarization angles @xmath24 . the average polarization percentage is 6.0%1.5% in the main core and 10.5%4.7% toward the eastern side of fir5 . the polarization percentage distribution is plotted in figure 5 . these plots show that the polarization percentage decreases toward regions of high intensity as well as toward the cloud center . based on the apparent close correlation in both figure 5(a ) and 5(b ) , we perform a least - squares fit on @xmath25 versus @xmath26 and @xmath15 versus the distance from the peak of fir5:lcgr 4 ( @xmath27 ) . we obtain the following results : ( 1 ) for all data points , @xmath28 with a correlation coefficient of @xmath00.86 , and ( 2 ) in the main core , @xmath29 with a correlation coefficient of 0.97 . because higher intensity and smaller radius both imply higher density , our results suggest that the polarization percentage decreases toward the high density region . this conclusion is consistent with what we reported in paper i for the w51 e1/e2 cores , and our interpretation was the decrease of the dust alignment efficiency toward high density regions . in 3.2 , we showed that the field lines in ngc2024 fir5 can be modeled with a set of parabolas which may represent part of an hour - glass shape . the reason that the rest of the hour - glass shape is not detected could be due to the relatively lower dust column density to the east of the region where polarization has been detected . the hour - glass geometry has been predicted by theoretical work and simulations ( fiedler & mouschovias 1993 ; galli & shu 1993 ) . when enough matter collapses along field lines and forms a disk - like structure , the gravity in the direction along the major axis of the disk would eventually overcome the supporting forces and pull the field lines into an hour - glass shape . observations of magnetic fields have found several cases consistent with this geometry , such as in w3 ( roberts et al . 1993 ; greaves , murray , & holland 1994 ) , omc-1 ( schleuning 1998 ) , and ngc1333 iras4a ( girart , crutcher , & rao 1999 ) . our results in ngc 2024 fir5 provide a possible supporting example to the theoretical models at a small scale of few thousand au . this scale is much smaller than the hour - glass shapes observed in w3 and omc-1 ( @xmath30 au ) , and is comparable to that in ngc1333 iras4a . the symmetry axis of our hour - glass model is approximately parallel to the binary disk previously proposed by wiesemeyer et al . such a coincidence could be an indirect evidence for the existence of a collapsing disk . although our high resolution map shows that fir5:main is more complicated than a binary disk , it is still possible that there is an east - west disk consisting of four clumps ( fir5:lcgr 1 , 4 , 6 , and 7 ) , with the remaining three clumps either collapsing into the disk or being ejected through outflows . fir5:lcgr 2 is at @xmath31145 direction with respect to fir5:lcgr 4 , which is close to the unipolar outflow at @xmath31170 . since outflows could be deflected by magnetic fields away from the protostellar cores ( girart , crutcher , & rao 1999 ) , it is possible that fir5:lcgr 2 traces the direction of the outflow near the central core . further high resolution kinematic study is needed to examine the above speculation . although the magnetic field strength can not be directly inferred from polarization of dust emission , ostriker , stone , & gammie ( 2001 ) show that the chandrasekhar - fermi formula ( chandrasekhar & fermi 1953 ) modified with a factor of 0.5 can provide accurate estimates of the plane - of - sky field strength under strong field cases ( @xmath32 ) for their simulations of magnetic turbulent clouds . therefore , the projected field strength ( @xmath33 ) can be expressed as @xmath34 where @xmath35 is the average density , @xmath36 is the rms line - of - sight velocity , the number density of molecular hydrogen @xmath37 , and the linewidth @xmath38 . to estimate @xmath33 in ngc2024 fir5 , we must first carefully determine the density of the dust core and the turbulent linewidth . mezger et al . ( 1992 ) derived high density ( @xmath39 - 3 ) in fir 1 - 7 from dust emission , but their value is significantly higher than that derived from molecular studies ( @xmath210@xmath40 - 3 from cs : schulz et al . 1991 ; @xmath210@xmath41 - 3 from c@xmath42o : wilson , mehringer , & dickel 1995 ) . it is possible that depletion of molecules in the dense core causes molecules to only trace the envelope of the dense core ; on the other hand , chandler & carlstrom ( 1996 ) showed that the kinetic temperature in the fir cores is higher than what mezger et al . ( 1992 ) assume , implying a lower density . mangum , wootten , & barsony ( 1999 ) studied the kinetic temperature in ngc2024 with multi - line observations of formaldehyde ( h@xmath43co ) , and their results supported the arguments of chandler & carlstrom ( 1996 ) . we therefore adopt @xmath44 - 3 for fir5 from mangum , wootten , & barsony s large velocity gradient ( lvg ) calculation . we also adopt a weighted linewidth of @xmath45 at the position at the peak of fir5 from mangum , wootten , & barsony ( 1999 ) . note that , just as for other molecules , the peak of h@xmath43co emission does not coincide with the continuum peak ; therefore , the h@xmath43co emission may come from the envelope of the fir5 . this may be an advantage because the h@xmath43co linewidth is less likely contaminated by the possible dynamical motions in the core and better represents the turbulent motion . in 3.2 , we calculated the angle dispersion @xmath24 after taking out the systematic field structure in order to identify the dispersion purely from the alfvnic motion . along with the parameters discussed in the previous paragraph , we obtain @xmath463.5 mg . we can also calculate the lower limit of @xmath33 using the largest angle dispersion , which is the dispersion before taking out the parabolic model ( @xmath4713146 ) ; thus , @xmath481.9 mg . our estimate of the plane - of - sky field strength is much larger than the line - of - sight field strength measured by crutcher et al . ( 1999 ) from oh zeeman observation , which is @xmath49 g at fir5 . except in the unlikely case that the magnetic field direction lies almost on the plane of the sky , the low field strength detected with the zeeman measurement can be explained by ( 1 ) the beam - averaging over small scale field structure , and/or ( 2 ) that oh line does not trace high density region of the dust cores . the small dispersion of the polarization angles observed by us may imply that the turbulent motion is not strong enough to disturb the magnetic field structure . in order to quantitatively discuss the the relative importance of the turbulent and magnetic energy in ngc2024 fir5 , we calculate the ratio of the turbulent to magnetic energy @xmath50 where @xmath51 is the turbulent linewidth and @xmath52 is the alfvn speed . statistically , @xmath53 , thus @xmath54 can be estimated from eq . ( 1 ) . therefore , @xmath55 in the general conditions prevalent in molecular clouds , the thermal linewidth @xmath56 is much smaller than the turbulent linewidth , so @xmath57 . the ratio of the turbulent to magnetic energy simply depends on the polarization angle dispersion , @xmath58 the 0.5 factor in eq ( 1 ) was obtained under the assumption that the dust alignment efficiency is uniform throughout the cloud . if dust alignment efficiency is lower in the regions with higher density as we suggest in 3.3 , @xmath59 would be larger for the same @xmath60 and the correction depends on the degree of depolarization . for our case , the most conservative estimate of the angle dispersion is @xmath61 , which leads to a small turbulent to magnetic energy ratio , @xmath62 . therefore , the magnetic field most likely dominates the turbulent motion in the core region of ngc2024 fir5 . this is also the case in w51 e1/e2 ( see paper i ) . high - resolution polarization observations are needed to explore the detailed magnetic field structure in star - forming cores . we have obtained a continuum map of ngc2024 fir5 and fir6 and a polarization map of ngc2024 fir5 with the highest resolution ever obtained ( @xmath22 ) . our observations resolve fir5 and fir6 into several continuum clumps . the information revealed by our polarization observations of fir5 are summarized below : * extended polarization is detected associated with the main core at @xmath0606 . compact polarization is also observed toward the eastern side of fir5 at 549 . the polarization is low between these two regions and the secondary intensity peak of fir5 has an upper limit of 6% . * the magnetic field lines in the core are systematically curved with a symmetry axis close to the major axis of a putative disk . this is consistent with an hour - glass morphology for the magnetic fields predicted by theoretical works . * the polarization percentage decreases toward regions with high intensity and short distance to the center of the core . the tight correlations imply that the depolarization is a global effect and may be caused by the decrease of the dust alignment efficiency in high density regions . * the small dispersion of the polarization angles in the core suggests that the magnetic field is strong ( @xmath63 2 mg ) and the ratio of the turbulent to magnetic energy is small ( @xmath62 ) . therefore , the magnetic field most likely dominates turbulent motions in ngc2024 fir5 . this research was supported by nsf grants ast 99 - 81363 and ast 98 - 20651 . we would like to thank the staff at hat creek , especially rick forster and mark warnock for assistance with the polarimeter control system . we also thank charles gammie for his helpful comments . akeson , r. l. , & carlstrom , j. e. 1997 , , 491 , 254 anthony - twarog , b. j. 1982 , , 87 , 1213 briggs , d. s. 1995 , phd dissertation , the new mexico institute of mining and technology ( available at the website http://www.aoc.nrao.edu/ftp/dissertations/dbriggs/diss.html ) chandler , c. j. , & carlstrom , j. e. 1996 , , 466 , 338 chandrasekhar , s. , & fermi , e. 1953 , , 118 , 113 crutcher , r. m. , roberts , d. a. , troland , t. h. , & goss , w. m. 1999 , , 515 , 275 davis , l. , & greenstein , j. l. 1951 , , 114 , 209 dotson , j. l. , davidson , j. , dowell , c. d. , schleuning , d. a. , & hildebrand , r. h. 2000 , , 128 , 335 fiedler , r. a. , & mouschovias , t. ch . 1993 , apj , 415 , 680 galli , d. , & shu , f. h. 1993 , apj , 417 , 243 girart , j. m. , rao , r. , & crutcher , r. m. 1999 , , 525 , 109 greaves , j. s. , murray , a. g. , & holland , w. s. 1994 , , 284 , l19 heiles , c. , goodman , a. a. , mckee , c. f. , & zweibel , e. g. 1993 , in protostars and planets iii , eds . m. matthews , & e. levy ( tuscon : university of arizona press ) , 279 holland , w. s. , greaves , j. s. , ward - thompson , d. , & andre , p. 1996 , , 309 , 267 lai , s .- p . , crutcher , r. m. , girart , j. m. , & rao , r. 2001 , to be published in the november 10 , 2001 issue ( paper i ) lazarian , a. , goodman , a. a. , & myers , p. c. 1997 , apj , 490 , 273 leahy , p. , vla scientific memoranda no . 161 mangum , j. g. , wootten , a. , & barsony , m. 1999 , , 526 , 845 mauersberger , r. , wilson , t. l. , mezger , p. g. , gaume , r. , & johnston , k. j. 1992 , , 256 , 640 mezger , p. g. , chini , r. , kreysa , e. , wink , j. e. , & salter , c. j. 1988 , , 191 , 44 mezger , p. g. , sievers , a. w. , haslam , c. g. t. , kreysa , e. , lemke , r. , mauersberger , r. , & wilson , t. l. 1992 , , 256 , 631 moore , t. j. t. , & chandler , c. j. 1989 , , 241 , 19p mouschovias , t. ch . 1976 , , 207 , 141 mouschovias , t. ch . , & ciolek , g. e. 1999 , in the origin of stars and planetary systems , eds . c. j. lada & n. d. kylafis ( kluwer academic press ) , 305 ostriker , e. c. , gammie , c. f. , & stone , j. m. 1999 , , 513 , 259 ostriker , e. c. , stone , j. m. , & gammie , c. f. 2001 , , 546 , 980 rao , r. , crutcher , r. m. , plambeck , r. l. , & wright , m. c. h. 1998 , , 502 , l75 rao , r. 1999 , ph.d . dissertation , university of illinois at urbana - champaign rao , r. , crutcher , r. m. , girart , j. m. , lai , s .- p . , wright , m. c. h. , & plambeck , r. l. 2001 , in preparation richer , j. s. , hills , r. e. , & padman , r. 1992 , , 254 , 525 roberge , w. g. 1996 , asp conf . ser . 97 : polarimetry of the interstellar medium , 401 roberts , d. a. , crutcher , r. m. , troland , t. h. , & goss , w. m. 1993 , , 412 , 675 sault , r. j. , & killeen , n. e. b. 1998 , miriad users guide ( available at the website http://www.atnf.csiro.au/computing/software/miriad ) sault , r. j. , teuben , p. j. , & wright , m. c. h. 1995 , asp conf . ser . 77 : astronomical data analysis software and systems iv , 4 , 433 schleuning , d. a. 1998 , apj , 493 , 811 schulz , a. , guesten , r. , zylka , r. , & serabyn , e. 1991 , , 246 , 570 shu , f. h. , allen , a. , shang , h. , ostriker , e. c. , & li , z. 1999 , in the origin of stars and planetary systems , eds . c. j. lada & n. d. kylafis ( kluwer academic press),193 thompson , a. r. , moran , j. m. , & swenson , g. w. 1986 , interferometry and synthesis in radio astronomy ( new york : wiley ) visser , a. e. , richer , j. s. , chandler , c. j. , & padman , r. 1998 , , 301 , 585 wiesemeyer , h. , guesten , r. , wink , j. e. , & yorke , h. w. 1997 , , 320 , 287 wilson , t. l. , mehringer , d. m. , & dickel , h. r. 1995 , , 303 , 840 lcccc & @xmath64 & @xmath65 & @xmath9 & @xmath15 + source & @xmath66 & & ( ) & ( % ) + fir5:lcgr 1 & 5 41 44.00 & -1 55 40.7 & 46 & @xmath67 17 + fir5:lcgr 2 & 5 41 44.10 & -1 55 44.0 & 64 & 10.3 + fir5:lcgr 3 & 5 41 44.23 & -1 55 36.8 & 39 & @xmath67 25 + fir5:lcgr 4 & 5 41 44.25 & -1 55 40.8 & 293 & 3.8 + fir5:lcgr 5 & 5 41 44.32 & -1 55 35.7 & 40 & @xmath67 21 + fir5:lcgr 6 & 5 41 44.48 & -1 55 42.2 & 107 & @xmath67 6 + fir5:lcgr 7 & 5 41 44.69 & -1 55 43.5 & 55 & @xmath67 14 + fir6 c & 5 41 45.13 & -1 56 04.2 & 67 & + fir6 n & 5 41 45.17 & -1 56 00.3 & 26 & + ccccl position & stokes @xmath9 & polarization & polarization & note + ( , ) & ( ) & percentage(% ) & angle ( ) & + ( -0.3 , 4.5 ) & 26.67.3 & 23.19.5 & -729 & + ( -1.1 , 3.0 ) & 132.87.3 & 18.71.9 & -712 & polarization peak + ( -1.2 , 1.2 ) & 300.67.3 & 3.90.7 & -645 & intensity peak + ( -1.8 , 4.5 ) & 41.97.3 & 19.56.1 & -667 & + ( -2.6 , 1.2 ) & 153.77.3 & 4.91.4 & -398 & + ( -2.6,-0.8 ) & 137.17.3 & 6.51.6 & -397 & + ( -2.6,-2.7 ) & 45.67.3 & 15.45.2 & - 89 & + ( -3.9,-0.9 ) & 53.27.3 & 20.04.8 & -426 & + ( -4.8,-2.4 ) & 39.17.3 & 17.36.3 & -399 & + ( 10.2 , 8.9 ) & 31.77.3 & 27.19.2 & 637 & fir5:ne + ( 4.1,-0.5 ) & 75.77.3 & 9.32.9 & 319 & near fir5:lcgr 7 + & & 6.01.5 & -606 & average of the main core + & & 10.54.7 & 549 & average of the eastern side of fir5 +
|
we present the first interferometric polarization maps of the ngc2024 fir5 molecular core obtained with the bima array at approximately 2 resolution .
we measure an average position angle of @xmath0606 in the main core of fir5 and 549 in the eastern wing of fir5 .
the morphology of the polarization angles in the main core of fir5 suggests that the field lines are parabolic with a symmetry axis approximately parallel to the major axis of the putative disk in fir5 , which is consistent with the theoretical scenario that the gravitational collapse pulled the field lines into an hour - glass shape .
the polarization percentage decreases toward regions with high intensity and close to the center of the core , suggesting that the dust alignment efficiency may decrease at high density .
the plane - of - sky field strength can be estimated with the modified chandrasekhar - fermi formula , and the small dispersion of the polarization angles in fir5 suggests that the magnetic field is strong ( @xmath1 mg ) and perhaps dominates the turbulent motions in the core .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
hl tau is a very young solar - type star surrounded by a dusty circumstellar disk and a remnant envelope . the object is located at a distance of @xmath3140 pc ( loinard et al . 2007 ) , within the taurus star - forming region . showing all ingredients of a young system in the earliest stages of planet formation , hl tau has attracted a lot of attention over the years . for a summary of the early observational data and the results of the first comprehensive radiative transfer modeling we refer to dalessio et al . ( 1997 ) and menshchikov , henning & fischer ( 1999 ) . hl tau drives an ionized jet indicating on - going accretion ( e.g. , pyo et al . 2005 , anglada et al . early interferometric observations revealed that emission at cm wavelengths traces the radio counterpart of this collimated jet , while the emission at wavelengths @xmath4 1.3 cm predominantly traces dust emission from a disk ( rodrguez et al . 1994 , wilner et al . this source attracted renewed interest after high angular resolution interferometric observations indicated that the hl tau disk , despite its youth , may already be forming planets . observations performed with the combined array for research in millimeter - wave astronomy ( carma ) at 1.3 and 2.7 mm ( @xmath320 - 120 au resolution ) suggested a gravitationally unstable disk which might undergo fragmentation ( kwon et al . 2011 ) . very large array ( vla ) observations at 1.3 cm ( @xmath312 au resolution ) revealed a compact structure in the disk at 65 au radius , interpreted as a protoplanet candidate ( greaves et al . subsequent high sensitivity vla observations at 7.0 mm ( @xmath37 au resolution ) could not confirm this putative proto - planet , but found evidence for a depression at radius @xmath5 au in the radial density profile of the disk , which was interpreted as being related to the presence of an orbiting protoplanet ( carrasco - gonzlez et al . 2009 ) . with the long baselines of the atacama large millimeter / submillimeter array ( alma ) becoming available ( alma partnership et al . 2015a ) , this facility produced iconic images of the dust emission at 2.9 , 1.3 , and 0.87 mm from the hl tau disk ( @xmath33.5 to 10 au resolution ) , showing a number of axisymmetric bright and dark rings , most probably corresponding to high and low density concentric dust structures in the disk ( alma partnership et al . the images immediately triggered numerous theoretical works in order to explain these remarkable structures . planet - related explanations range from the presence of embedded sub - jupiter mass planets ( picogna & kley 2015 , dipierro et al . 2015 , dong et al . 2015 ) to individual more massive planets ( gonzalez et al . alternative explanations include magnetized disks without planets ( flock et al . 2015 ) , fast pebble growth near condensation fronts ( zhang et al . 2015 ) , and sintering - induced dust rings ( okuzumi et al . 2016 ) . the presence of massive planets ( @xmath310 - 15 m@xmath6 ) in two prominent dips in the dust distribution at @xmath370 au was excluded utilizing deep direct l band imaging with the large binocular telescope ( lbt ) ( testi et al . 2015 ) , but the presence of lower - mass planets in the disk is not yet excluded and remains an interesting possibility . detailed radiative transfer analysis of the alma data shows that the emission from the various bright rings is probably optically thick , even at the longest alma wavelength of 2.9 mm ( pinte et al . 2016 , jin et al . the challenge of deriving density profiles and grain size distributions can only be circumvented by observations at even longer wavelengths where the disk will be optically thinner . in this letter , we present new high - sensitivity karl g. jansky vla observations at 7.0 mm of the hl tau disk . this data provide a deeper view of the hl tau disk , with an angular resolution comparable to the alma images . we observed hl tau with the vla of the national radio astronomy observatory ( nrao ) using the q band receivers in the c , b , and a configurations ( see table [ tab1 ] for details ) . we observed the frequency range 39 - 47 ghz ( central wavelength @xmath7 7.0 mm ) . calibration of the data was performed with the data reduction package common astronomy software applications ( casa ; version 4.4.0 ) , using a modified version of the nrao calibration pipeline . images were made with the casa task clean using multi - scale multi - frequency synthesis that fits the emission with a taylor series with nterms=2 during the deconvolution ( rau & cornwell 2011 ) . since our multi - configuration observations are sensitive to emission at very different scales ( from @xmath316@xmath82240 au to 0@xmath905@xmath77 au ) , we made images with different angular resolutions by adjusting the briggs robust parameter ( briggs 1995 ) and the gaussian uv - taper in clean . we also made images by splitting the 8 ghz band in two sub - bands of 4 ghz each ( central wavelengths 6.7 and 7.3 mm ) . for comparison , images were aligned by assigning to the position of the central peak of the alma images the same absolute coordinates than in the vla images , @xmath10(j2000)=04@xmath1131@xmath1238@xmath13426 , @xmath14(j2000)=18@xmath1513@xmath1657@xmath923 . in figures [ fig1 ] and [ fig2 ] we present the vla images with different angular resolutions and comparisons with the alma images . we obtained radial profiles of the intensity of the alma images and our most sensitive vla image at 7.0 mm ( natural weighting ; beam size@xmath70@xmath9067 ) . for a proper comparison , we convolved all images to a common circular beam size of 0@xmath907 ( the smallest beam size obtained from the public 2.9 mm alma uv - data with a uniform weighting ) . from these images , we obtained the average intensity within concentric elliptical rings of 0@xmath901 width at different radii , using the task iring of the astronomical image processing system ( aips ) . the dimension and orientation of the elliptical rings match those of the hl tau disk derived from the alma images ( inclination angle , @xmath1746.72@xmath15 , and position angle , [email protected]@xmath15 ; alma partnership et al . 2015b ) . some contamination from free - free emission from the hl tau jet is expected at 7.0 mm . from our 6.7 and 7.3 mm sub - band images we obtain a spectral index of @xmath31 at the center of the disk , consistent with comparable contributions from free - free and dust emission . to correct for the free - free contamination , we calibrated vla a configuration archival data at 6 and 2 cm ( project code : 12b-272 ) where emission is dominated by a partially optically thick radio jet in the ne - sw direction . from these images , the frequency dependence of the jet s flux density and angular size ( major axis ) can be expressed as @xmath18^{0.4}$ ] @xmath19jy and @xmath20^{-1}$ ] , respectively , consistent with previous 7 mm observations ( e.g. wilner & lay 2000 ) . thus , an upper limit to the free - free contribution at 7.0 mm is obtained by extrapolating the flux density from cm wavelengths with a spectral index of 0.4 , while a lower limit can be obtained by assuming free - free emission becomes optically thin at 2 cm ( i.e. , spectral index of @xmath210.1 from 2 cm to shorter wavelengths ) . therefore , we expect unresolved ( @xmath220@xmath907 ) free - free emission with a flux density in the range @xmath3200 - 400 @xmath19jy corresponding to @xmath335 - 65% of the 7.0 mm emission at the disk center . this correction implies a larger uncertainty in the dust intensity at the center of the disk . from the corrected intensity profiles , we derived brightness temperatures at each wavelength using the planck equation for blackbody radiation , and we computed spectral indices between different wavelengths . we also derived profiles of optical depth and column density ( assuming dust temperature power - laws and opacity , see 3.1 ) . radial profiles are shown in figure 3 . the recent alma images of the hl tau disk revealed several dark and bright rings ( named d1-d7 and b1-b7 , respectively ) ( fig . [ fig1 ] ) . our new 7.0 mm vla observations are the most sensitive and highest angular resolution observations of the hl tau disk performed to date at such a long wavelength . the low - angular resolution 7.0 mm image shows an elliptical source with similar size and orientation as the alma images ( fig . [ fig1 ] ) . at higher angular resolution , the vla is able to image with high signal - to - noise ( s / n ) ratio ( @xmath234-@xmath24 ) the 7.0 mm emission from the inner half of the disk ( @xmath450 au ; see figs . [ fig1 ] and [ fig2 ] ) . in our 7.0 mm images we clearly identify several of the features seen in the alma images : the central disk and the first pair of dark ( d1 ) and bright ( b1 ) rings ( figs . [ fig1]-[fig3 ] ) . the importance of our sensitive 7.0 mm images is that , at such long wavelength , the emission has lower optical depth than in the alma data . this is especially critical for the study of the innermost part of the disk , where dust becomes opaque at all alma wavelengths and , as a consequence , physical properties are poorly constrained even with detailed modeling ( e.g. pinte et al . 2016 , jin et al . 2016 ) . at the positions of the most opaque regions , the center of the disk and the first bright ring , the 7.0 mm brightness temperatures ( @xmath345 and 15 k , resp . ) are @xmath34 times lower than those of the alma 0.87 mm image ( @xmath3130 and 60 k , resp . ) . therefore , assuming that at 0.87 mm the emission from these two structures is optically thick , we obtain optical depths @xmath40.4 at 7.0 mm . this imply that dust emission at 7.0 mm is well optically thin at all radii , even in the densest parts of the disk . our new high sensitivity 7.0 mm images of the hl tau disk are an excellent basis for future comprehensive radiative transfer modeling to accurately obtain the physical properties of the disk . they are especially necessary in order to better constrain properties in the inner disk regions , where terrestial planets are thought to form , in principle . in the following , we analyze our vla images of the hl tau disk to obtain direct rough estimates of the different physical parameters ( e.g. , mass and grain size distributions ) . we also analyze possible substructure in the disk and discuss our results in the context of planet formation . an accurate determination of the mass distribution in the hl tau disk requires detailed radiative transfer modeling . for this paper , we obtain first estimates by assuming a simple power - law for the dust temperature in the form @xmath25 . while the exponent seems to be well constrained in the range @xmath26=0.5 - 0.6 by previous studies , there is large uncertainty in the reference temperature , with different proposed values in the range @xmath2770 - 140 k at @xmath28=10 au ( e.g. , menshchikov et al . 1999 , kwon et al . 2011 , pinte et al . 2016 ) . for the dust opacity at 7.0 mm , we use a range of typical values for the disk - averaged opacity , @xmath290.13 - 0.2 @xmath30 g@xmath31 ( e.g. , menshchikov et al . 1999 , prez et al . 2012 ) . thus , at each radius , we calculate ranges for the optical depth and the dust column density taking into account these uncertainties ( see fig . [ fig3 ] ) . our calculations are consistent with the inner features of the disk being optically thick at all alma wavelengths , while at 7.0 mm the emission is optically thin at all radii ( see fig . [ fig3]b ) . we estimated values of the dust column density around @xmath31 g @xmath32 at the center of the disk ( see fig . [ fig3]c ) . this suggest a denser disk at inner radii ( @xmath2250 au ) than previously obtained by detailed modeling ( e.g. , pinte et al . 2016 predict @xmath40.2 g @xmath32 at the center of the disk ) . we also estimated dust masses for the inner disk ( i d ) and the bright rings ( b1 to b6 ; see table [ tab2 ] ) . for those features which are optically thin in the alma images , i.e. b2-b6 , we obtain dust masses consistent with previous estimations ( pinte et al . however , our optically thin 7.0 mm data suggest large dust masses for the inner disk and the first bright ring ( b1 ) for which only lower limits were obtained previously ( see table [ tab2 ] ) . finally , we estimate that the total dust mass of the disk is within the range ( 1 - 3)@xmath010@xmath1 m@xmath2 , which is also somewhat larger than previous estimates , @xmath3(0.3 - 1)@xmath010@xmath1 m@xmath2 ( e.g. , menshchikov et al . 1999 , dalessio et al . 1997 , kwon et al . 2008 , pinte et al . 2016 ) . grain growth and mixing lead to changes in particle - size distribution and dust composition throughout the disk ( henning & meeus 2011 ) . this has been recently studied in several objects for which segregation by particle - size ( e.g. menu et al . 2014 ) and radial changes in dust optical properties ( e.g. guilloteau et al . 2011 , prez et al . 2012 , 2015 ) are observed . the fully resolved alma and vla images of the hl tau disk offer now an excellent opportunity for a detailed study of the properties of the particle - size distribution in a very young disk . in particular , changes in the dust properties can be inferred from changes in the spectral index of the emission , @xmath10 , but only for optically thin emission in the rayleigh - jeans regime ( e.g. beckwith et al . 2000 ) . when derived from the short alma wavelengths , the observed radial variations of @xmath10 , from @xmath32 to 2.5 ( figure [ fig3]d ) , reflect high optical depths inwards of @xmath350 au . thus , these alma observations can not be used to infer grain growth in the densest , inner disk regions . in contrast , the observed radial variations of @xmath10 derived from the two most optically thin wavelengths , 7.0 and 2.9 mm , show a different behavior : ( 1 ) at all radii , except at the location of the dark gap d5 , we obtain @xmath33 , consistent with the emission at shorter wavelengths being more optically thick and not in the r - j regime , and ( 2 ) a clear gradient in @xmath34 is observed between @xmath310 - 50 au , consistent with a change in the dust optical properties and a differential grain - size distribution , with larger grains at smaller radii . similar results have been inferred for a number of more evolved disks ( e.g. prez et al . 2012 , 2015 ; tazzari et al . 2016 , menu et al . 2014 ) . some more evolved transitional disks , when observed at long wavelengths , show knotty rings of dust emission ( e.g. , hd 169142 ; see fig . 1c in osorio et al . 2014 ) . our highest angular resolution vla images of the young hl tau disk also reveal an interesting knotty and not axisymmetric morphology of the first bright ring ( b1 ) ( see fig . [ most of the knots seem to be consistent with small ( 1-@xmath24 ) fluctuations of the brightness ( due to the rms noise of the map ) , suggesting a structure with a roughly uniform brightness . however , to the ne , there is a compact knot ( labeled as clump candidate " in figs . [ fig2]b - d ) whose morphology is very different to the rest of the ring . we made several images with different angular resolutions and different bandwidths ( see 2 ) , and noted that , while the knotty emission changes significantly in different images ( consistent with being small fluctuations in a uniform structure ) , the ne knot is clearly identifiable and the most compact knot in all the images ( see figs . [ fig2]c and [ fig2]d ) . this knot also coincides with a local intensity maximum in the 1.3 mm alma image ( see fig . all this suggests that the 7.0 mm ne knot traces a real structure in the first bright ring . we note that the position of the ne knot is coincident with the direction of the jet , and thus it could be related to a local increase in the flux density because of the foreground free - free emission . however , at 7.0 mm , the free - free emission from the jet is expected to be confined within the inner disk ( see 2 ) . moreover , from our sub - band images we derive a spectral index @[email protected] , suggesting that the emission from this knot is dominated by thermal dust emission . we also noted that this spectral index is slightly smaller than the average spectral index obtained for b1 from 7.0 and 2.9 mm ( see fig . [ fig3 ] ) , suggesting an accumulation of larger dust grains in the clump . it is known that dense rings could undergo vortex formation by the rossby wave instability and efficiently concentrate large particles ( meheut 2012 ) , then , we speculate that the ne knot traces a dense dust clump formed within the massive bright ring . in this case , we estimate a dust mass in the range 3 - 8 m@xmath37 for this clump . the presence of dark and bright concentric rings has been commonly interpreted as the result of planet formation already ongoing in the hl tau disk . however , since hl tau is a very young t tauri star , the presence of several ( proto)planets sufficiently massive to carve holes in the disk at this early stage is somewhat surprising . on the other hand , alternative formation mechanisms , not requiring the presence of protoplanets , seem also possible . moreover , sensitive searches for massive ( proto)planets in the outer dark rings have yielded negative results ( see 1 and references therein ) . we propose a scenario in which the hl tau disk may have not formed planets yet , but rather is in an initial stage of planet formation . instead of being caused by ( proto)planets , the dense rings could have been formed by an alternative mechanism . our 7.0 mm data suggest that the inner rings are very dense and massive , and then , they can be gravitationally unstable and fragment . it is then possible that the formation of these rings result in the formation of dense clumps within them like the one possibly detected in our 7.0 mm image . these clumps are very likely to grow in mass by accreting from their surroundings , and then they possibly represent the earliest stages of protoplanets . in this scenario , the concentric holes observed by alma and vla would not be interpreted as a consequence of the presence of massive ( proto)planets . instead , planets may be just starting to form in the bright dense rings of the hl tau disk . _ acknowledgments . _ cc - g , lfr and rg - m acknowledge support from unam - dgapa papiit ia101715 and ia102816 . lmp acknowledges support from the alexander van humboldt foundation . tb acknowledges support from the dfg grant ( kl 1469/13 - 1 ) . ga , em , mo , and jmt acknowledge support from mineco and feder funds . this paper makes use of the following alma data : ads / jao.alma#2011.0.00015.sv . alma is a partnership of eso ( representing its member states ) , nsf ( usa ) and nins ( japan ) , together with nrc ( canada ) , nsc and asiaa ( taiwan ) , and kasi ( republic of korea ) , in cooperation with the republic of chile . the joint alma observatory is operated by eso , aui / nrao and naoj . lccc obs . & project & & on - source + date & code & conf . & total time + 2014-dec-07 & 14b-485 & c & 1.7 h + 2015-feb-15 & 14b-485 & b & 1.6 h + 2015-aug-13 & 14b-487 & a & 1.1 h + 2015-aug-25 & 14b-487 & a & 1.1 h + 2015-sep-19 & 14b-487 & a & 1.7 h + 2015-sep-20 & 14b-487 & a & 3.8 h + 2015-sep-21 & 14b-487 & a & 3.4 h + ccccc & & flux density & & + covered disk & radius & at 7.0 mm & + feature & ( au ) & ( mjy ) & this paper@xmath38 & pinte et al.@xmath39 + i d & @xmath2213 & 0.61 @xmath36 0.04 & 10 - 50 & @xmath232.3 + b1 & 13 - 32 & 1.45 @xmath36 0.02 & 70 - 210 & @xmath2347 + b2 & 32 - 42 & 0.48 @xmath36 0.01 & 30 - 90 & 30 - 69 + b3 & 42 - 50 & 0.35 @xmath36 0.01 & 20 - 80 & 14 - 37 + b4 & 50 - 64 & 0.36 @xmath36 0.01 & 30 - 90 & 40 - 82 + b5 & 64 - 74 & 0.18 @xmath36 0.01 & 10 - 50 & 5.5 - 8.7 + b6 & 74 - 90 & 0.45 @xmath36 0.01 & 40 - 140 & 84 - 129 +
|
the first long - baseline alma campaign resolved the disk around the young star hl tau into a number of axisymmetric bright and dark rings . despite the very young age of hl tau these structures
have been interpreted as signatures for the presence of ( proto)planets .
the alma images triggered numerous theoretical studies based on disk - planet interactions , magnetically driven disk structures , and grain evolution .
of special interest are the inner parts of disks , where terrestrial planets are expected to form .
however , the emission from these regions in hl tau turned out to be optically thick at all alma wavelengths , preventing the derivation of surface density profiles and grain size distributions . here
, we present the most sensitive images of hl tau obtained to date with the karl g. jansky very large array at 7.0 mm wavelength with a spatial resolution comparable to the alma images . at this long wavelength
the dust emission from hl tau is optically thin , allowing a comprehensive study of the inner disk .
we obtain a total disk dust mass of ( 1 - 3)@xmath010@xmath1 m@xmath2 , depending on the assumed opacity and disk temperature .
our optically thin data also indicate fast grain growth , fragmentation , and formation of dense clumps in the inner densest parts of the disk .
our results suggest that the hl tau disk may be actually in a very early stage of planetary formation , with planets not already formed in the gaps but in the process of future formation in the bright rings .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
physical , chemical , and biological systems are investigated in many laboratories using single molecule spectroscopy @xcite . the investigation of the distribution of the number of photons emitted from a single molecule source is the topic of extensive theoretical research e.g. @xcite and @xcite for a review . since optical properties of single molecules are usually very sensitive to dynamics and statics of their environment , and since the technique removes the many particle averaging found in conventional measurement techniques , single molecule spectroscopy reveals interesting fluctuation phenomena . an important mechanism responsible for the fluctuations in the number of photons emitted from a single molecule source is spectral diffusion e.g. @xcite . in many cases the absorption frequency of the molecule will randomly change due to different types of interactions between the molecule and its environment ( e.g. @xcite and ref . therein ) . for example for single molecules embedded in low temperature glasses , flipping two level systems embedded in the glassy environment , induce stochastic spectral jumps in the absorption frequency of the single molecule under investigation @xcite . in this way the molecule may come in and out of resonance with the continuous wave laser field with which it is interacting . obviously a second mechanism responsible for fluctuations of photon counts is the quantum behavior of the spontaneous emission process @xcite . in his fundamental work mandel @xcite showed that a single atom in the process of resonance fluorescence , _ in the absence of spectral diffusion _ , exhibits sub - poissonian photon statistics @xcite . photon statistics is characterized by mandel s @xmath0 parameter @xmath1 where @xmath2 is the number of emitted photons within a certain time interval . the case @xmath3 is called sub - poissonian behavior , while @xmath4 is called super - poissonian . sub - poissonian statistics has no classical analog @xcite . briefly , the effect is related to anti - bunching of photons emitted from a single source and to rabi - oscillations of the excited state population which favors an emission process with some periodicity in time ( see details below ) . sub - poissonian photon statistics and photon anti - bunching were measured in several single molecule , and single quantum dots experiments @xcite . while sub - poissonian statistics is well understood in the context of resonance fluorescence of an isolated electronic transition of a simple atom in the gas phase , our theoretical understanding of sub - poissonian statistics for a molecule embedded in a fluctuating condensed phase environment is still in its infant stages . in this paper we obtain an exact analytical expression for the @xmath0 parameter in the long time limit , for a single molecule undergoing a stochastic spectral diffusion process . to obtain the exact solution we use the zheng - brown generating function method for single molecule photon statistics @xcite . for the spectral diffusion we use a simple stochastic approach , in the spirit of the kubo andersen line shape theory @xcite . the model exhibits generic behaviors of line shapes of molecules embedded in a condensed phase environment , e.g. motional narrowing when the stochastic fluctuations are fast , power broadening etc . we show that the @xmath0 parameter exhibits rich types of behaviors , in particular it reveals the quantum nature of the emission process in the sub - poissonian regime , while the corresponding model line - shape exhibits a classical behavior . a brief summary of our results was recently published @xcite . our analytical expressions for @xmath0 classify the transitions between sub and super poissonian statistics . they give the conditions on the spectral diffusion time scale for sub - poissonian behavior . motional narrowing type of effect is revealed also for the @xmath0 parameter . our exact result is valid for weak and strong excitation ( i.e arbitrary rabi frequency ) . it yields the lower bound on @xmath0 . the solution shows how we may choose the rabi frequency so that the quantum nature of the photon emission process becomes larger , namely how to minimize @xmath0 in the sub - poissonian regime . this is important for the efficient detection of quantum effects in single molecule spectroscopy , since choosing too small or too large values of the rabi frequency results in very small and hence undetectable values of @xmath0 . finally our exact result is used to test the domain of validity of the generalized wiener khintchine which yields @xmath0 in terms of a fourier transform of a three time dipole correlation ( as well known the wiener khintchine theorem yields the line shape in terms of a one time dipole correlation function ) . the theorem @xcite is based on the semi - classical theory of interaction of light with matter , and on linear response theory ( i.e. , weak rabi frequency ) , it yields @xmath4 . as pointed out in @xcite such a behavior is expected to be valid only for slow enough spectral diffusion processes . we briefly explain some of the main ideas of sub - poissonian statistics . the general idea is that the photons emitted from a single particle , e.g. a molecule , a nano - crystal or atom are correlated in time . consider first a hypothetical molecule , interacting with an exciting laser field , which emits photons with a constant time interval @xmath5 between successive emission events . then @xmath6 , @xmath7 , and hence @xmath8 . due to quantum uncertainty the photon emission process is always random and therefore @xmath9 . sub - poissonian behavior where @xmath10 implies that the stream of photons emitted from a single source maintain correlations in their arrival times to a detector . usually when many molecules interact with a continuous wave laser the emission events are not correlated , and the fluorescence exhibits poissonian statistics @xmath11 . in contrast a single molecule , once it emits a photon , is collapsed to its ground state . hence immediately after an emission event the molecule can not emit a second time ( it has to be re - excited by the laser ) . hence successive photons emitted from a single molecule , seem to repel each other on the time axis , a non - poissonian behavior . this well known effect is called anti - bunching @xcite which is related to sub - poissonian statistics . a second effect related to sub - poissonian behavior are rabi - oscillations . consider a simple atom in the process of resonance fluorescence . when the electronic transition ( frequency @xmath12 ) is in resonance with a continuous wave laser field ( frequency @xmath13 ) the electronic transition can be approximated by a two level system . first let us mentally switch off the spontaneous emission , i.e. set the inverse life time of the transition @xmath14 . for zero detuning @xmath15 the transition will exhibit well known rabi oscillations : the population of the excited state will behave like @xmath16 . since the population in the excited state attains its maximum ( minimum ) periodically , also the emission times of successive photons maintain certain degree of periodicity in time , which implies sub - poissonian statistics . mandel showed @xcite , that for a two level atom in the process of resonance fluorescence @xmath17 when @xmath18 we have @xmath19 since then successive photon emission times are not correlated , because the time between successive emissions becomes very large . while when @xmath20 the excited state is populated swiftly , and only the finite spontaneous emission rate delays the emission , hence @xmath19 also in this case . ( [ eqmold ] ) the lower bound @xmath21 is easily obtained , and the minimum @xmath22 is obtained when @xmath23 . let @xmath2 be the random number of photons emitted by a single molecule source in a time interval @xmath24 , and @xmath25 is the probability of @xmath2 emission events . the information about the photon statistics is contained in the moment generating function @xcite @xmath26 which yields the moments of @xmath2 @xmath27 with which the @xmath0 parameter can in principle be obtained . in eq . ( [ eqa02 ] ) , and in what follows , we use the notation @xmath28 and similarly for second order derivatives with respect to @xmath29 . the over - line in eq . ( [ eqa02 ] ) describes an average over the process of photon emission , later we will consider a second type of average with respect to the spectral diffusion process , which we will denote with @xmath30 . the equations of motion for the generating function was given in @xcite and are called generalized optical bloch equation . for a chromophore with single excited and ground states , and interacting with a continuous wave laser field @xmath31 these equations are exact within the rotating wave approximation and optical bloch equation formalism . they yield the same type of information on photon statistics contained in the quantum jump approach to quantum optics which is used in quantum monte carlo simulations @xcite . in eq . ( [ eqa01 ] ) @xmath32 is the spontaneous emission rate of the electronic transition and @xmath33 is the rabi frequency . the time evolving detuning is @xmath34 where @xmath13 @xmath35 is the laser frequency ( the molecule s bare frequency ) , and @xmath36 is the stochastic spectral diffusion process . in eq . ( [ eqa01 ] ) it is assumed that the molecule in its excited and ground state have no permanent dipole moments , hence the system is described only by the transition dipole moment via the rabi frequency . the physical meaning of @xmath37 , @xmath38 , and @xmath39 and their relation to the standard bloch equation was given in @xcite , some discussion on this issue will follow eq . ( [ eqmt ] ) . for related work on the foundations of these equations see @xcite and ref . therein . note that when @xmath40 the damping terms in eq . ( [ eqa01 ] ) become small [ i.e. the @xmath41 terms ] , hence relaxation of the generalized bloch equations in the important limit of @xmath40 is slow . in what follows we will consider the moments @xmath42 , @xmath43 . for this aim it is useful to derive equations of motion for the vector @xmath44 @xmath45 taking the first and the second derivative of eq . ( [ eqa01 ] ) with respect to @xmath29 and setting @xmath46 , we find @xmath47 where @xmath48 is a @xmath49 matrix @xmath50 @xmath51 the first three lines of @xmath48 describe the evolution of @xmath52 , these are the standard optical bloch equations in the rotating wave approximation @xcite . these equations yield @xmath53 which in turn gives the mean number of photons using eq . ( [ eqa02 ] ) @xmath54,\ ] ] and us - usual @xmath55 is the population in the excited state . the fourth line of @xmath48 is zero , it yields @xmath56 , this equation describes the normalization condition of the problem namely @xmath57 for all times @xmath58 [ to see this use eq . ( [ eq02 ] ) and @xmath59 . the evolution of the other terms @xmath60 are of current interest since they describe the fluctuation of the photon emission process . in particular using eqs . ( [ eqa02],[eqmt ] ) @xmath61 } = \gamma\left [ \overline{n\left(t\right ) } + 2 { \cal w}'(1 ) \right].\ ] ] solutions of time dependent equations like eq . ( [ eqmtz ] ) are generally difficult to obtain , a formal solution is given in terms of the time ordering operator @xmath62 , @xmath63 z(0)$ ] . ( [ eqmtz ] ) yields a general method for the calculation of @xmath0 for a single molecule undergoing a spectral diffusion process . the aim of this paper is to obtain an exact solution of the problem for an important stochastic process used by kubo and anderson @xcite to investigate characteristic behaviors of line shapes . we assume @xmath64 where @xmath65 describe frequency shifts , and @xmath66 describes a random telegraph process : @xmath67 or @xmath68 . the transition rate between state up ( + ) and state down ( - ) and vice versa is @xmath69 . this dichotomic process is sometimes called the kubo - anderson process @xcite . it was used to describe generic behaviors of line shapes @xcite , here our aim is to calculate @xmath0 describing the line shape fluctuations . we use burshtein s method @xcite of marginal averages , to solve the stochastic differential matrix equation ( [ eqmtz ] ) . the method yields the average @xmath70 with respect to the stochastic process . we will calculate @xmath70 in the limit of long times , and then obtain the steady state behavior of the line shape @xmath71 and the @xmath0 parameter . let @xmath72 be the average of @xmath73 under the condition that at time @xmath58 the value of @xmath74 respectively . @xmath72 are called marginal averages , the complete average is @xmath75 . the equation of motion for the marginal averages is an @xmath76 matrix equation @xmath77 in eq . ( [ eqmat18 ] ) the matrix @xmath78 is identical to matrix @xmath79 in eq . ( [ eqmt ] ) when @xmath80 is replaced by @xmath81 , and @xmath82 is a @xmath49 identity matrix . in the next subsection we obtain the long time solution of eq . ( [ eqmat18 ] ) , the reader not interested in the mathematical details may skip to subsection [ secexact ] , where the solution for the line shape and @xmath0 is presented . in three main steps , we now find the long time behavior of the marginal averages , with which the line and @xmath0 are then obtained . + @xmath83 as mentioned , from normalization condition we have @xmath57 for all times . ( [ eqmat18 ] ) yields the marginal averages @xmath84 , in the steady state . inserting these identities in eq . ( [ eqmat18 ] ) we obtain an equation of motion for the vector @xmath85 @xmath86 where @xmath87 @xmath88 and @xmath89 . in the long time limit the solution of eq . ( [ eqstep1 ] ) reaches a steady state ( ss ) given by @xmath90 and @xmath91 is the inverse of @xmath92 . from eq . ( [ eqstep11 ] ) we find @xmath93 @xmath94 where @xmath95 , and @xmath96 is the @xmath97 matrix element of @xmath98 , @xmath99 . we note that @xmath100 yields the steady state marginal averages of the population difference between the excited and ground state . from eq . ( [ eqa02 ] ) we see that we need @xmath101 to obtain the average number of photon emissions @xmath102 . we use eq . ( [ eqmat18 ] ) and show @xmath103 @xmath104 and @xmath105 the average number of emitted photon , in the long time limit is ( [ eqa02 ] , [ eqyd ] ) @xmath106 namely @xmath107 is proportional to the steady state occupation in the excited state . the line shape is @xmath108 where @xmath109.\ ] ] @xmath110 we use the solutions obtained in previous step to obtain inhomogeneous equations for @xmath111 , @xmath112 where and @xmath113 with @xmath114 \right\ } \pm { \gamma^2 \over 8 r } \left [ \langle { \cal w}^{ss}(1)\rangle_{- } - \langle{\cal w}^{ss}(1)\rangle_{+ } \right ] . \label{eqbb}\ ] ] in the long time limit we obtain @xmath115 where @xmath116 and @xmath117 are column vectors @xmath118 with @xmath119 \mp { \gamma^2 \over 8 r } \left [ \langle { \cal w}^{ss } \left(1 \right ) \rangle_{- } - \langle { \cal w}^{ss } \left(1 \right ) \rangle_{+ } \right]\ ] ] we therefore obtain @xmath120\ ] ] @xmath121.\ ] ] @xmath122 from eq . ( [ eqa02 ] ) , @xmath123 . in steady state we have @xmath124 from eq . ( [ eqmat18 ] ) one can show that in the long time limit @xmath125 t + \ ] ] @xmath126 t\ ] ] @xmath127t^2 . \label{eqydd}\ ] ] finally we obtain the @xmath0 parameter using : @xmath128 using eqs . ( [ eqyd],[eqydd],[eqdef ] ) we obtain the main result of this manuscript @xmath129 ^ 2 + { \gamma \over i\left ( \omega_l \right ) } \sum_{k=\pm } \left\ { \langle { \cal w}^{ss}(1)\rangle_{k } \left [ 1 + 4 \langle { \cal w}^{ss}(1)\rangle_{k } \right ] \right\ } , \label{eqmain}\ ] ] which is valid when measurement time @xmath130 . the @xmath0 parameter in eq . ( [ eqmain ] ) is expressed in terms of @xmath91 . to obtain the solution in terms of the original parameters of the problem @xmath131 , we found analytical expressions for @xmath91 using mathematica . the formula for the @xmath0 parameter is given in the following subsection . without loss of generality we set @xmath132 , hence @xmath13 is the detuning . we find @xmath133 \over \mbox{denominator}\left[q\right ] } , \label{eqq}\ ] ] @xmath134 = \ ] ] @xmath135 @xmath136 = \ ] ] @xmath137 + the line shape is @xmath138 \over \mbox{denominator}[i\left(\omega_l \right ) ] } , \label{eqls}\ ] ] @xmath139 = \ ] ] @xmath140 @xmath141 = \ ] ] @xmath142 when the detuning is zero we find @xmath143 \over \left [ 4 \gamma \nu^2 + \left ( \gamma^2 + 2 \omega^2 \right ) \left ( \gamma + 4 r \right ) \right]^2 } . \label{eqzt}\ ] ] using eq . ( [ eqzt ] ) the lower bound @xmath144 is obtained . the absolute minimum of @xmath0 , i.e. @xmath145 is found when @xmath146 , or when @xmath147 . namely the absolute minimum is found for a molecule whose absorption frequency is fixed , or for a very fast spectral modulation . * remark : * eq . ( [ eqzt ] ) indicates a transition from sub - poissonian statistics @xmath148 to super - poissonian statistics @xmath149 when @xmath150 . if @xmath151 , i.e. if the bath is fast compared with the radiative life time , we find sub - poissonian behavior for all values of @xmath65 and @xmath33 . the @xmath0 parameter is a function of two control parameters @xmath152 and @xmath33 , and three model parameters @xmath153 . the classification of different types of physical behaviors , based on the relative magnitude of the parameters is investigated in this section . the limiting behaviors of @xmath0 are obtained from the exact solution using mathematica . in the slow modulation regime , the bath fluctuation process ( r ) is very slow compared with the radiative decay rate ( @xmath32 ) , frequency fluctuation amplitude ( @xmath65 ) and the rabi frequency ( @xmath33 ) . this case is similar to situations in many single molecule experiments , for example single molecules in low temperature glasses . the exact solution can be simplified in the limit @xmath155 , using eq . ( [ eqls ] ) @xmath156 and @xmath157 the line is a sum of two lorentzians centered on @xmath158 , namely it exhibits splitting behavior when @xmath159 . using eq . ( [ eqq ] ) we find a simple super - poissonian behavior for @xmath0 in the slow modulation limit @xmath160 @xmath161 and since @xmath160 , @xmath0 may become very large ( e.g. @xmath162 in fig . [ fig1 ] ) . a simple picture can be used to understand these results . in the slow modulation limit the molecule jumps between two states @xmath163 and @xmath164 , the time between successive jumps is very long , in such a way that many photons are emitted between jump events . in each of these two states the molecule emits photons at a rate @xmath165 , eq . ( [ eqlspm1 ] ) . these rates are determined by the familiar steady state solutions of the optical bloch equation , for a two level atom with the absorption frequency @xmath166 fixed in time @xcite . in this slow limit the random number of emitted photons , in time interval @xmath24 , is @xmath167 , and @xmath168 is a stochastic intensity that jumps between two states @xmath169 with the rate @xmath69 . using this simple random walk picture it is straightforward to derive eqs . ( [ eqsl01 ] , [ eqq1 ] ) . for mathematical details @xcite who considered a similar slow modulation limit which is valid only for weak rabi frequency . within the slow modulation limit we distinguish between two cases . the case @xmath171 is called the slow modulation strong coupling limit . in this case the line and @xmath0 have two well separated peaks and the broadening of the two peaks due to the finite life time , and power of the laser field is small compared with the frequency shifts ( see fig . [ fig1 ] ) . from fig . [ fig1 ] , and as expected from eqs . ( [ eqsl01 ] , [ eqq1 ] ) , @xmath0 decreases when @xmath69 is increased , while @xmath172 is independent of @xmath69 . thus it is @xmath0 not @xmath173 that yields information on the dynamics . in fig . [ fig1 ] the agreement between the exact solution and the approximation eq . ( [ eqq1 ] ) is good . + the limit @xmath175 is called the weak coupling slow modulation limit . in this case the two peaks of the line , discussed in the previous sub - section , are overlapping and the line is approximated by @xmath176 this result is exact when @xmath177 , for arbitrary @xmath69 , the behavior of @xmath0 is demonstrated in fig . [ fig2 ] , where we observe both super - poissonian and sub - poissonian behaviors . in the slow modulation weak coupling limit , we must distinguish between the cases of large and small detuning . first note that according to eq . ( [ eqq1 ] ) when the detuning is zero we find @xmath178 , namely the leading @xmath179 term in our asymptotic expansion vanishes . we must therefore consider the higher order terms in our asymptotic expansion of eq . ( [ eqq ] ) and we find @xmath180 where @xmath181 eq . ( [ eqqapp ] ) has a simple meaning , the first term is a contribution to @xmath0 from spectral diffusion , which is identical to eq . ( [ eqq1 ] ) . the second term @xmath182 is identical to the result obtained by mandel , for the @xmath0 parameter in the absence of spectral diffusion @xcite , and @xmath183 provided that the detuning is not too large . the second term is dominating over the first when the detuning is small , and for zero detuning we obtain in eq . ( [ eqqapp ] ) sub - poissonian statistics . more explicitly , we taylor expand ( [ eqlspm1 ] ) using @xmath65 as a small parameter , and obtain for the slow modulation weak coupling limit @xmath184 one may say that for zero detuning , the molecule behaves as if its absorption frequency is fixed . to conclude , we see that in the slow modulation limit @xmath0 is a sum of two additive contributions : i ) a part related to spectral diffusion @xmath185 and ii ) and a part related to quantum fluctuations , i.e. @xmath186 . such a behavior was very recently discussed in @xcite ( see also @xcite for related discussion ) . the quantum fluctuations are however bounded from above and below @xmath187 , while the contribution from spectral diffusion is not @xmath188 . hence detection of the quantum fluctuations is possible only when @xmath189 is small , which for our case implies zero detuning and weak coupling limit . we now consider the fast modulation limit . if we take @xmath191 keeping @xmath33 , @xmath32 , and @xmath65 fixed we find from eq . ( [ eqq ] ) @xmath192 given in eq . ( [ eqmandel ] ) . hence in this limit we find a sub - poissonian behavior , provided that the detuning @xmath13 is not too large . the line shape is identical to the expression on the right hand side of eq . ( [ eqs02 ] ) . this behavior is expected , when the spectral diffusion is very fast the emitting single molecule can not respond to the stochastic fluctuations . a more interesting case is to let @xmath191 and @xmath193 but keep , @xmath194 finite . we call this limit the fast modulation limit , using eq . ( [ eqls ] ) the line shape is @xmath195 and when the rabi frequency is small @xmath196 in this limit we have the well known effect of motional narrowing : the width of the line is determine by @xmath197 and as the process becomes faster the line becomes narrower , namely @xmath198 decreases when @xmath69 is increased . the @xmath0 parameter is obtained from eq . ( [ eqq ] ) @xmath199 @xmath200 \over \left [ \gamma^3 + \gamma \gamma_{{\rm sd}}\ ^2 + 2 \gamma \omega^2 + 2 \gamma_{{\rm sd } } \left ( \gamma^2 + \omega^2 \right ) + 4 \gamma \omega_l ^2 \right]^2 } . \label{eqq3}\ ] ] thus in the fast modulation limit the photon statistics is sub - poissonian provided that the detuning is not too large . when @xmath201 the result for @xmath0 reduces to mandel s result eq . ( [ eqmandel ] ) . in fig . [ fig3 ] , we show the line shape and the @xmath0 parameter , for three values of the jump rate ( r ) in the fast modulation regime while @xmath202 are kept fix . we see that as the stochastic spectral diffusion process gets faster , both the line shape and the @xmath0 parameter become narrow . thus both @xmath172 and @xmath0 exhibit a motional narrowing effect . also , as the stochastic process gets faster , a stronger quantum behavior is obtained , in the sense that the minimum of @xmath0 decreases . to investigate the strong coupling limit we consider the value of @xmath0 for @xmath204 , and @xmath205 . from eq . ( [ eqq ] ) we obtain @xmath206 \over 2 r \left [ \gamma^3 + 4 \gamma r^2 + 2 \gamma \omega^2 + 2 r \left ( 2\gamma^2 + \omega^2 \right ) \right]^2 } . \label{eqwwww}\ ] ] this equation exhibits both sub - poissonian and super poissonian behaviors . when the process is very _ slow _ , namely @xmath160 , we obtain @xmath207 a super - poissonian behavior . in the _ intermediate modulation limit _ , when @xmath208 , we obtain @xmath209 a sub - poissonian behavior . when @xmath191 we find that @xmath0 is small @xmath210 eqs . ( [ eqnu1]-[eqnu3 ] ) are valid only in the limit of @xmath193 . however behaviors similar to the predictions of these equations are found also for finite values of @xmath65 . on @xmath211 we have three typical behaviors : @xmath212 @xmath213 when the process is slow , see fig . [ fig1 ] ) . @xmath214 @xmath215 when @xmath216 , see fig . [ fig4 ] for @xmath217 , and @xmath218 when @xmath191 we find @xmath219 , fig . [ fig3 ] . when @xmath221 we obtain interesting behaviors for @xmath0 . in fig . [ fig4 ] the @xmath0 parameter shows a transition from sub - poissonian to super poissonian , photon statistics . in this regime of parameters , the shape of @xmath0 when plotted as a function of @xmath13 is very sensitive to the value of the control parameters e.g. in fig . [ fig4 ] we change @xmath69 only moderately still we see very different types of behaviors for @xmath0 . for certain values of parameters @xmath0 attains more than two peaks ( see fig . [ fig4 ] for @xmath222 ) . in contrast @xmath172 exhibits a simple splitting behavior with two peaks on @xmath158 , which is similar to the slow modulation case . besides the transition from sub to super poissonian behavior , a second type of transition is observed as @xmath69 is increased . in our problem we have two types of sub - poissonian behavior . we noticed already that when the stochastic modulation becomes very fast , @xmath0 has one minimum on zero detuning ( see fig . [ fig2 ] ) , while when @xmath220 , @xmath0 has two minima on @xmath223 ( see fig . [ fig3 ] and @xmath224 . the transition between these two types of sub - poissonian behaviors is shown in fig . we now investigate the dependence of @xmath0 on the excitation field . in fig . [ fig6 ] we consider an example line shape and @xmath0 parameter , where we fix the model parameters @xmath225 and vary the rabi frequency . for the line we see well known power broadening : as the rabi frequency is increased the line becomes wider , and as expected the photon emission rate @xmath172 increases monotonically when @xmath33 is increased . for the @xmath0 parameter we have a turn - over behavior , as we increase @xmath33 the value of @xmath0 on zero detuning decreases then increases . generally this type of turn - over is expected , since as discussed in sec . [ secintrosub ] , @xmath11 when @xmath226 or @xmath227 . thus there exists an optimal rabi frequency which yields an extremum of @xmath0 . obviously it is important to obtain the values of @xmath33 which yield the extremum of @xmath0 , since then the fluctuations are the largest . the extremum can be either a minimum or a maximum , as we shall show now . we now consider the quantum regime @xmath3 . in fig . [ fig8](b ) we demonstrate the turn - over behavior of @xmath228 for an example where the stochastic fluctuations are fast . in this fast modulation case @xmath3 hence @xmath228 has a minimum . for the same parameters the photon emission rate @xmath229 saturates as @xmath33 is increased , and the emission rate is never faster than @xmath32 [ see fig . [ fig8](a ) ] . let @xmath230 be the rabi frequency which minimizes @xmath0 in the sub - poissonian case @xmath3 , and @xmath231 the corresponding value of @xmath0 . ( [ eqzt ] ) we find for zero detuning , and for @xmath232 @xmath233 and @xmath234 eqs . ( [ eqmin1],[eqmin2 ] ) yield @xmath230 and @xmath231 in terms of @xmath65 and @xmath69 . due to motional narrowing effect , for fast processes satisfying @xmath235 , @xmath65 and @xmath69 are not easily obtained from experiment , while the parameter @xmath198 is in principle easy to obtain from the measurement of the line width . ( [ eqq3 ] ) , we find in the fast modulation limit and for zero detuning @xmath236 @xmath237 these simple equations relate between the width of the line given in eq . ( [ eqls3a ] ) and @xmath231 and @xmath230 . from eqs . ( [ eqsim1],[eqsim2 ] ) we see that when @xmath238 , @xmath239 and @xmath22 . when @xmath240 we find @xmath241 and @xmath242 . * remark : * when the detuning is not zero , we find using eq . ( [ eqq3 ] ) @xmath243 similar turn - over behaviors of @xmath228 are found also in other non fast parameter regimes . in fig . [ fig7 ] we show @xmath228 versus @xmath33 for the slow modulation weak coupling limit @xmath244 and for zero detuning . [ fig7 ] shows that @xmath0 exhibits a minimum as function of @xmath33 , this minimum is found in the vicinity of @xmath245 . such a behavior is understood based on eq . ( [ eqqapp ] ) , the spectral diffusion contribution for @xmath0 is not important at zero detuning , while the contribution of @xmath246 yields @xmath239 . to demonstrate that the turnover behavior of @xmath228 is generic , we consider also the intermediate modulation limit in fig . [ fig9 ] . here we choose the detuning according to @xmath247 , since the @xmath0 parameter on zero detuning is relatively small ( see fig . [ fig5 ] ) . in contrast to the behaviors in the quantum regime @xmath3 , in the slow modulation limit where @xmath4 , @xmath228 obtains a maximum , whose location is easy to calculate with eq . ( [ eqq1 ] ) . such a behavior is demonstrated in fig . [ fig10 ] for a case when the spectral shift @xmath65 is not very large . if @xmath248 then in the slow modulation limit @xmath249 when the detuning is equal to @xmath247 . in eq . ( [ eqyh1 ] ) the second term on the right hand side is supposed to be a correction to the first term , namely @xmath250 . let @xmath251 be the value of @xmath33 which maximizes @xmath0 in the super - poissonian regime , and @xmath252 the corresponding maximum . this maximum always exists since as mentioned @xmath178 when @xmath227 or @xmath226 . then using eq . ( [ eqyh1 ] ) @xmath253 which is independent of @xmath69 and @xmath254 note that when the frequency shifts are very large @xmath255 we find using eq . ( [ eqyh2 ] ) @xmath256 . hence the value of @xmath251 may become very large and then in experiment it is impossible to reach @xmath251 ( e.g. @xmath257 ) . if we impose the condition @xmath258 we have @xmath259 for the laser detuning @xmath211 . hence @xmath185 monotonically increases and eventually saturates , similar to the behavior of the average emission rate . the @xmath0 parameter yields informations not contained in the line shape . the most obvious is the transition from super ( i.e. classical ) to sub ( quantum ) poissonian behavior . such a quantum signature of the photon emission process is not obtained from the line shape . in comparison with the @xmath0 parameter of a single atomic transition , the @xmath0 parameter investigated here exhibits rich behaviors . these include splitting , both in the sub and in the super poissonian regime , a transition from a fast to a slow modulation limit , and effects related to motional narrowing . the most non - trivial behavior is obtained in the intermediate modulation limit when @xmath260 where @xmath0 attains more then two peaks . since @xmath0 contains the new information on single molecule experiments , namely information beyond the line shape , it is important to emphasize that @xmath0 attains an extremum for a particular value of the rabi frequency . in particular in the sub - poissonian regime @xmath228 has a minimum . hence we optimize the rabi frequency in such a way that @xmath261 is increased , e.g. we obtain @xmath230 . in other words there exist an `` ideal '' choice of the rabi frequency in single molecule experiments . in the quantum sub - poissonian regime this optimal rabi frequency can not be considered weak , neither strong , hence perturbative approaches to single molecule spectroscopy are not likely to yield it . this is in complete contrast to most theories of line shapes which are based on the assumption of weak external fields , e.g. the wiener khintchine theorem and linear response theory . single molecule theories should be able to predict the turn - over behavior of @xmath228 based on different models , since such a behavior is not expected to be limited to the model under investigation . of - course the exact solution presented in this manuscript is very valuable in this direction , since it predicts precisely the details of this transition for the kubo - anderson stochastic process . it would be interesting to investigate further how general are our results . from line shape theory , we know that in the fast modulation limit , line shapes have lorentzian shapes under very general conditions . from experiment we know that motional narrowing effect , and lorentzian behavior of lines is wide spread . thus at - least in this limit certain general features of line shapes , which are not sensitive to model assumptions are found . similarly , we expect , that in the fast modulation limit , some of our results are general . for example the motional narrowing behavior of @xmath0 and its approach to mandel s behavior @xmath186 is likely to be general . it would be interesting to check if the relation between @xmath231 and @xmath230 eq . ( [ eqsim1],[eqsim2 ] ) , and the width of the line given by @xmath262 and @xmath32 is valid for other models , both stochastic and hamiltonian . these simple equations are important since they yield the optimal rabi frequency @xmath230 in terms of the width of the line shape , which in turn is easily determined in usual line shape measurement . in ref . ( @xcite,@xcite ) a semi - classical framework for the mathematical calculation of @xmath0 for single molecule spectroscopy was investigated . the approach yields the @xmath0 parameter in terms of a fourier transform of a three time dipole correlation function . as mentioned in the introduction , the approach in ( @xcite,@xcite ) is based on two main approximations ( i ) external fields are weak @xmath227 , i.e. linear response theory , and the ( ii ) semi - classical approach to photon counting statistics . the second assumption implies that @xmath4 , and as pointed out in @xcite such an approach is expected to be valid for slow processes . the approach is useful since most single molecule experiments report on slow fluctuations . the results obtained in this manuscript reduce to those in ( @xcite,@xcite ) in the limit of @xmath227 , and in the slow modulation limit , as they should . the quantum behavior of @xmath0 becomes important when @xmath263 or for faster processes . it is left for future work to construct a general quantum linear response theory , based on the eqs . of motion [ eqmtz ] , which would yield both super and sub - poissonian statistics . finally , also the investigation of the time dependence of @xmath0 is timely .
|
we investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave laser field .
the spectral diffusion is modeled based on a stochastic approach , in the spirit of the anderson - kubo line shape theory . using a generating function formalism
we solve the generalized optical bloch equations , and obtain an exact analytical formula for the line shape and mandel s q parameter .
the line shape exhibits well known behaviors , including motional narrowing when the stochastic modulation is fast , and power broadening .
the mandel parameter , describing the line shape fluctuations , exhibits a transition from a quantum sub - poissonian behavior in the fast modulation limit , to a classical super - poissonian behavior found in the slow modulation limit .
our result is applicable for weak and strong laser field , namely for arbitrary rabi frequency .
we show how to choose the rabi frequency in such a way that the quantum sub - poissonian nature of the emission process becomes strongest . a lower bound on @xmath0 is found , and simple limiting behaviors are investigated .
a non - trivial behavior is obtained in the intermediate modulation limit , when the time scales for spectral diffusion and the life time of the excited state , become similar .
a comparison is made between our results , and previous ones derived based on the semi - classical generalized wiener khintchine formula .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
accurate measurements of flow velocities are essential for exploring and understanding fluid dynamics . such measurements raise important technical problems whenever one needs them to be nonintrusive . indeed , many applications where direct access to the fluid is impossible perclude the use of local probes such as miniature piezoelectric pressure probes , hot films or hot wires . in other cases , the introduction of a probe inside the fluid may perturb the flow or even the fluid structure itself , and may lead to incorrect interpretations of the measurements . thus , in general , much insight can be gained from some remote , noninvasive sensing of the flow field . in the following , we focus on the case of complex fluids under simple shear . a `` complex '' fluid may be characterized by the existence of a `` mesoscopic '' length located somewhere between the size of individual molecules and the size of the sample @xcite . for instance , this intermediate length scale is the particle diameter in a colloidal suspension , the diameter of an oil droplet in an emulsion , or the radius of gyration of polymer coils . the existence of such a supramolecular organization ( or `` microstructure '' ) can lead to very complicated behaviors under flow and to inhomogeneous velocity profiles under simple shear @xcite . since the microstructure of a complex fluid is very sensitive to local deformations , it is quite obvious that a nonintrusive technique is required in order to measure velocity profiles . the most popular nonintrusive techniques , namely particle imaging velocimetry ( piv ) or laser doppler velocimetry ( ldv ) , are based on the interaction between light and seeding particles following the flow @xcite . however , many complex fluids , such as emulsions , slurries , or pastes , may not be transparent enough to allow the use of piv or ldv . nuclear magnetic resonance ( nmr ) offers the possibility to image opaque media @xcite but requires the use of powerful magnets and remains expensive and tricky to set up . on the other hand , ultrasound appears as an efficient , cost - effective tool to measure velocity profiles in a large range of fluids . in this paper , we adapt a speckle - based ultrasonic velocimetry technique @xcite to complex fluid flows . although based on the classical principle of backscattering by particles suspended in the flow , our technique brings an original contribution to both the fields of acoustical flow measurements and of complex fluid studies . indeed , by using high - frequency pulses ( frequencies larger than 20 mhz ) , we show that we are able to measure velocity profiles in complex fluids sheared between two plates separated by 1 mm with a spatial resolution of about 40 @xmath0 m . depending on the required accuracy , a full velocity profile can be obtained typically in 0.02 s to 2 s , which makes it possible to resolve transient regimes or temporal fluctuations of the flow . the paper is organized as follows . we first explain in more details why local velocity measurements are crucial to the understanding of complex fluids . examples are given that show that inhomogeneous flows may occur even in simple shear geometries and at small imposed stresses . the third section is devoted to a brief review of existing ultrasonic techniques for measuring flow velocities . we then present the electronic setup and the data analysis used for high - frequency ultrasonic speckle velocimetry ( usv ) . section [ s.calib ] deals with the calibration step necessary to obtain quantitative estimates of the velocity . in section [ s.lamellar ] , high - frequency usv is applied to a particular complex fluid : an aqueous solution of surfactant . the ultrasonic data reveal the existence of inhomogeneous flows , with both wall slip and shear bands , as well as complex spatio - temporal behaviors during transient regimes . finally , in light of the present results , the technique is compared to other nonintrusive tools used in the field of complex fluid flows . we emphasize on the promising spatio - temporal resolution of usv and discuss its possible future applications . it is well - known that shear flows of simple fluids are laminar at low velocities and may become unstable above a critical reynolds number due to hydrodynamic instabilities @xcite . a classical shear geometry is the _ couette geometry _ in which the fluid is sheared between two concentric cylinders as sketched on fig . [ f.couette ] . usually the inner cylinder of radius @xmath1 is rotating while the outer cylinder of radius @xmath2 remains fixed . the two cylinders are thus called `` rotor '' and `` stator '' respectively . the distance @xmath3 between the two walls is called the `` gap '' of the couette cell . this couette geometry will be used throughout the paper as well as the cylindrical coordinates @xmath4 . we will denote @xmath5 the distance from the inner cylinder and @xmath6 the velocity vector . a couette device can work in two different modes : either a torque @xmath7 is imposed on the rotor and the rotor angular velocity @xmath8 is measured , or @xmath8 is imposed and the torque @xmath7 is measured . for a newtonian fluid at small velocities , the flow is stationary and purely orthoradial @xmath9 . the tangential velocity @xmath10 decreases from the rotor velocity @xmath11 down to zero on the outer wall . in the small gap approximation _ i.e. _ when @xmath12 , the velocity profile is linear as shown in fig . [ f.exs](a ) . the shear rate @xmath13 ( @xmath14-component of the rate - of - strain tensor ) and the shear stress @xmath15 ( tangential force per unit surface _ i.e. _ @xmath14-component of the stress tensor ) are then almost uniform across the gap and are simply given by : @xmath16 where @xmath17 is the height of the couette cell . the viscosity @xmath18 of the sample can thus be computed from the rheological data @xmath13 and @xmath15 by @xmath19 , as long as the flow remains laminar and stationary . in complex fluids , the picture of the couette flow may be radically different . for instance , when a complex fluid is confined between two plates and depending on the roughness properties of the plates , the fluid velocity close to the walls may strongly differ from that of the walls ( see fig . [ f.exs](b ) ) : the fluid _ slips_. slippage is usually explained by the existence of two thin lubricating layers in which the fluid structure is very different from that in the bulk . for instance , in a colloidal suspension , electrostatic or steric effects next to the walls may induce a depletion of the particles on a small distance , leaving a much less viscous fluid film at the walls and leading to this apparent wall slip @xcite . in concentrated oil - in - water emulsions , wall slip is due to the presence of very thin water layers close to the walls . even if the bulk fluid remains newtonian , the velocity profile between the two plates may be quite different from that expected in a uniformly sheared fluid that does not slip ( see fig . [ f.exs](a ) ) . wall slip in complex fluids raises important industrial problems and is often difficult to detect and to assess quantitatively . in particular , computing the viscosity from eqs . ( [ e.gammarheo ] ) and ( [ e.sigmarheo ] ) can be very misleading since the effective shear rate in the bulk is sometimes orders of magnitude smaller than @xmath20 @xcite . moreover , even in the absence of wall slip and even for newtonian fluids , above a critical rotation speed of the rotor , the laminar flow becomes unstable and vortices develop due to the centrifugal force ( _ taylor - couette instability _ ) . upon further increasing the rotation speed , this three - dimensional , inhomogeneous but still stationary flow turns into a time - dependent turbulent flow @xcite . the procedure based on eqs . ( [ e.gammarheo ] ) and ( [ e.sigmarheo ] ) for measuring @xmath13 and @xmath15 and computing the viscosity then completely fails because the flow is no longer laminar and stationary . about fifteen years ago , complex fluids such as entangled polymer solutions were shown to present inhomogeneous flows even when sheared at rotor velocities far below the onset of the taylor - couette instability . these unstable flows result from purely _ elastic instabilities _ @xcite . although due to the elasticity of the fluid rather than visco - inertial effects , the phenomenology of such instabilities is very similar to classical hydrodynamic instabilities but occur at very low reynolds numbers . beside those inertial and elastic instabilities , more subtle effects may come into play that are due to the very nature of complex fluids . indeed , application of shear to a complex fluid may completely change its microstructure leading to a new _ shear - induced structure_. such a phenomenon may be called a _ shear - induced transition _ or a _ material instability _ in the sense that it results from a strong coupling between the shear flow and the microstructure of the fluid @xcite . a material instability usually shows up on rheological data as a sudden jump in the viscosity when shear is increased . for instance , if the shear - induced structure is much less viscous than the complex fluid at low shear , then a large drop of the visocity may be observed and the transition is called a shear - thinning transition . during such a transition , the fluid is assumed to phase - separate into two states : a shear - induced state that flows at a given shear rate and coexists with the old structure , thus giving rise to two shear bands @xcite . the corresponding _ shear - banded _ velocity profile is sketched in fig . [ f.exs](c ) . in order to investigate those transitions , most previous studies have focused on global measurements such as the viscosity of the sample as a function of time @xcite . indeed , the time series @xmath21 sometimes display complex temporal fluctuations on time scales characteristic of the microstructure rearrangements . however , such global measurements do not provide any information on the spatial structure of the flow @xcite . thus , other studies have focused on the local characterization of the fluid structure using birefringence neutron , x - ray or light scattering and showed the coexistence of bands of different microstructures in the vicinity of shear - induced transitions @xcite . as far as the flow field is concerned , rather few local velocity measurements are reported in the complex fluid literature @xcite . the presence of inhomogeneous flows in some surfactant mixtures ( `` wormlike micelles '' ) was first unveiled using nmr @xcite . however , the existence of shear bands in the velocity profiles was firmly ascertained only recently using dynamic light scattering ( dls ) in heterodyne mode , a technique close to ldv @xcite . from the various examples cited above , it is quite clear that local measurements are essential , for instance to precisely follow the shear bands in both space and time . in general , a velocity profile with about 20 points accross a 1 mm gap , _ i.e. _ with a resolution of 50 @xmath0 m , will carry enough spatial information on the flow to allow quantitative conclusions on possible inhomogeneous flow profiles . contrary to electromagnetic waves , the _ phase _ of acoustic waves is easily accessible using piezoelectric transducers . moreover , ultrasound may propagate deeply into optically opaque media . these two properties have led to the spectacular development of ultrasonic imaging techniques since the early 1950s particularly in the _ biomedical _ domain . when an ultrasonic wave travels through biological tissues , it gets scattered by density and/or compressibility inhomogeneities @xcite . if the scatterer moves in a flow , like for instance red blood cells inside blood vessels , its motion induces a doppler shift in the wave frequency . early systems for measuring blood velocities were based on the estimation of this doppler frequency shift using a monochromatic ultrasonic wave @xcite . unfortunately , even if these `` true doppler '' systems helped to detect the occlusion of blood vessels , they did not offer any spatial resolution . indeed , if one uses a continuous wave , one gets a very good resolution on the frequency shift ( and thus on the velocity ) but one also loses any temporal information on the echo arrival time ( and thus on the scatterer position , as explained below ) . continuous doppler systems may thus be very efficient in the case of a single scatterer suspended in a flow whose position is measured independently @xcite . in order to discriminate between different scatterers in space and to measure velocity profiles , a solution is to use short acoustic _ pulses_. indeed , if an ultrasonic pulse is sent by a transducer through a scattering medium , a series of backscattered echoes can be recorded on the same transducer . the precise nature of the backscattered ( bs ) signal will be discussed at length in sec . [ s.usv ] . at this point , it is only important to note that the arrival times @xmath22 of the echoes are directly linked to the position @xmath23 of the scatterer along the acoustic beam by @xmath24 , where @xmath25 is the effective sound speed and the factor 2 accounts for the round trip from the transducer to the @xmath26 inhomogeneity and back to the transducer . the idea is then to follow the scatterer motion through the evolution of the bs signals : between successive pulses , echoes will move along with the flow and their positions may be tracked in time . although the doppler effect is not used in pulsed wave systems , these are still mistermed `` pulsed doppler systems '' in the literature and the technique is referred to as `` ultrasonic doppler velocimetry '' ( udv ) or `` ultrasonic velocity profiler '' ( uvp ) @xcite . this confusion may be due to the fact that , owing to the constraint of real - time display in commercial devices , bs signals are sampled only at a few given depths and that these sampled signals are analyzed in the frequency domain @xcite . udv has been successfully applied to various classical problems in hydrodynamics such as the taylor - couette instability @xcite , cylinder wakes @xcite , or magnetic fluid flows @xcite . the fluid has to be seeded with small particles that scatter ultrasound . such acoustic `` contrast agents '' are assumed to follow the flow as lagrangian tracers . the uvp has also been used fairly recently for in - line rheological studies of concentrated suspensions in pipe flows @xcite . if one drops the constraint of real - time display , the whole bs signals may be stored for post - processing . direct tracking of the bs signals in the time domain is the subject of recent technical developments such as ultrasonic speckle velocimetry which will be addressed in the next section . when the moving medium contains a lot of scatterers per unit volume , the bs signal results from the interferences of all the backscattered waves and appears as a complex , high - frequency signal called _ ultrasonic speckle _ ( see fig . [ f.signal ] for a typical speckle signal and sec . [ s.analysis ] for more details , in particular about the possibility of multiple scattering ) . the development of fast digitizers with large memories has allowed the full recording of backscattered bs signals for post - processing . various time - domain algorithms for tracking the motion of ultrasonic echoes have been proposed @xcite . so far , such _ speckle tracking _ techniques have mainly been applied to the measurement of tissue motion or blood flow @xcite . they have also been adapted to the uvp system and tested on pipe flows to provide high time resolution ( of about 0.5 ms per profile ) @xcite . recently , one of us introduced a combination of a 2d echographic system and speckle tracking that we called 2d ultrasonic speckle velocimetry ( usv ) @xcite . this system allowed us to image strong vortical flows in newtonian fluids in two dimensions @xcite . here we propose to adapt such a technique to complex fluids confined in a couette cell of gap @xmath27 mm . as explained in section [ ss.inhom ] , one needs to measure velocity profiles with a spatial resolution of typically 50 @xmath0 m . however , frequencies commonly used in commercial ultrasonic velocimeters are in the range @xmath2810 mhz , and with a typical sound speed @xmath29 m.s@xmath30 , this yields a resolution of about @xmath311.5 mm . thus , conventional ultrasound does not permit very fine measurements and one has to turn to utrasound with frequencies larger than 20 mhz . again , high frequencies have first attracted the biomedical community . high - resolution ultrasonic images have been recently obtained in dermatology , ophtalmology and stomatology @xcite or for measuring blood flow in the microcirculation @xcite . to our knowledge , no such high - frequency measurements ( above 20 mhz ) have been reported so far in classical hydrodynamics nor in the field of complex fluids . note that the resolution can not be increased indefinitely by increasing the frequency because _ attenuation _ of sound waves sharply increases with frequency ( as @xmath32 in water ) . attenuation is due to the combination of thermo - viscous absorption and scattering . the signal - to - noise ratio rapidly deteriorates as @xmath33 increases . we found that for our application to complex fluids , @xmath34 mhz realizes a good compromise between resolution and attenuation over propagation distances of a couple of centimeters . finally , since transducer arrays are not available at such a high frequency , the technique will have to be restricted to one - dimensional measurements . figure [ f.system ] presents our usv electronic system . focused ultrasonic pulses are generated by a pvdf piezo - polymer immersion transducer of central frequency @xmath34 mhz ( panametrics pi 50 - 2 ) . the focal distance is 11.6 mm and the active element diameter is 6.3 mm . the transducer bandwidth is 11 mhz at -3 db . the axial and lateral resolutions given by the manufacturer at -3 db are 30 @xmath0 m and 65 @xmath0 m respectively and the depth of field is about 1 mm . the transducer is controlled by a pulser - receiver unit ( panametrics 5900pr ) . the pulser generates 220 v pulses with a rise time of about 1 ns . the pulse repetition frequency ( prf ) is tunable from 0 to 20 khz . the receiver is equipped with a 200 mhz broadband amplifier of maximum voltage gain 54 db as well as a set of selectable high - pass and low - pass filters . bs signals are sampled at @xmath35 mhz , stored on a high - speed pci digitizer with 8 mb on - board memory ( acqiris dp235 ) , and later transferred to the host computer for post - processing . the shear flow is generated in a plexiglas couette cell with @xmath36 mm , @xmath37 mm , and @xmath38 mm . the rotation of the inner cylinder is controlled by a standard rheometer ( ta instruments ar1000 ) . the whole cell is surrounded by water whose temperature is kept constant to within @xmath39c . the thickness of the stator is 2 mm everywhere except for a small rectangular window where the minimal thickness is 0.5 mm in order to avoid additional attenuation due to the plexiglas . ultrasonic pulses are incident on the stator with a given angle @xmath40 relative to the normal to the window in the stator as sketched in fig . [ f.setup ] . they travel through plexiglas and enter the gap with an angle @xmath41 that is given by the law of refraction . since the sound speed of the working fluid may differ from that of water , and since the precise value of @xmath41 depends on the exact arrangement of the acoustic beam relative the stator , @xmath41 will be taken as an unknown until a careful calibration procedure is completed ( see section [ s.calib ] ) . once inside the fluid , ultrasonic pulses get scattered by inhomogeneities that can be either naturally present ( oil droplets in an emulsion for instance ) or artificially introduced to enhance the acoustic contrast . the total round trip for a pulse travelling from the transducer to the rotor and back to the transducer lasts about 15 @xmath0s . the position of the transducer is tuned so that the 1 mm gap lies into the focal spot in order to optimize the signal - to - noise ratio . typical recorded bs signals are 1000 point long , which corresponds to a transit time of 2 @xmath0s or equivalently to a depth of roughly 1.5 mm inside the medium . allowing for a fraction of a millimeter before the stator and after the rotor helps to locate the two walls . note that spurious reflections on water plexiglas or plexiglas fluid interfaces are minimized by carefully choosing the angle @xmath42 . in any case , even in the absence of interfaces , such an angle is necessary to get a non - zero projection of the velocity vector along the acoustic axis as explained below . figure [ f.signal ] shows a typical signal backscattered by a dilute suspension of polystyrene spheres and normalized by the maximum amplitude . the noise level is about 10 mv in all our experiments and typical signal amplitudes are in the range 50200 mv corresponding to a signal - to - noise ratio of 1030 db . under the assumption of _ single scattering _ , the speckle signal received between times @xmath43 and @xmath44 can be interpreted as interferences coming from scatterers located between @xmath45 and @xmath46 , where @xmath47 , @xmath48 , and @xmath45 is the distance from the transducer along the acoustic beam . the approximation that no multiple scattering takes place is thus essential in order to use the echographic conversion rule @xmath47 . multiple scattering is avoided by carefully controlling the amount and the properties of the scatterers ( see sec . [ s.calib ] ) . when the fluid is submitted to a shear flow , the interference pattern changes as the scatterers move along . figure [ f.signal_zoom ] shows two signals corresponding to two pulses separated by @xmath49 ms and plotted over 4 acoustic periods . a global shift to the right of @xmath50 ns is clearly visible . with @xmath29 m.s@xmath30 , this corresponds to @xmath51 m and to a velocity @xmath52 mm.s@xmath30 projected along the acoustic axis . in order to get an accurate estimate of the time - shift @xmath53 , we use the simple cross - correlation algorithm described below . two consecutive signals @xmath54 and @xmath55 recorded after pulses sent at times @xmath56 and @xmath57 are cross - correlated over small time windows of width @xmath58s , where @xmath59 is the acoustic period . more precisely , the following cross - correlation coefficient is computed according to : @xmath60 where the @xmath61 time window is centered around @xmath62 and @xmath63s is a reference time that corresponds to the beginning of the gap . @xmath64 depends on experimental parameters such as the working temperature and on the exact arrangement of the transducer relative to the couette cell ( see sec . [ s.calib ] ) . the times @xmath22 correspond to the centers of the various time windows over which the signals are split . they will be later converted into positions @xmath23 at which the velocity is measured using @xmath65 . such windows correspond to slices of width @xmath66 m along the acoustic axis separated by @xmath67 m . as shown in fig . [ f.correl ] , the correlation function @xmath68 is then interpolated around its main maximum by a parabola and the value @xmath53 of @xmath69 that maximizes this parabola is extracted : @xmath70 where @xmath71 denotes the parabolic interpolation of @xmath72 . note that @xmath53 depends both on the position @xmath22 of the time window ( fast `` ultrasonic time '' ) and on the time @xmath56 for which the cross - correlation is performed ( slow `` pulse time '' ) . thus @xmath53 will be noted as a function of the two variables @xmath22 and @xmath56 below . finally , it is straightforward to convert time - shifts @xmath53 into velocities using : @xmath73 where @xmath74 is the position of the center of the window along the acoustic axis . the origin @xmath75 is taken at the stator fluid interface . equation ( [ e.vy ] ) thus yields the projection @xmath76 of the velocity vector along the acoustic axis at the discrete positions @xmath23 and at time @xmath56 . the procedure based on the interpolation of @xmath68 allows us to measure time - shifts as small as @xmath77 ns that correspond to displacements @xmath78 m . since the prf @xmath79 can be made as small as desired , infinitesimal velocities could in principle be measured . however , at very small shear rates , decorrelation of the bs signals may occur due to brownian motion of the scatterers or to low - frequency mechanical vibrations in the experimental setup . consequently , we will avoid using prfs smaller than 10 hz and we estimate the minimum measurable velocity at about 1 @xmath0m.s@xmath30 . on the other hand , the maximum measurable velocity is reached when @xmath80 m . indeed , displacements greater than the acoustic wavelength can not be measured using the cross - correlation algorithm described above unless phase unwrapping and continuity conditions are implemented , which will not be considered here @xcite . with the highest achievable prf @xmath81 khz , a 40 @xmath0 m maximum displacement means that velocities above 0.8 m.s@xmath30 may not be accessed using usv . note however that this limiting range of [ 1 @xmath0m.s@xmath30 0.8 m.s@xmath30 ] applies to the velocity @xmath76 projected along the acoustic axis . this range may be extended by varying the incidence angle @xmath41 ( see sec . [ s.conversion ] and eqs . ( [ e.v])([e.x ] ) below ) . finally , eq . ( [ e.vy ] ) shows that a velocity profile could in principle be obtained by cross - correlating only two successive bs signals _ i.e. _ within a time interval @xmath83 that can be made very small depending of the prf . in practice , as seen on fig . [ f.signal ] , the speckle amplitude is never uniform . locally , destructive interferences or the absence of scatterers may lead to signal levels too small to be analyzed ( see the signal of fig . [ f.signal ] around @xmath84 or @xmath85s for instance ) . time windows where the signal amplitude does not reach a given threshold ( typically 20 % of the maximum amplitude ) are thus left out of the analysis . to recover a full velocity profile , some averaging is then performed over several sucessive cross - correlations : @xmath86 figure [ f.data ] shows twenty successive bs signals recorded in a sheared newtonian suspension with @xmath87 khz . the ultrasonic pulses leave the stator and enter the gap at @xmath63s and a small fixed echo corresponding to the rotor position may be seen at @xmath88s ( as well as a stronger echo at about 15.2 @xmath0s that corresponds to a multiple reflection from the plexiglas window in the stator ) . the slopes of the traces left by the echoes in this two - dimensional @xmath89 diagram are inversely proportional to the local velocities according to eq . ( [ e.vy ] ) . thus , the signature of shear is rather clear : velocities increase from the stator , where the scatterers remain almost fixed , to the rotor . moreover , the presence of moving echoes for times @xmath90 is not surprising : these echoes simply correspond to scattering of the wave reflected on the rotor . one can see that the whole range of times from @xmath64 to @xmath91 gets covered by some significant speckle signal in about 20 ms in that case . if better accuracy is required , more averaging can be performed to improve the statistical convergence of the method ( see fig . [ f.converg ] below ) . depending on the trade - off between accuracy and temporal resolution , we found that usv allows us to obtain a full velocity profile every 0.022 s. in order to test and calibrate the experiment , much effort was devoted to developing `` contrast agents '' that feature good scattering properties for high - frequency usv . indeed , too weak a scatterer yields a signal level too low to work with , whereas too strong a scatterer leads to multiple scattering . when the scatterer is small compared to the acoustic wavelength ( rayleigh approximation ) , three parameters control the amount of scattering : the scatterer diameter , its compressibility and its density @xcite . moreover , if one wants the contrast agents to remain lagrangian tracers of the flow , their density has to be matched to that of the fluid . we found that homemade polystyrene spheres of diameter @xmath9210 @xmath0 m were large and hard enough to scatter 36 mhz pulses efficiently but soft enough to prevent multiple scattering when diluted at a weight concentration of 1 % in water . such polystyrene spheres were obtained by polymerization following refs . a 1 % wt . solution of benzoyl peroxide in divynylbenzene is first prepared . then , a polydisperse emulsion is obtained by stirring 40 g of the previous mixture in 59 g of water and 1 g of sodium dodecyl sulfate at room temperature . finally , the emulsion is polymerized at 90@xmath93c for two hours . the prepared polystyrene spheres were washed with water three times before drying . the 1 % wt . suspension of such polystyrene spheres is newtonian and has the same viscosity as water . in order to use eqs . ( [ e.vy ] ) and ( [ e.yk ] ) , the sound speed @xmath25 of the working fluid has to be measured . we used a classical _ transmission _ setup consisting of two transducers facing each other . one of them , the emitter , remains fixed while the other works as a receiver and can be moved by a computer - controlled actuator . pulses are recorded for various displacements @xmath94 of the receiver and averaged over 100 sweeps . the time - shifts @xmath53 between the arrival times of the pulses are measured and the sound speed @xmath25 is given by the slope of @xmath94 vs. @xmath53 ( see fig . [ f.soundspeed ] ) . due to technical constraints , the measurements of @xmath25 were performed with 25 mhz transducers . temperature is controlled to within 0.1@xmath93c by a water circulation around the sample . figure [ f.soundspeed ] shows that @xmath95 m.s@xmath30 leads to a very good approximation of the data obtained in the suspension of polystyrene spheres at our working temperature of 32@xmath93c . this value of @xmath25 is in good agreement with the measurements reported in the literature for the sound speed of water vs. temperature @xcite . in the following , we will assume that this value does not vary significantly with frequency between 25 and 36 mhz and all the data will be taken at the same temperature of 32@xmath93c . finally , note that the measurement of @xmath25 using the transmission of ultrasonic pulses between two transducers also allows us to check that no significant multiple scattering takes place in our fluids by looking at the transmitted waveform : the received signal remains always as short as the emitted pulse and no long - lasting echoes typical of multiply scattered waves were ever recorded during the measurements of @xmath25 . another important parameter is @xmath64 that gives the position of the stator fluid interface and strongly depends on the experimental conditions . figure [ f.brut ] presents the results of our speckle tracking algorithm averaged over 50 series of 20 pulses like that shown in fig . [ f.data ] . the rotor velocity @xmath96 was varied between 2.9 and 47.0 mm.s@xmath30 . the linear behavior of the time - shift @xmath53 with the position @xmath43 of the correlation window is a direct signature of the uniform shear flow inside the newtonian suspension . indeed , all the velocities considered here are far below the onset of the taylor - couette instability which occurs when @xmath97 ( where @xmath98 [email protected]@xmath30 is the kinematic viscosity of our calibration fluid ) @xcite _ i.e. _ when @xmath100 mm.s@xmath30 . for our dilute suspension , no wall slip is expected and @xmath53 should go to zero at the stator : @xmath101 . by linearly fitting @xmath102 vs. @xmath43 for various values of @xmath96 , different estimates of @xmath64 are obtained that yield an average value @xmath103s . the uncertainty on @xmath64 is due to the presence of the stator which leads to a small fixed echo on the signals and to small edge effects around @xmath64 ( see inset of fig . [ f.brut ] ) . it corresponds to an uncertainty of about @xmath104 m on the position of the stator fluid interface . inserting the values of @xmath25 and @xmath64 found above in eqs . ( [ e.vy ] ) and ( [ e.yk ] ) yields the velocity profile @xmath106 . thus , so far , we obtained measurements of the velocity vector projected along the acoustic axis . in order to get some physically more relevant information , we will assume from now on that the radial velocity is zero ( or at least negligible when compared to the orthoradial velocity ) . this is of course the case for our newtonian suspension below the onset of the taylor - couette instability . in complex fluids , inhomogeneous flows with wall slip or shear banding are purely orthoradial ( see sec . [ ss.inhom ] ) so that the above approximation remains valid as long as hydrodynamic or elastic instabilities do not occur . note that in the geometry discussed here , usv measurements are not affected by a non - zero vertical component of the velocity field . assuming that @xmath107 and using standard trigonometric relations ( see fig . [ f.convert ] ) , one gets : @xmath108 where @xmath109 in the small gap approximation , eqs . ( [ e.v ] ) and ( [ e.x ] ) reduce to @xmath110 and @xmath111 . thus the last parameter to calibrate is the angle @xmath41 . this is done by looking for a value of @xmath41 for which @xmath105 coincides with the velocity profile expected for a newtonian fluid : @xmath112\simeq{v_{\scriptscriptstyle 0}}\left(1-\frac{x}{e}\right)\ , , \label{e.vnewt}\ ] ] where the last term results from the small gap approximation . figure [ f.calib ] shows that @xmath113 leads to very good results for all the rotor velocities @xmath96 considered here . normalized data @xmath114 vs. @xmath115 show that the newtonian velocity profiles @xmath116 are recovered for all our data with the set of parameters @xmath25 , @xmath64 , and @xmath41 inferred from the above calibration ( see inset of fig . [ f.calib ] ) . this calibration method is very sensitive to small variations of @xmath41 and allows its estimation to within @xmath117 . note that @xmath41 depends on both the temperature and the nature of the fluid under study through @xmath25 . as long as the temperature remains the same , the new value @xmath118 of the incidence angle in a fluid whose sound speed @xmath119 differs from that of the calibration fluid is simply given by snell s law : @xmath120 where @xmath25 and @xmath41 are the parameters measured for the calibration fluid . finally , fig . [ f.converg ] shows the value of the @xmath76 measured at two different positions along the acoustic beam as more and more averaging is performed . it is clear that convergence is reached when @xmath121 pulses are taken into account . the standard deviation of the measurements decreases from about 6 % for @xmath122 to less than 3 % for @xmath123 . let us now turn to an application of high - frequency usv to a complex fluid . we chose to test usv in a lyotropic lamellar phase composed of sds ( 6.5 % wt . ) , octanol ( 7.8 % wt . ) and brine ( 20 g.l@xmath30 nacl ) . this surfactant mixture is known to display a series of structural shear - induced transitions @xcite . at equilibrium , the system is composed of surfactant bilayers stacked with a smectic distance of about 15 nm . under shear flow , this lamellar phase was shown to form a closely packed assembly of multilamellar vesicles @xcite . the diameter of these vesicles , commonly called `` onions , '' is typically a few microns . at low shear velocities , the diffraction pattern observed with light scattering is an isotropic ring indicating the absence of any positional order . however , when the velocity @xmath96 is increased , bragg peaks appear in the diffraction pattern showing the apparition of long - range order in the vesicle positions . as sketched in fig . [ f.transit ] , this disorder - to - order transition corresponds to a sharp drop in the sample viscosity by a factor of about 3 @xcite . such a shear - thinning transition is also observed in colloidal suspensions and is called a `` layering '' transition @xcite . as explained in sec . [ ss.inhom ] , inhomogeneous velocity profiles are expected in the vicinity of this shear - induced transition where the two states ( ordered and disordered states ) may coexist in the gap of a couette cell . as for its ultrasonic behavior , the `` onion '' assembly does not scatter 36 mhz pulses . thus , the lamellar phase was seeded with the polystyrene spheres used in sec . [ s.calib ] in order to get some speckle signal from the fluid . we checked that the `` layering '' transition is not significantly modified by the addition of such contrast agents . figure [ f.photos ] presents two pictures of the seeded lamellar phase as seen under crossed polarizers and after shearing : the granular texture characteristic of disordered multilamellar vesicles does not seem perturbed by the addition of polystyrene spheres at 1 % wt . as shown on fig . [ f.soundspeed ] , the sound speed of the seeded lamellar phase was measured to be @xmath124 m.s@xmath30 . since the experiments and the calibration are performed at the same temperature and since the sound speeds of both fluids are the same ( within less than 1 % ) , we used @xmath95 m.s@xmath30 , @xmath125s and @xmath113 to obtain the velocity profiles shown in figs . [ f.lam1 ] and [ f.lam2 ] . the two main features of the velocity profiles are as follows : [ [ wall - slip . ] ] wall slip . + + + + + + + + + + as clearly shown by the insets of figs . [ f.lam1 ] and [ f.lam2 ] , the velocity close to the walls is never equal to that of the walls . on the contrary , @xmath126 may differ from @xmath96 by more than 30 % . one can also notice that slip velocities are comparable at the rotor and at the stator for the smallest rotor velocity but become highly dissymmetric when @xmath127 mm.s@xmath30 . a detailed study of wall slip in this system is left for future work . [ [ shear - banding . ] ] shear banding . + + + + + + + + + + + + + + figure [ f.lam1 ] also unveils the presence of shear bands when @xmath128 mm.s@xmath30 . a highly sheared band nucleates close to the rotor and progressively fills the gap as @xmath96 is increased ( see fig . [ f.lam2 ] for the highest velocities ) . the shear rate in the high shear band is about three times larger than that in the low shear band . this ratio remains roughly constant as long as two bands coexist . for @xmath129 mm.s@xmath30 , the highly sheared region has invaded the whole gap . when @xmath130 mm.s@xmath30 , the flow is laminar again and looks similar to that observed before the `` layering '' transition at @xmath131 mm.s@xmath30 . simultaneously to the present ultrasonic measurements , dls data obtained in our group have confirmed the existence of shear - banded flows and strong wall slip effects in the same lamellar phase without seeding polystyrene particles @xcite . this last observation shows _ a posteriori _ that our `` contrast agents '' do not perturb the flow and can be treated as simple lagrangian tracers . both usv and dls data are in good qualitative agreement with classical theoretical pictures of shear - banded flows @xcite . our purpose here is only to show the relevance of high - frequency usv in the field of sheared complex fluids and a more quantitative analysis of shear banding in light of dls results can be found in ref . @xcite . finally let us show how usv may be used to follow the dynamics of a complex fluid under shear . until now , we have considered time - averaged velocity profiles . however , during the `` layering '' transition , noticeable fluctuations of the flow field occur and result in large error bars for @xmath132 or 29.4 mm.s@xmath30 for instance . actually , some very rich dynamics were reported in this sheared lamellar phase when the _ torque _ applied on the rotor is imposed instead of the _ velocity _ @xcite . in that case , the viscosity @xmath21 displays large , complex fluctuations on long time scales of the order of 100 s. similar ( although faster ) temporal behaviors were also observed in other surfactant systems @xcite . the origin of these fluctuations and their precise statistical properties remain unclear . one possibility is that the flow presents shear bands that may couple to each other and lead to the observed behavior of the viscosity . thus the dynamics could be not only temporal as observed on @xmath21 but also spatial as suggested in ref . @xcite . figure [ f.lam3 ] shows a transient regime recorded after the torque applied to the rotor was suddenly increased from 83 to 87 mn at time @xmath133 . the response of the rotor velocity @xmath134 is shown in fig . [ f.lam3](a ) @xcite . a maximum of about 38 mm.s@xmath30 is reached at @xmath135 s before @xmath134 slowly relaxes with small fluctuations . velocity profiles recorded during that transient show that : ( i ) during the initial increase of @xmath134 , a shear - banded flow is generated with _ three _ bands : two highly sheared bands at both the rotor and the stator and a low shear region in between ( see fig . [ f.lam3](b)(d ) ) ; ( ii ) the band located at the stator disappears when @xmath134 decreases , leaving only _ two _ bands similar to figs . [ f.lam1 ] and [ f.lam2 ] ( see fig . [ f.lam3](e)(f ) ) ; ( iii ) states where the highly sheared band almost disappears can be observed ( see fig . [ f.lam3](g ) ) even though the rotor velocity @xmath136 mm.s@xmath30 indicates that a band should extend over almost half the gap if the average picture at imposed @xmath96 remained valid when the torque is imposed ( see the @xmath137 symbols corresponding to @xmath138 mm.s@xmath30 in fig . [ f.lam1 ] ) . the above results show that high - frequency usv is a powerful tool for measuring velocity profiles in sheared complex fluids . we have successfully tested the technique on a lyotropic lamellar phase that undergoes a shear - induced `` layering '' transition . when the rotor velocity , _ i.e. _ the global shear rate , is imposed , shear banding is observed : time - averaged velocity profiles display a highly sheared region that nucleates from the rotor and progressively invades the gap as @xmath96 is increased . wall slip can also be very large especially during the transition , which makes global measurements such as the sample viscosity tricky to analyze . moreover , usv allowed us to record velocity profiles in 1 s typically . this acquisition time is short enough ( compared to the intrinsic time scales of our complex fluid ) to follow the velocity profiles in time . we showed the existence of more complex , three - band flows during a transient at imposed torque _ i.e. _ imposed shear stress . this raises the question of the structure of the flow during the spontaneous oscillations of the viscosity observed in the same lamellar phase @xcite . usv may thus provide a `` time - resolved '' tool for investigating spatially the dynamical behaviors of @xmath21 . in this paper , we have shown that we were able to measure velocity profiles with a spatial resolution of about @xmath139 m and every 0.022 s. high - frequency usv compares very well with other local techniques for measuring flow fields . ldv , or equivalently dls , provides in principle a slightly better spatial resolution of 1050 @xmath0 m but only yields a pointlike measurement . in order to obtain a full velocity profile with ldv or dls , one has to mechanically scan the whole gap of the cell @xcite . this usually takes about 1 min and may raise major interpretation problems when the flow evolves on time scales shorter than 1 min . nmr appears as a very promising technique for complex fluids @xcite since no seeding is required and two - dimensional images of the flow are obtained . however , acquisition times are rather long too ( from about 20 s to more than an hour depending on the number of points in the image ) , making it rather difficult to access dynamical local information . the main drawback of usv is that it needs a good control of the scattering properties of the fluid and that it may require that the flow be seeded with contrast agents . unless their mechanical properties is strictly controlled , most concentrated suspensions or emulsions will lead to multiple scattering of high - frequency pulses . thus , further application of usv to other complex fluids will involve a careful study of their acoustic properties . however , we believe that usv is likely to extend the range of complex fluids in which local measurements may be performed , in particular to optically opaque fluids . the biggest advantage of usv lies undoubtedly in its ability to perform nonintrusive velocity measurements fast enough to follow the dynamics of complex fluids under low shear . in principle , in an ideal scattering environment , the temporal resolution of usv could even get as short as 1 ms per measurement . future research directions on usv are as follows : ( i ) more measurements on seeded lamellar phases will be performed and compared to dls data ; more precisely , we will focus on the spatio - temporal flow dynamics in regimes where sustained oscillations of the viscosity are observed ; ( ii ) we will investigate shear flows of other fluids such as emulsions , gels , or granular pastes ; ( iii ) we also intend to apply high - frequency usv to the case of very small deformations in order to perform some `` local linear rheology '' and understand how a complex fluid starts to flow in the transition from linear to nonlinear rheology . the authors wish to thank the `` cellule instrumentation '' at crpp for designing and building the mechanical parts of the experimental setup . we are very grateful to d. roux , j .- b . salmon , and r. wunenburger for fruitful discussions . e. cappelaere , j .- f berret , j .- p . decruppe , r. cressely , and p. lindner . rheology , birefringence , and small - angle neutron scattering in a charged micellar system : evidence of a shear - induced phase transition . , 56:18691878 , 1997 . d. h. turnbull , b. g. starkoski , k. a. harasiewicz , j. l. semple , l. from , a. k. gupta , d. n. sauder , and f. s. foster . a 40100 mhz b - scan ultrasound backscatter microscope for skin imaging . , 21:7988 , 1995 . d. a. christopher , p. n. burns , b. g. starkoski , and f. s. foster . a high - frequency pulsed - wave doppler ultrasound system for the detection and imaging of blood flow in the microcirculation . , 23:9971015 , 1997 .
|
high - frequency ultrasonic pulses at 36 mhz are used to measure velocity profiles in a complex fluid sheared in the couette geometry .
our technique is based on time - domain cross - correlation of ultrasonic speckle signals backscattered by the moving medium .
post - processing of acoustic data allows us to record a velocity profile in 0.022 s with a spatial resolution of 40 @xmath0 m over 1 mm . after a careful calibration using a newtonian suspension ,
the technique is applied to a sheared lyotropic lamellar phase seeded with polystyrene spheres of diameter 310 @xmath0 m .
time - averaged velocity profiles reveal the existence of inhomogeneous flows , with both wall slip and shear bands , in the vicinity of a shear - induced `` layering '' transition .
slow transient regimes and/or temporal fluctuations can also be resolved and exhibit complex spatio - temporal flow behaviors with sometimes more than two shear bands .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
[ sec:1 ] * remark . * notation @xmath0 is equivalent to @xmath1 where @xmath2 } _ { n\times 1 } \ , & \rho & = & \rho_{ij } & = & \underbrace { \left [ \begin{array}{c } \rho_{1j } \\ \vdots \\ \rho_{nj } \\ \end{array } \right ] } _ { n \times 1 } \ . \end{array}\ ] ] notation @xmath3 is equivalent to @xmath4 where @xmath5}_{n \times n } } \ . \end{array}\ ] ] let us consider ( e.g. for @xmath6 ) variance of the difference @xmath7 of two random variables @xmath8 and @xmath9 , where @xmath10 , in terms of covariance @xmath11 introducing the estimation statistics @xmath12 @xmath13 in terms of correlation function @xmath14 @xmath15 if @xmath16 and @xmath17 or @xmath18 if @xmath19 and @xmath20 the unbiasedness constraint ( i condition ) @xmath21 is equivalent to @xmath22 the minimization constraint @xmath23 where @xmath24 produces @xmath25 equations in the @xmath26 unknowns : kriging weights @xmath27 and a lagrange parameter @xmath28 ( ii condition ) @xmath29}_{n\times(n+1 ) } } & \cdot & \underbrace { \left [ \begin{array}{c } \omega_j^1 \\ \vdots \\ \omega_j^n \\ \mu_j \\ \end{array } \right ] } _ { ( n+1)\times 1 } & = & \underbrace { \left [ \begin{array}{c } \rho_{1j } \\ \vdots \\ \rho_{nj } \\ \end{array } \right ] } _ { n \times 1 } \end{array}\ ] ] multiplied by @xmath30 @xmath31 and substituted into @xmath32 @xmath33 ^ 2\}-\underbrace{e^2\{v_j-\hat{v}_j\}}_0 \\ & = & e\{[(v_j - m)-(\hat{v}_j - m)]^2\ } \\ & = & e\{[v_j - m]^2\}-2(e\{v_j\hat{v}_j\}-m^2)+e\{[\hat{v}_j - m]^2\ } \\ & = & \sigma^2 -2 \sigma^2 |\omega^i_j \rho_{ij}| + \sigma^2 |\omega^i_j \rho_{ii } \omega^i_j| \\ & = & \sigma^2 \pm 2 \sigma^2 \omega^i_j \rho_{ij } \mp \sigma^2 \omega^i_j \rho_{ii } \omega^i_j \end{array}\ ] ] give the minimized variance of the field @xmath8 under estimation @xmath34 ^ 2\ } = \sigma^2 ( 1 \pm ( \omega^i_j \rho_{ij } + \mu_j ) ) \ ] ] and these two conditions produce @xmath26 equations in the @xmath26 unknowns @xmath35}_{(n+1)\times(n+1 ) } } & \cdot & \underbrace { \left [ \begin{array}{c } \omega_j^1 \\ \vdots \\ \omega_j^n \\ \mu_j \\ \end{array } \right ] } _ { ( n+1)\times 1 } & = & \underbrace { \left [ \begin{array}{c } \rho_{1j } \\ \vdots \\ \rho_{nj } \\ 1 \\ \end{array } \right ] } _ { ( n+1 ) \times 1 } \ . \end{array}\ ] ] since @xmath36 then @xmath37 and ( since ) @xmath38 then @xmath39 the minimized variance of the field @xmath8 under estimation @xmath34 ^ 2\ } = \sigma^2 ( 1\pm(\omega^i_j \rho_{ij } + \mu_j))\ ] ] has known asymptotic property @xmath40 ^ 2\ } = \lim_{n \rightarrow \infty } e\{[v_j-\omega^i_j v_i]^2\ } = e\{[v_j - m]^2\ } = \sigma^2 \ .\ ] ] let us consider the field @xmath8 under estimation @xmath41 where for auto - estimation holds @xmath42 with minimized variance of the estimation statistics @xmath43 ^ 2\ } & = & cov\{(\omega^i_j v_i)(\omega^i_j v_i)\ } \\ & = & \sum_i\sum_l\omega^i_j \omega^l_j cov\{v_i v_l\ } \\ & = & \sigma^2 |\omega^i_j \rho_{ii } \omega^i_j| \\ & = & \mp\sigma^2(\omega^i_j \rho_{ij}-\mu_j ) \ , \end{array}\ ] ] where for auto - estimation holds @xmath44 ^ 2\ } = e\{[v_i - m]^2\ } = \sigma^2\ ] ] that means outcoming of input value is unknown for mathematical model , with minimized variance of the field @xmath8 under estimation @xmath45 ^ 2\ } & = & \sigma^2(1\pm(\omega^i_j \rho_{ij } + \mu_j ) ) \end{array}\ ] ] where for auto - estimation holds @xmath46 ^ 2\ } = \underbrace{e\{[v_i - m]^2\}}_{\sigma^2 } - \underbrace{2(e\{v_i\hat{v}_i\}-m^2)}_{2\sigma^2 } + \underbrace{e\{[\hat{v}_i - m]^2\}}_{\sigma^2 } = 0\ ] ] that means variance of the field is equal to variance of the ( auto-)estimation statistics ( not that auto - estimation matches observation ) . for @xmath47 @xmath48 } _ { n \times 1 } = \xi \underbrace { \left [ \begin{array}{c } 1 \\ \vdots \\ 1 \\ \end{array } \right ] } _ { n \times 1 } \qquad \xi \rightarrow 0 ^ - ~(\mbox{or } ~\xi \rightarrow 0^+ ) \ ] ] and a disjunction of the minimized variance of the field @xmath8 under estimation @xmath49 ^ 2\ } - \underbrace{(e\{v_j\hat{v}_j\}-m^2)}_{\mp\sigma^2\xi } + \underbrace{e\{\hat{v}_j[\hat{v}_j - v_j]\}}_{\mp\sigma^2\xi } \quad \mbox{if } \quad \rho_{ij } \omega^i_j + \mu_j = \xi+ \mu_j=0\ ] ] which fulfills its asymptotic property the kriging system @xmath50}_{(n+1)\times(n+1 ) } } & \cdot & \underbrace { \left [ \begin{array}{c } \omega^1 \\ \vdots \\ \omega^n \\ - \xi \\ \end{array } \right ] } _ { ( n+1)\times 1 } & = & \underbrace { \left [ \begin{array}{c } \xi \\ \vdots \\ \xi \\ 1 \\ \end{array } \right ] } _ { ( n+1 ) \times 1 } & \end{array}\ ] ] equivalent to @xmath51 and @xmath52 where : @xmath53 , @xmath54 , @xmath55 , has the least squares solution @xmath56 and @xmath57 with a mean squared error of mean estimation @xmath58 ^ 2\ } = \mp\sigma^2 2\xi \ .\ ] ] for white noise @xmath45 ^ 2\ } & = & e\{[v_j - m]^2\}+e\{[\hat{v}_j - m]^2\ } \\ * remark . * precession of arithmetic mean can not be identical to @xmath62 cause a straight line fitted to high - noised data by ordinary least squares estimator can not have the slope identical to @xmath62 . for this reason the estimator of an unknown constant variance @xmath63 in fact is the lower bound for precession of the minimized variance of the field under estimation @xmath34 ^ 2\ } = \sigma^2\left(1+\frac{1}{n}\right)\ ] ] to increase ` a bit ' the lower bound for @xmath64 we can effect on weight and reduce total counts @xmath25 by @xmath65 because @xmath65 is the closest positive integer number to @xmath62 so it is easy to find the closest weight such that @xmath66 then the so - called unbiased variance @xmath67 in fact is the simplest estimator of minimized variance of the field under estimation .
|
we present statistics ( s - statistics ) based only on random variable ( not random value ) with a mean squared error of mean estimation as a concept of error .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
_ ab initio _ treatments of electronic systems become unworkable for sufficiently complex systems . on the other hand , the kohn - sham formulation @xcite of density functional theory ( dft ) @xcite incorporates many - body correlations ( beyond hartree - fock ) , while only single - particle ( sp ) equations must be solved . due to this simplicity dft is the only feasible approach in some modern applications of electronic structure theory . there is therefore a continuing interest both in developing new and more accurate functionals and in studying conceptual improvements and extensions to the dft framework . in particular it is found that dft can handle short - range interelectronic correlations quite well , while there is room for improvements in the description of long - range ( van der waals ) forces and dissociation processes . microscopic theories offer some guidance in the development of extensions to dft . orbital dependent functionals can be constructed using many - body perturbation theory ( mbpt ) @xcite . more recently , the development of general _ ab initio _ dft @xcite addressed the lack of a systematic improvement in dft methods . in this approach one considers an expansion of the exact ground - state wave function ( e.g. , mbpt or coupled cluster ) from a chosen reference determinant . requiring that the correction to the density vanishes at a certain level of perturbation theory allows one to construct the corresponding approximation to the kohn - sham potential . a different route has been proposed in ref . @xcite by developing a quasi - particle ( qp)-dft formalism . in the qp - dft approach the full spectral function is decomposed in the contribution of the qp excitations , and a remainder or background part . the latter part is complicated , but does not need to be known accurately : it is sufficient to have a functional model for the energy - averaged background part to set up a single - electron selfconsistency problem that generates the qp excitations . such an approach is appealing since it contains the well - developed standard kohn - sham formulation of dft as a special case , while at the same time emphasis is put on the correct description of qps , in the landau - migdal sense @xcite . hence , it can provide an improved description of the dynamics at the fermi surface . given the close relation between qp - dft and the green s function ( gf ) formulation of many - body theory @xcite , it is natural to employ _ ab initio _ calculations in the latter formalism to investigate the structure of possible qp - dft functionals . in this respect it is imperative to identify which classes of diagrams are responsible for the correct description of the qp physics . some previous calculations , based on gf theory , have focused on a self - consistent treatment of the self - energy at the second order @xcite for simple atoms and molecules . for the atomic binding energies it was found that the bulk of correlations , beyond hartree - fock , are accounted for while significant disagreement with experiment persists for qp properties like ionization energies and electron affinities . the formalism beyond the second - order approximation was taken up in ref . @xcite by employing a self - energy of the @xmath0 type @xcite . in this approach , the random phase approximation ( rpa ) in the particle - hole ( _ ph _ ) channel is adopted to allow for possible collective effects on the atomic excited states . the latter are coupled to the sp states by means of diagrams like the last two in fig . [ fig : sigr](c ) . two variants of the @xmath1 formalism were employed in ref . @xcite ( where the subscript `` @xmath2 '' indicates that non - dressed propagators are used ) . in the first only the direct terms of the interelectron coulomb potential are taken into account . in the second version , also the exchange terms are included when diagonalizing the _ ph _ space [ generalized rpa ( grpa ) ] and in constructing the self - energy [ generalized @xmath0 ( @xmath3 ) ] . although the exchange terms are known to be crucial in order to reproduce the experimentally observed rydberg sequence in the excitation spectrum of neutral atoms , they were found to worsen the agreement between the theoretical and experimental ionization energies @xcite . in the @xmath0 approach the sp states are directly coupled with the two - particle one - hole ( _ 2p1h _ ) and the two - hole one - particle ( _ 2h1p _ ) spaces . however , only partial diagonalizations ( namely , in the _ ph _ subspaces ) are performed . this procedure unavoidably neglects pauli correlations with the third particle ( or hole ) outside the subspace . in the case of the @xmath3 approach , this leads to a double counting of the second order self - energy which must be corrected for explicitly @xcite . we note that simply subtracting the double counted diagram is not completely satisfactory here , since it introduces poles with negative residues in the self - energy . more important , the interaction between electrons in the two - particle ( _ pp _ ) and two - hole ( _ hh _ ) subspaces are neglected altogether in ( @xmath4)@xmath0 . clearly , it is necessary to identify which contributions , beyond @xmath3 , are needed to correctly reproduce the qp spectrum . in this respect , it is known that highly accurate descriptions of the qp properties in finite systems can be obtained with the algebraic diagrammatic construction ( adc ) method of schirmer and co - workers @xcite . the most widely used third - order version [ adc(3 ) ] is equivalent to the so - called extended @xmath5 tamm - dancoff ( tda ) method @xcite and allows to predict ionization energies with an accuracy of 10 - 20 mh in atoms and small molecules . upon inspection of its diagrammatic content , the adc(3 ) self - energy is seen to contain all diagrams where tda excitations are exchanged between the three propagator lines of the intermediate @xmath5 or @xmath6 propagation . the tda excitations are constructed through a diagonalization in either @xmath5 or @xmath6 space , and neglect ground - state correlations . however , it is clear that use of tda leads to difficulties for extended systems . in the high - density electron gas _ e.g. _ , the correct plasmon spectrum requires the rpa in the @xmath7 channel , rather than tda . in order to bridge the gap between the qp description in finite and extended systems , it seems therefore necessary to develop a formalism where the intermediate excitations in the @xmath8 propagator are described at the rpa level . this can be achieved by a formalism based on employing a set of faddeev equations , as proposed in ref . @xcite and subsequently applied to nuclear structure problems @xcite . in this approach the grpa equations are solved separately in the _ ph _ and _ pp / hh _ subspaces . the resulting polarization and two - particle propagators are then coupled through an all - order summation that accounts completely for pauli exchanges in the _ 2p1h_/_2h1p _ spaces . this faddeev - rpa ( f - rpa ) formalism is required if one wants to couple propagators at the rpa level or beyond . apart from correctly incorporating pauli exchange , f - rpa takes the explicit inclusion of ground - state correlations into account , and can therefore be expected to apply to both finite and extended systems . the adc(3 ) formalism is recovered as an approximation by neglecting ground - state correlations in the intermediate excitations ( _ i.e. _ replacing rpa with tda phonons ) . in this work we consider the neon atom and apply the f - rpa method to a nonrelativistic electronic problem for the first time . the relevant features of the f - rpa formalism ( also extensively treated in ref . @xcite ) , are introduced in sect . [ theory ] . the application to the neon atom is discussed in sec . [ results ] , where we also investigate the separate effects of the ladder and ring series on the self - energy , as well as the differences between including tda and rpa phonons . our findings are summarized in sec . [ conclusions ] . some more technical aspects are relegated to the appendix , where the interested reader can find the derivation of the faddeev expansion for the _ 2p1h_/_2h1p _ propagator , adapted from ref . in particular , the approach used to avoid the multiple - frequency dependence of the green s functions is discussed in app . [ app_time ] , along with its basic assumptions . the explicit expressions of the faddeev kernels are given in app . [ app_kernel ] . together with ref . @xcite , the appendix provides sufficient information for an interested reader to apply the formalism . the theoretical framework of the present study is that of propagator theory , where the object of interest is the sp propagator , instead of the many - body wave function . in this paper we will employ the convention of summing over repeated indices , unless specified otherwise . given a complete orthonormal basis set of sp states , labeled by @xmath9,@xmath10 , ... , the sp propagator can be written in its lehmann representation as @xcite @xmath11 where @xmath12 ( @xmath13 ) are the spectroscopic amplitudes , @xmath14 ( @xmath15 ) are the second quantization destruction ( creation ) operators and @xmath16 ( @xmath17 ) . in these definitions , @xmath18 , @xmath19 are the eigenstates , and @xmath20 , @xmath21 the eigenenergies of the ( @xmath22)-electron system . therefore , the poles of the propagator reflect the electron affinities and ionization energies . the sp propagator solves the dyson equation @xmath23 which depends on the irreducible self - energy @xmath24 . the latter can be written as the sum of two terms @xmath25 where @xmath26 represents the hartree - fock diagram for the self - energy . in eqs . ( [ eq : dys ] ) and ( [ eq : sigma1 ] ) , @xmath27 is the sp propagator for the system of noninteracting electrons , whose hamiltonian contains only the kinetic energy and the electron - nucleus attraction . the @xmath28 represent the antisymmetrized matrix elements of the interelectron ( coulomb ) repulsion . note that in this work we only consider antisymmetrized elements of the interaction , hence , our result for the ring summation always compare to the generalized @xmath0 approach . equation ( [ eq : sigma1 ] ) introduces the _ 2p1h_/_2h1p_-irreducible propagator @xmath29 , which carries the information concerning the coupling of sp states to more complex configurations . both @xmath24 and @xmath29 have a perturbative expansion as a power series in the interelectron interaction @xmath30 . some of the diagrams appearing in the expansion of @xmath29 are depicted in fig . [ fig : sigr ] , together with the corresponding contributions to the self - energy . note that already at zero order in @xmath29 ( three free lines with no mutual interaction ) the second order self - energy is generated . in terms of the ( antisymmetrized ) coulomb interaction and undressed propagators . b ) @xmath29 is related to the self - energy according to eq . ( [ eq : sigma1 ] ) . c ) by substituting the diagrams a ) in the latter equation , one finds the perturbative expansion of the self - energy . ] different approximations to the self - energy can be constructed by summing particular classes of diagrams . in this work we are interested in the summation of rings and ladders , through the ( g)rpa equations . in order to include such effects in @xmath29 , we first consider the polarization propagator describing excited states in the @xmath31-electron system @xmath32 and the two - particle propagator , that describes the addition / removal of two electrons @xmath33 we note that the expansion of @xmath29 arises from applying the equations of motion to the sp propagator ( [ eq : g1 ] ) , which is associated to the ground state @xmath34 . hence , all the green s functions appearing in this expansion will also be ground state based , including eqs . ( [ eq : pi ] ) and ( [ eq : g2 ] ) . however the latter contain , in their lehmann representations , all the relevant information regarding the excitation of _ ph _ and _ pp / hh _ collective modes . the approach of ref . @xcite consists in computing these quantities by solving the ring - grpa and the ladder - rpa equations @xcite , which are depicted for propagators in fig . [ fig : rpaeq ] . in the more general case of a self - consistent calculation , a fragmented input propagator can be used and the corresponding dressed ( g)rpa [ d(g)rpa ] equations @xcite solved [ see eqs . ( [ eq : rpa_ph ] ) and ( [ eq : rpa_ii ] ) ] . since the propagators ( [ eq : pi ] ) and ( [ eq : g2 ] ) reflect two - body correlations , they still have to be coupled to an additional sp propagator in order to obtain the corresponding approximation for the _ 2p1h _ and _ 2h1p _ components of @xmath29 . this is achieved by solving two separate sets of faddeev equations . taking the _ 2p1h _ case as an example , one can split @xmath35 in three different components @xmath36 ( @xmath37 ) that differ from each other by the last pair of lines that interact in their diagrammatic expansion , @xmath38 + \sum_{i=1,2,3 } \bar{r}^{(i)}_{\alpha \beta \gamma , \mu \nu \lambda}(\omega ) \ ; , \label{eq : faddfullr } \ ] ] where @xmath39 is the _ 2p1h _ propagator for three freely propagating lines . these components are solutions of the following set faddeev equations @xcite @xmath40 \ ; , ~ ~ i=1,2,3 ~ \nonumber\end{aligned}\ ] ] where ( @xmath41 ) are cyclic permutations of ( @xmath42 ) . the interaction vertices @xmath43 contain the couplings of a _ ph _ or _ pp / hh _ collective excitation and a freely propagating line . these are given in the appendix in terms of the polarization ( [ eq : pi ] ) and two - particle ( [ eq : g2 ] ) propagators . equations . ( [ eq : faddtda ] ) include rpa - like phonons and fully describe the resulting energy dependence of @xmath29 . however , they still neglect energy - independent contributions even at low order in the interaction that also correspond to relevant ground - state correlations . the latter can be systematically inserted according to @xmath44 where @xmath29 is the propagator we employ in eq . ( [ eq : sigma1 ] ) , @xmath45 is the one obtained by solving eqs . ( [ eq : faddtda ] ) , @xmath46 , and @xmath47 is the identity matrix . following the algebraic diagrammatic construction method @xcite , the energy independent term @xmath48 was determined by expanding eq . ( [ eq : uru ] ) in terms of the interaction and imposing that it fulfills perturbation theory up to first order ( corresponding to third order in the self - energy ) . the resulting @xmath48 , employed in this work , is the same as in ref . @xcite and is reported in app . [ app_kernel ] for completeness . it has been shown that the additional diagrams introduced by this correction are required to obtain accurate qp properties . equations . ( [ eq : faddtda ] ) and ( [ eq : uru ] ) are valid only in the case in which a mean - field propagator is used to expand @xmath29 . this is the case of the present work , which employs hartree - fock sp propagators as input . the derivation of these equations for the general case of a fragmented propagator is given in the appendix . more details about the actual implementation of the faddeev formalism to _ 2p1h_/_2h1p _ propagation have been presented in ref . the calculation of the _ 2h1p _ component of @xmath29 follows completely analogous steps . ) ( left ) . the corresponding contribution to the self - energy , obtained upon insertion into eq . ( [ eq : sigma1 ] ) , is also shown ( right ) . ] it is important to note that the present formalism includes the effects of _ ph _ and _ pp / hh _ motion to be included simultaneously , while allowing interferences between these modes . these excitations are evaluated here at the rpa level and are then coupled to each other by solving eqs . ( [ eq : faddtda ] ) . this generates diagrams as the one displayed in fig . [ fig : faddex ] , with the caveat that two phonons are not allowed to propagate at the same time . equations . ( [ eq : faddtda ] ) also assure that pauli correlations are properly taken into account at the _ 2p1h_/_2h1p _ level . in addition , one can in principle employ dressed sp propagators in these equations to generate a self - consistent solution . if we neglect the ladder propagator @xmath49 ( [ eq : g2 ] ) in this expansion , we are left with the ring series alone and the analogous physics ingredients as for the generalized @xmath0 approach . however , this differs from @xmath3 due to the fact that no double counting of the second - order self - energy occurs , since the pauli exchanges between the polarization propagator and the third line are properly accounted for ( see fig . [ fig : faddex ] ) . alternatively , one can suppress the polarization propagator to investigate the effects of _ pp / hh _ ladders alone . it is instructive to replace in the above equations all rpa phonons with tda ones ; this amounts to allowing only forward - propagating diagrams in fig . [ fig : rpaeq ] , and is equivalent to separate diagonalisations in the spaces of @xmath7 , @xmath50 and @xmath51 configurations , relative to the hf ground state . it can be shown that using these tda phonons to sum all diagrams of the type in fig . [ fig : faddex ] reduces to one single diagonalization in the @xmath5 or @xmath6 spaces . therefore , eqs . ( [ eq : faddtda ] ) and ( [ eq : uru ] ) with tda phonons lead directly to the `` extended '' @xmath5 tda of ref . @xcite , which was later shown to be equivalent to adc(3 ) in the general adc framework @xcite . the faddeev expansion formalism of ref . @xcite creates the possibility to go beyond adc(3 ) by including rpa phonons . this is more satisfactory in the limit of large systems . at the same time , the computational cost remains modest since only diagonalizations in the @xmath8 spaces are required . note that complete self - consistency requires the use of fragmented ( or dressed ) propagators in the evaluation of all ingredients leading to the self - energy . this is outside the scope of the present paper , but we included partial selfconsistency by taking into account the modifications to the hf diagram by employing the correlated one - body density matrix and iterating to convergence . this is relatively simple to achieve , since the @xmath8 propagator is only evaluated once with the input hf propagators . below we will give results with and without this partial selfconsistency at the hf level . .parameters that define the sp basis : radius of the confining wall @xmath52 ( in atomic units ) and number of orbits @xmath53 used for different partial waves @xmath54 . the value of @xmath55 is always set to 5 a.u .. [ cols="<,^,^,^,^,^,^,^,^",options="header " , ] . [ tab:4 ] as discussed in sec . [ intro ] , the the f - rpa self - energy contains rpa excitations of both @xmath7 type ( ring diagrams ) and @xmath56 type ( ladder diagrams ) . it is instructive to analyze their separate contributions to the final ionization energies , in order to understand how the f - rpa self - energy is related to the standard @xmath57 self - energy . table [ tab:4 ] compares the results for the ionization energies , obtained with the second - order self - energy , to different approximations for including the ring summations . as one can see , the second - order self - energy generates an @xmath54=1 sp energy of -0.747 mh , which is 46 mh above the empirical @xmath58 ionization energy . the @xmath1 self - energy , which includes the ring summation with only direct coulomb matrix elements , improves this result and brings it close to experiment . the @xmath59 behaves in a similar way . unfortunately , including the exchange terms of the interelectron repulsion in the @xmath60 method turns out to have the opposite effect ( the @xmath58 ionization energy becomes -0.712 h @xcite and the @xmath60 results of ref . @xcite were obtained by retaining only the diagonal part of the electron self - energy , @xmath61 in the hf+continuum basis . this approximation was _ not _ made in the present work . the error in the ionization energies by retaining the diagonal approximation is quite small ( about @xmath622 mh @xcite for the ne atom ) , but larger effects are possible for the total binding energy . ] ) and the agreement with experiment is lost . obviously , @xmath60 is too simplistic to account for exchange in the @xmath7 channel . with the f - rpa(ring ) self - energy one can go one step further and employ the faddeev expansion to also force proper pauli exchange correlations in the _ 2p1h_/_2h1p _ spaces . as shown in table [ tab:4 ] , this enhances the screening due to the exchange interaction terms , leading to even less binding for the @xmath59 and @xmath58 . the corrections relative to the second - order self - energy can be large ( 100 mh for the @xmath59 state ) and in the direction away from the experimental value . we also note that the larger shift , in the 2@xmath63 orbit , is accompanied by an increase of the fragmentation ( see fig . [ fig : sh ] and tab . [ tab:4 ] ) . similar observations were also made in ref . @xcite for other atoms : in general ring summations in the direct channel alone bring the quasihole peaks close to the experiment . this agreement is then spoiled as soon as one includes proper exchange terms in the self - energy . on the other hand , exchange in the _ ph _ channel is required to reproduce the correct rydberg sequence in the excitation spectrum of neutral atoms . so further corrections must arise from other diagrams , and obviously the summation of ladder diagrams can play a relevant role , since these contribute to the expansion of the self - energy at the same level as that of the ring diagrams . the result when only including ladder - type rpa phonons in the f - rpa self - energy is also shown in table [ tab:4 ] . one can see that _ pp / hh _ ladders do actually work in the opposite way as the _ ph _ channel ring diagrams , and have the same order of magnitude with , _ e.g. _ , a shift of 66 mh for the @xmath59 relative to the second - order result . when combined with the ring diagrams in the full f - rpa self - energy , the agreement with experiment is restored again . note that the final result can not be obtained by adding the contributions of rings and ladders , but depends nontrivially on the interplay between these classes of diagrams thereby pointing to significant interference effects . with the f - rpa(ring ) self - energy , where only the contributions of the @xmath7 channel are included , the main peaks listed in table [ tab:4 ] are not only considerably shifted but also strongly depleted , _ e.g. _ a strength of only 0.56 for the main @xmath59 peak . the complete spectral function for the @xmath64 strength in fig . [ fig : sh ] shows that the depletion of the main fragment is accompanied by strong fragmentation over several states . while correlation effects are overestimated in f - rpa(ring ) , they are suppressed in f - rpa(ladder ) , where only the @xmath56 ladders are included in the self - energy . in this case one finds a spectral distribution closer to the hf one , with a main @xmath59 fragment of strength 0.95 and less fragmentation than the the second - order self - energy . the spectral distribution generated by the complete f - rpa self - energy is again a combination of the above effects . the strength of the deeply bound @xmath65 orbital behaves in an analogous way . the strength of the main peak is reduced but several satellite levels appear due to the mixing with @xmath6 configurations . in all the calculations reported in fig . [ fig : sh ] we found a summed @xmath64 strength exceeding 0.98 in the interval [ -40 h , -30 h ] which can be associated with the @xmath65 orbital , and this remains true even in the presence of strong correlations using the f - rpa(ring ) self - energy . of course , the mixing with @xmath66 configurations , not included in this work , may further contribute to the fragmentation pattern in this energy region . in conclusion , the electronic self - energy for the ne atom was computed by the f - rpa method which includes simultaneously the effects of both ring and ladder diagrams . this was accomplished by employing an expansion of the self - energy based on a set of faddeev equations . this technique was originally proposed for nuclear structure applications @xcite and is described in the appendix . at the level of the self - energy one sums all diagrams where the three propagator lines of the intermediate @xmath5 or @xmath6 propagation are connected by repeated exchange of rpa excitations in both the @xmath7 and the @xmath56 channel . this differs from the adc(3 ) formalism in the fact that the exchanged excitations are of the rpa type , rather than the tda type , and therefore take ground - state correlations effects into account . the coupling to the external points of the self - energy uses the same modified vertex as in adc(3 ) , which must be introduced to include consistently all third - order perturbative contributions . the resulting main ionization energies in the neon atom are at least of the same quality , and even somewhat improved , compared to the adc(3 ) result . note that , numerically , f - rpa can be implemented as a diagonalization in @xmath8 space implying about the same cost as adc(3 ) . the present study also shows that in localized electronic systems subtle cancellations occur between the ring and ladder series . in particular , only a combination of the ring and ladder series leads to sensible results , as the separate ring series tends to correct the second - order result in the wrong direction . since the limit to extended systems requires an rpa treatment of excitations , the f - rpa method holds promise to bridge the gap between an accurate description of quasiparticles in both finite and extended systems . in particular , the @xmath0 treatment of the electron gas has been shown to yield excellent binding energies , but poor quasiparticle properties @xcite . further progress beyond @xmath0 theory requires a consistent incorporation of exchange in the @xmath7 channel . the f - rpa technique may be highly relevant in this respect . a common framework for calculating accurate qp properties in both finite and extended systems , is also important for constraining functionals in quasiparticle density functional theory ( qp - dft ) @xcite . finally , complete self - consistency requires sizable computational efforts for bases as large as the hf+continuum basis used here . it would nevertheless represent an important extension of the present work , since it is related to the fulfillment of conservation laws @xcite . these issues will be addressed in future work . this work was supported by the u.s . national science foundation under grant phy-0652900 . although only the one - energy ( or two - time ) part of the _ 2p1h_/_2h1p _ propagator enters the definition of the self energy , eq . ( [ eq : sigma1 ] ) , a full resummation of all its diagrammatic contributions would require to treat explicitly the dependence on three separate frequencies , corresponding to the three final lines in the expansion of @xmath29 . for example , inserting the rpa ring ( ladder ) series in @xmath29 implies the propagation of a _ ph _ ( _ pp / hh _ ) pair of lines both forward and backward in time , while the third line remains unaffected . a way out of this situation is to solve the bethe - salpeter - like equations for the polarization and ladder propagators separately and then to couple them to the additional line . if it is assumed that different phonons do not overlap in time , the three lines in between phonon structures will propagate only in one time direction [ see figures ( [ fig : faddex ] ) and ( [ fig : int_ex ] ) ] . in this situation the integration over several frequencies can be circumvented following the prescription detailed in the next subsection . this approach will be discussed in the following for the general case of a fully fragmented propgator , in order to derive a set of faddeev equations capable of dressing the sp propagator self - consistently . since the forward ( _ 2p1h _ ) and the backward ( _ 2h1p _ ) parts of @xmath29 decouple in two analogous sets of equations , it is sufficient to focus on the first case alone . ) . double lines represent fully dressed sp green s funcions which , however , are restricted to propagate only in one time direction [ i.e. , only one of the two terms on the r.h.s . ( [ eq : g1 ] ) is retained ] . the faddeev eqs . ( [ eq : fadd_1w ] ) and ( [ eq : faddtda ] ) allow for both forward and backward propagation of the phonons @xmath67 and @xmath68 as long as these do not overlap in time . for the propagators , time ordereing is asumed with forwad propagation in the upward direction . ] we start by considering the effective interactions in the _ ph _ and _ pp_/_hh _ channels that correspond to eqs . ( [ eq : pi ] ) and ( [ eq : g2 ] ) stripped of the external legs . in the present work , these are the following two - time objects : where the residues and poles for the ring series are @xmath70 and @xmath71 . for the ladders , @xmath72 and @xmath73 , with poles @xmath74 and @xmath75 . equations . ( [ eq : dg ] ) solve the ring and ladder rpa equations , respectively to display how the phonons ( [ eq : dg_ph ] ) and ( [ eq : dg_ii ] ) enter the expansion of @xmath29 , we perform explicitly the frequency integrals for the diagram of fig . [ fig : int_ex ] . since it is assumed that the separate propagators lines evolve only in one time direction , only the forwardgoing ( @xmath77 ) or the backwardgoing ( @xmath78 ) part of eq . ( [ eq : g1 ] ) must be included for particles and holes , respectively . after some algebra , one obtains @xmath79 \ ; \omega^{n'_\pi}_{\gamma_1 \beta_1 } \left ( \omega^{n'_\pi}_{\lambda_1 \sigma_1 } \right)^ * } { [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_2 } + \varepsilon^-_{k_3 } ] [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_4 } + \varepsilon^-_{k_7 } ] } \right\ } \nonumber \\ & & \times ~ \frac { \left ( { \cal x}^{n_4}_{\sigma_1 } \right)^ * \ ; { \cal x}^{n_4}_{\sigma_2 } } { \omega - ( \varepsilon^+_{n_1 } + \varepsilon^+_{n_4 } - \varepsilon^-_{k_7 } ) + i \eta } \ ; \frac{1}{2 } \left\ { v_{\alpha_1 \sigma_2 , \mu_1 \nu_1 } ~+~ \frac{\left ( \delta^{+,n_{ii}}_{\alpha_1 \sigma_2 } \right)^ * \delta^{+,n_{ii}}_{\mu_1 \nu_1 } } { \omega - ( \varepsilon^{\gamma+}_{n_{ii } } - \varepsilon^-_{k_7 } ) + i \eta } ~+~ \frac { [ \omega + \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_1 } - \varepsilon^+_{n_4 } - \varepsilon^+_{n_5 } - \varepsilon^+_{n_6 } + \varepsilon^-_{k_7 } ] \ ; \delta^{-,k_{ii}}_{\alpha_1 \sigma_2 } \left ( \delta^{-,k_{ii}}_{\mu_1 \nu_1 } \right)^ * } { [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_1 } - \varepsilon^+_{n_4 } ] [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_5 } - \varepsilon^+_{n_6 } ] } \right\ } \nonumber \\ & & ~ \times ~ \frac { \left ( { \cal x}^{n_5}_{\mu_1 } { \cal x}^{n_6}_{\nu_1 } { \cal y}^{k_7}_{\lambda_1 } \right)^ * \ ; { \cal x}^{n_5}_{\mu } { \cal x}^{n_6}_{\nu } { \cal y}^{k_7}_{\lambda } } { \omega - ( \varepsilon^+_{n_5 } + \varepsilon^+_{n_6 } - \varepsilon^-_{k_7 } ) + i \eta } \nonumber \\ & - & \frac{1}{\omega - ( \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_4 } - \varepsilon^{\pi}_{n'_\pi } ) - i \eta } \frac { \left ( { \cal x}^{n_1}_{\alpha } { \cal x}^{n_2}_{\beta } { \cal y}^{k_3}_{\gamma } \right)^ * \ ; { \cal x}^{n_1}_{\alpha_1 } { \cal x}^{n_2}_{\beta_1 } { \cal y}^{k_3}_{\gamma_1 } ~ \omega^{n'_\pi}_{\gamma_1 \beta_1 } \left ( \omega^{n'_\pi}_{\lambda_1 \sigma_1 } \right)^ * ~ \left ( { \cal x}^{n_4}_{\sigma_1 } \right)^ * \ ; { \cal x}^{n_4}_{\sigma_2 } ~ \delta^{-,k_{ii}}_{\alpha_1 \sigma_2 } \left ( \delta^{-,k_{ii}}_{\mu_1 \nu_1 } \right)^ * ~ \left ( { \cal x}^{n_5}_{\mu_1 } { \cal x}^{n_6}_{\nu_1 } { \cal y}^{k_7}_{\lambda_1 } \right)^ * \ ; { \cal x}^{n_5}_{\mu } { \cal x}^{n_6}_{\nu } { \cal y}^{k_7}_{\lambda } } { [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_2 } + \varepsilon^-_{k_3 } ] \ ; [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_4 } + \varepsilon^-_{k_7 } ] \ ; [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_1 } - \varepsilon^+_{n_2 } ] \ ; [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_5 } - \varepsilon^+_{n_6 } ] } \nonumber\end{aligned}\ ] ] the last term in this expression contains an energy denominator that involves the simultaneous propagation of two phonons . thus , it will be discarded in accordance with our assumptions . it must be stressed that similar terms , with overlapping phonons , imply the explicit contribution of at least _ 3p2h_/_3h2p_. a proper treatment of these would require a non trivial externsion of the present formalism , which is beyond the scope of this paper . the remaining part in eq . ( [ eq : int_ex ] ) is the relevant contribution for our purposes . this has the correct energy dependence of a product of denominators that correspond to the intermediate steps of propagation . all of these involve configurations that have at most _ although , ground state correlations are implicitely included by having already resummed the rpa series . still , this term does not factorize in a product of separate green s functions due to the summations over the fragmentation indices @xmath80 and @xmath81 [ labeling the eigenstates of the ( n@xmath821)-electron systems ] . this is overcome if one defines the matrices @xmath83 , @xmath84 and @xmath85 , with elements ( no implicit summation used ) [ eq : g0gmforward ] @xmath86 \ ; \omega^{n'_\pi}_{\gamma \beta } \left ( \omega^{n'_\pi}_{\lambda \nu } \right)^ * } { [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_\beta } + \varepsilon^-_{k_\gamma } ] [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_\nu } + \varepsilon^-_{k_\lambda } ] } \right\ } \ ; , \label{eq : gm12f } \\ \gamma^{(3)>}_{\alpha n_\alpha \beta n_\beta \gamma k_\gamma ; \ ; \mu n_\mu \nu n_\nu \lambda k_\lambda}(\omega ) & = & \nonumber \\ & & \hspace{-3 cm } = ~ \frac{\delta_{\gamma , \lambda } \ ; \delta_{k_\gamma , k_\lambda } } { 2 \ ; \sum_\sigma \left| { \cal y}^{k_\gamma}_\sigma \right|^2 } \left\ { v_{\alpha \beta , \mu \nu } ~+~ \sum_{n_{ii } } \frac{\left ( \delta^{+,n_{ii}}_{\alpha \beta } \right)^ * \delta^{+,n_{ii}}_{\mu \nu } } { \omega - ( \varepsilon^{\gamma+}_{n_{ii } } - \varepsilon^-_{k_\gamma } ) + i \eta } ~+~ \sum_{k_{ii } } \frac { [ \omega + \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_\alpha } - \varepsilon^+_{n_\beta } - \varepsilon^+_{n_\mu } - \varepsilon^+_{n_\nu } + \varepsilon^-_{k_\gamma } ] \ ; \delta^{-,k_{ii}}_{\alpha \beta } \left ( \delta^{-,k_{ii}}_{\mu \nu } \right)^ * } { [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_\alpha } - \varepsilon^+_{n_\beta } ] [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_\mu } - \varepsilon^+_{n_\nu } ] } \right\ } \ ; . \hspace{1.5 cm } \label{eq : gm3f}\end{aligned}\ ] ] in these definitions , the row and column indices are ordered to represent at first two quasiparticle lines and then a quasihole . the index ` @xmath87 ' in @xmath88 refer to the line that propagates independently along with the phonon . using eqs.([eq : g0gmforward ] ) , the first term on the r.h.s . ( [ eq : int_ex ] ) can be written as @xmath89_{\alpha n_\alpha \beta n_\beta \gamma k_\gamma ; \ ; \mu n_\mu \nu n_\nu \lambda k_\lambda } \nonumber\end{aligned}\ ] ] eq . ( [ eq : factor ] ) generalizes to diagrams involving any number of phonon insertions , as long as the terms involving two or more simultaneous phonons are dropped . based on this relation , we use the following prescription to avoid performing integrals over frequencies . one extends all the green s functions to objects depending not only on the sp basis indices ( @xmath90 ) but also on the indices labeling quasi - particles and holes ( @xmath80 and @xmath81 ) . whether a given argument represents a particle or an hole depends on the type of line being propagated . at this point one can perform calculations working with only two - time quantities . the standard propagator is recovered at the end by summing the `` extended '' one over the quasi - particle / hole indices . the 2p1h/2h1p propagator that includes the full resummation of both the ladder and ring diagrams at the ( g)rpa level is the solution of the following bethe - salpeter - like equation , @xmath91 \ ; g_{\lambda \gamma}(-\omega_3 ) ~ + \nonumber \\ & & \hspace{-0.8 cm } \left\ { g_{\beta \beta_1}(\omega_2 ) g_{\gamma_1 \gamma}(-\omega_3 ) v_{\beta_1 \sigma , \gamma_1 \rho } \int \frac{d s}{2\pi i } r_{\alpha \rho \sigma , \mu \nu \lambda}(\omega_1 , s , \omega_2 + \omega_3 - s ) \right . \nonumber \\ & & \hspace{-0.8 cm } + \left . g_{\alpha \alpha_1}(\omega_1 ) g_{\gamma_1 \gamma}(-\omega_3 ) v_{\alpha_1 \sigma , \gamma_1 \rho } \int \frac{d s}{2\pi i } r_{\rho \beta \sigma , \mu \nu \lambda}(s , \omega_2 , \omega_1 + \omega_3 - s ) \right . \nonumber \\ & & \hspace{-1 cm } + \left . \frac{1}{2 } g_{\alpha \alpha_1}(\omega_1 ) g_{\beta \beta_1}(\omega_2 ) v_{\alpha_1 \beta_1 , \rho \sigma } \int \frac{d s}{-2\pi i } r_{\rho \sigma\gamma , \mu \nu \lambda}(s , \omega_1 + \omega_2 - s , \omega_3 ) \right\ } . \nonumber\end{aligned}\ ] ] if this equation is solved , a double integration of @xmath92 would yield the two - time propagator @xmath29 contributing to eq . ( [ eq : sigma1 ] ) . however , the numerical solution of eq . ( [ eq : r_bse ] ) appears beyond reach of the present day computers and one needs to avoid dealing directly with multiple frequencies integrals . the strategy used is to first solve the rpa equations ( [ eq : rpa_ph ] ) and ( [ eq : rpa_ii ] ) separately . once this is done it is necessary to rearrange the series ( [ eq : r_bse ] ) in such a way that only the resummed phonons appear . following the formalism introduced by faddeev @xcite , we identify the components @xmath93 with the three terms between curly brakets in eq . ( [ eq : r_bse ] ) . by employing eqs . ( [ eq : rpa_ph ] ) and ( [ eq : rpa_ii ] ) one is lead to the following set of equations differ from the ones of ref . @xcite which contain the additional term @xmath94 $ ] . the two different forms of the faddeev equations that result can be easily related into each other and are completely equivalent . the definition used here agrees with the standard literature on the subject @xcite . ] , @xmath95 \ ; g_{\lambda \lambda_1}(-s_3 ) \right . \nonumber \\ & & \left . + r^{(j)}_{\mu_1 \nu_1 \lambda_1 , \mu \nu \lambda}(s_1,s_2 , s_3 ) + r^{(k)}_{\mu_1 \nu_1 \lambda_1 , \mu \nu \lambda}(s_1,s_2 , s_3 ) \right\ } \ ; , \ ; i=1,2,3 \ ; , \nonumber \end{aligned}\ ] ] where ( i , j , k ) are cyclic permutations of ( 1,2,3 ) and the interaction vertices @xmath96 are given by finally , we apply the prescription of sec . [ app_time ] and substitute @xmath92 with its extended but two - time version @xmath29 . this leads to the following set of faddeev equations which propagate 2p1h forward in time , @xmath98 \ ; , \nonumber \\ & & \hspace{4 cm } i=1,2,3 \ ; . \end{aligned}\ ] ] since the full energy dependence is retained in eq . ( [ eq : fadd_3w ] ) , the self - energy corresponding to its solution , @xmath92 , is complete up to third order [ see eq . ( [ eq : sigma1 ] ) ] . this is no longer the case after the reduction to a two - time propagator . in particular , the approximation that only forward @xmath5 propagation is allowed between different phonons implies that all diagrams with different time propagation of their external lines are neglected in eqs . ( [ eq : fadd_1w ] ) . however , these terms are not energy dependent and can be can be reinserted in a systematic way a posteriori as in eq . ( [ eq : uru ] ) . in the general case , @xmath99 and @xmath100 where the correction @xmath101 can be determined by comparison with perturbation theory . the vertices ( [ eq : g0gmforward ] ) , that appear in eqs . ( [ eq : fadd_1w ] ) , and @xmath102 are expressed in terms of the fully fragmented propagator . therefore , this approach allows to obtain self - consistent solutions of the sp green s function @xcite . whenever , like in this work , only a mean - field propagator is employed as input there exist a one - to - one correspondence between the fragmentation indices and the sp basis . this is expressed by the relations @xmath103 and @xmath104 , where @xmath105 represents the set of occupied orbits . in this case , it is possible to drop one set of indices so that eqs . ( [ eq : fadd_1w ] ) and ( [ eq : urudressed ] ) simplify into the form ( [ eq : faddtda ] ) and ( [ eq : uru ] ) . in practical applications , it is worth to note that the poles of the free propagator @xmath106 , eq . ( [ eq : g0f ] ) , do not contribute to the kernel of eqs . ( [ eq : fadd_1w ] ) . this can be proven by employing the closure relations for the rpa problem , in the form obtained by extracting the free poles in eqs . ( [ eq : rpa ] ) . as an example , for the forward poles of the ladder propagator these are @xmath107 } ~ ~\longrightarrow } & & \\ & ~ & \left ( { \cal x}^{n_1}_{\alpha } { \cal x}^{n_2}_{\beta } \right)^ * { \cal x}^{n_1}_{\mu } { \cal x}^{n_2}_{\nu } \ ; \gamma^{(ii)}_{\mu \nu , \gamma \delta}(\omega = \varepsilon^+_{n_1 } + \varepsilon^+_{n_2 } ) ~=~ 0 \ ; , ~ ~ \forall n_1 , n_2 \ ; , \nonumber\end{aligned}\ ] ] and similarly for other cases . making use of these relations one can derive the following working expression of the kernels of the _ 2p1h _ faddeev equations ( no implicit summations used ) [ eq : vert2p1h ] @xmath108_{\alpha n_\alpha \beta n_\beta \gamma k_\gamma ; \ ; \mu n_\mu \nu n_\nu \lambda k_\lambda } & = & \left [ { \bf g^{0>}}(\omega){\bf \gamma^{(2)>}}(\omega ) \right]_{\beta n_\beta \alpha n_\alpha \gamma k_\gamma ; \ ; \nu n_\nu \mu n_\mu \lambda k_\lambda } ~=~ \\ & & \hspace{-5 cm } = ~ \delta_{n_\alpha , n_\mu } \ ; \frac{\left ( { \cal x}^{n_\alpha}_{\alpha } { \cal x}^{n_\beta}_{\beta } { \cal y}^{k_\gamma}_{\gamma } \right)^ * { \cal x}^{n_\alpha}_{\mu } } { \sum_\sigma \left| { \cal x}^{n_\alpha}_\sigma \right|^2 } \left\ { \sum_{n_\pi } \frac { \sum_{\beta_1 \gamma_1 } { \cal x}^{n_\beta}_{\beta_1 } { \cal y}^{k_\gamma}_{\gamma_1 } \left ( \omega^{n_\pi}_{\beta_1 \gamma_1 } \right)^ * \omega^{n_\pi}_{\nu \lambda } } { [ \varepsilon^{\pi}_{n_\pi } - \varepsilon^+_{n_\beta } + \varepsilon^-_{k_\gamma } ] [ \omega - ( \varepsilon^+_{n_\alpha } + \varepsilon^{\pi}_{n_\pi } ) + i \eta ] } ~+~ \sum_{n'_\pi } \frac { \sum_{\beta_1 \gamma_1 } { \cal x}^{n_\beta}_{\beta_1 } { \cal y}^{k_\gamma}_{\gamma_1 } \omega^{n'_\pi}_{\gamma_1 \beta_1 } \left ( \omega^{n'_\pi}_{\lambda \nu } \right)^ * } { [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_\beta } + \varepsilon^-_{k_\gamma } ] [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^+_{n_\nu } + \varepsilon^-_{k_\lambda } ] } \right\ } \ ; , \nonumber \\ \nonumber \\ \label{eq : vert2p1h_3 } \left [ { \bf g^{0>}}(\omega){\bf \gamma^{(3)>}}(\omega ) \right]_{\alpha n_\alpha \beta n_\beta \gamma k_\gamma ; \ ; \mu n_\mu \nu n_\nu \lambda k_\lambda } & = & \\ & & \hspace{-5 cm } = ~ \delta_{k_\gamma , k_\lambda } \ ; \frac{\left ( { \cal x}^{n_\alpha}_{\alpha } { \cal x}^{n_\beta}_{\beta } { \cal y}^{k_\gamma}_{\gamma } \right)^ * { \cal y}^{k_\gamma}_{\lambda}\ ; } { 2 \ ; \sum_\sigma \left| { \cal y}^{k_\gamma}_\sigma \right|^2 } \left\ { \sum_{n_{ii } } \frac{\sum_{\alpha_1 \beta_1 } { \cal x}^{n_\alpha}_{\alpha_1 } { \cal x}^{n_\beta}_{\beta_1 } \left ( \delta^{+,n_{ii}}_{\alpha_1 \beta_1 } \right)^ * \delta^{+,n_{ii}}_{\mu \nu } } { [ \varepsilon^{\gamma+}_{n_{ii } } - \varepsilon^+_{n_\alpha } - \varepsilon^+_{n_\beta } ] [ \omega - ( \varepsilon^{\gamma+}_{n_{ii } } - \varepsilon^-_{k_\gamma } ) + i \eta ] } ~+~ \sum_{k_{ii } } \frac{\sum_{\alpha_1 \beta_1 } { \cal x}^{n_\alpha}_{\alpha_1 } { \cal x}^{n_\beta}_{\beta_1 } \delta^{-,k_{ii}}_{\alpha_1 \beta_1 } \left ( \delta^{-,k_{ii}}_{\mu \nu } \right)^ * } { [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_\alpha } - \varepsilon^+_{n_\beta } ] [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_\mu } - \varepsilon^+_{n_\nu } ] } \right\ } \ ; . \nonumber\end{aligned}\ ] ] after substituting eq . ( [ eq : urudressed ] ) into ( [ eq : sigma1 ] ) , one needs the working expression for the matrix product @xmath109 ( where @xmath110 is the interelectron interaction ) . the minimum correction that guaranties to reproduce all third order self - energy diagrams is @xmath111_{\alpha ; \ ; \mu n_\mu \nu n_\nu \lambda k_\lambda } & = & v_{\alpha \lambda , \mu \nu } ~+~ \frac { v_{\alpha \lambda , \gamma_1 \delta_1 } \ ; { \cal y}^{k_\gamma}_{\gamma_1 } { \cal y}^{k_\delta}_{\delta_1 } \left ( { \cal y}^{k_\gamma}_{\gamma_2 } { \cal y}^{k_\delta}_{\delta_2 } \right)^ * \ ; v_{\gamma_2 \delta_2 , \mu \nu } } { 2 \ ; [ \varepsilon^-_{k_\gamma } + \varepsilon^-_{k_\delta } - \varepsilon^+_{n_\mu } - \varepsilon^+_{n_\nu } ] } \\ & & ~~+~ \frac { v_{\alpha \delta_1 , \mu \gamma_1 } \ ; { \cal y}^{k_\gamma}_{\gamma_1 } { \cal x}^{n_\delta}_{\delta_1 } \left ( { \cal y}^{k_\gamma}_{\gamma_2 } { \cal x}^{n_\delta}_{\delta_2 } \right)^ * \ ; v_{\gamma_2 \lambda , \delta_2 \nu } } { [ \varepsilon^-_{k_\gamma } + \varepsilon^-_{k_\lambda } - \varepsilon^+_{n_\delta } - \varepsilon^+_{n_\nu } ] } ~ - ~ \frac { v_{\alpha \delta_1 , \nu \gamma_1 } \ ; { \cal y}^{k_\gamma}_{\gamma_1 } { \cal x}^{n_\delta}_{\delta_1 } \left ( { \cal y}^{k_\gamma}_{\gamma_2 } { \cal x}^{n_\delta}_{\delta_2 } \right)^ * v_{\gamma_2 \lambda , \delta_2 \mu } } { [ \varepsilon^-_{k_\gamma } + \varepsilon^-_{k_\lambda } - \varepsilon^+_{n_\delta } - \varepsilon^+_{n_\mu } ] } \ ; . \nonumber\end{aligned}\ ] ] the case of _ 2h1p _ is handled in a completely analogous way along the steps of secs . ( [ app_time ] ) and ( [ app_faddeqs ] ) . after extending @xmath92 to depend on the fragmentation indices ( @xmath112,@xmath113,@xmath114 ) , the _ 2h1p _ equivalent of eq . ( [ eq : fadd_1w ] ) is obtained with the following definitions of the kernels , [ eq : vert2h1p ] @xmath115_{\alpha k_\alpha \beta k_\beta \gamma n_\gamma ; \ ; \mu k_\mu \nu k_\nu \lambda n_\lambda } & = & \left [ { \bf g^{0>}}(\omega){\bf \gamma^{(2)>}}(\omega ) \right]_{\beta k_\beta \alpha k_\alpha \gamma n_\gamma ; \ ; \nu k_\nu \mu k_\mu \lambda n_\lambda } ~=~ \\ & & \hspace{-5 cm } = ~ \delta_{k_\alpha , k_\mu } \ ; \frac { { \cal y}^{k_\alpha}_{\alpha } { \cal y}^{k_\beta}_{\beta } { \cal x}^{n_\gamma}_{\gamma } \left ( { \cal y}^{k_\alpha}_{\mu } \right)^ * } { \sum_\sigma \left| { \cal y}^{k_\alpha}_\sigma \right|^2 } \left\ { \sum_{n'_\pi } \frac { \sum_{\beta_1 \gamma_1 } \left ( { \cal y}^{k_\beta}_{\beta_1 } { \cal x}^{n_\gamma}_{\gamma_1 } \right)^ * \omega^{n'_\pi}_{\gamma_1 \beta_1 } \left ( \omega^{n'_\pi}_{\lambda \nu } \right)^ * } { [ -\varepsilon^{\pi}_{n'_\pi } - \varepsilon^-_{k_\beta } + \varepsilon^+_{n_\gamma } ] [ \omega - ( \varepsilon^-_{k_\alpha } - \varepsilon^{\pi}_{n'_\pi } ) - i \eta ] } ~+~ \sum_{n_\pi } \frac { \sum_{\beta_1 \gamma_1 } \left ( { \cal y}^{k_\beta}_{\beta_1 } { \cal x}^{n_\gamma}_{\gamma_1 } \omega^{n_\pi}_{\beta_1 \gamma_1 } \right)^ * \omega^{n_\pi}_{\nu \lambda } } { [ \varepsilon^{\pi}_{n_\pi } - \varepsilon^-_{k_\beta } + \varepsilon^+_{n_\gamma } ] [ \varepsilon^{\pi}_{n_\pi } - \varepsilon^-_{k_\nu } + \varepsilon^+_{n_\lambda } ] } \right\ } \ ; , \nonumber \\ \nonumber \\ \label{eq : vert2h1p_3 } \left [ { \bf g^{0>}}(\omega){\bf \gamma^{(3)>}}(\omega ) \right]_{\alpha k_\alpha \beta k_\beta \gamma n_\gamma ; \ ; \mu k_\mu \nu k_\nu \lambda n_\lambda } & = & \\ & & \hspace{-5 cm } = ~ \delta_{n_\gamma , n_\lambda } \ ; \frac { { \cal y}^{k_\alpha}_{\alpha } { \cal y}^{k_\beta}_{\beta } { \cal x}^{n_\gamma}_{\gamma } \left ( { \cal x}^{n_\gamma}_{\lambda}\ ; \right)^ * } { 2 \ ; \sum_\sigma \left| { \cal x}^{n_\gamma}_\sigma \right|^2 } \left\ { \sum_{k_{ii } } \frac{\sum_{\alpha_1 \beta_1 } \left ( { \cal y}^{k_\alpha}_{\alpha_1 } { \cal y}^{k_\beta}_{\beta_1 } \right)^ * \delta^{-,n_{ii}}_{\alpha_1 \beta_1 } \left ( \delta^{-,n_{ii}}_{\mu \nu } \right)^ * } { [ \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^-_{k_\alpha } - \varepsilon^-_{k_\beta } ] [ \omega - ( \varepsilon^{\gamma-}_{k_{ii } } - \varepsilon^+_{n_\gamma } ) - i \eta ] } ~+~ \sum_{n_{ii } } \frac{\sum_{\alpha_1 \beta_1 } \left ( { \cal y}^{k_\alpha}_{\alpha_1 } { \cal y}^{k_\beta}_{\beta_1 } \delta^{+,k_{ii}}_{\alpha_1 \beta_1 } \right)^ * \delta^{+,k_{ii}}_{\mu \nu } } { [ \varepsilon^{\gamma+}_{n_{ii } } - \varepsilon^-_{k_\alpha } - \varepsilon^-_{k_\beta } ] [ \varepsilon^{\gamma+}_{n_{ii } } - \varepsilon^-_{k_\mu } - \varepsilon^-_{k_\nu } ] } \right\ } \ ; , \nonumber\end{aligned}\ ] ] and correction to the external legs , @xmath116_{\alpha ; \ ; \mu k_\mu \nu k_\nu \lambda n_\lambda } & = & v_{\alpha \lambda , \mu \nu } ~+~ \frac { v_{\alpha \lambda , \gamma_1 \delta_1 } \ ; { \cal x}^{n_\gamma}_{\gamma_1 } { \cal x}^{n_\delta}_{\delta_1 } \left ( { \cal x}^{n_\gamma}_{\gamma_2 } { \cal x}^{n_\delta}_{\delta_2 } \right)^ * \ ; v_{\gamma_2 \delta_2 , \mu \nu } } { 2 \ ; [ \varepsilon^-_{k_\mu } + \varepsilon^-_{k_\nu } - \varepsilon^+_{n_\gamma } - \varepsilon^+_{n_\delta } ] } \\ & & ~~+~ \frac { v_{\alpha \delta_1 , \mu \gamma_1 } \ ; { \cal x}^{n_\gamma}_{\gamma_1 } { \cal y}^{k_\delta}_{\delta_1 } \left ( { \cal x}^{n_\gamma}_{\gamma_2 } { \cal y}^{k_\delta}_{\delta_2 } \right)^ * \ ; v_{\gamma_2 \lambda , \delta_2 \nu } } { [ \varepsilon^-_{k_\delta } + \varepsilon^-_{k_\nu } - \varepsilon^+_{n_\gamma } - \varepsilon^+_{n_\lambda } ] } ~ - ~ \frac { v_{\alpha \delta_1 , \nu \gamma_1 } \ ; { \cal x}^{n_\gamma}_{\gamma_1 } { \cal y}^{k_\delta}_{\delta_1 } \left ( { \cal x}^{n_\gamma}_{\gamma_2 } { \cal y}^{k_\delta}_{\delta_2 } \right)^ * v_{\gamma_2 \lambda , \delta_2 \mu } } { [ \varepsilon^-_{k_\delta } + \varepsilon^-_{k_\mu } - \varepsilon^+_{n_\gamma } - \varepsilon^+_{n_\lambda } ] } \ ; . \nonumber\end{aligned}\ ] ] it should be pointed out that while the prescription of sec . [ app_time ] allows sp lines to propagate only in one time direction , it allows for backward propagation of the phonons . these contributions translate directly into the energy independent terms of eqs . ( [ eq : vert2p1h ] ) and ( [ eq : vert2h1p ] ) and are a direct consequence of the inversion pattern typical of rpa theory . these terms have normally a weaker impact than the direct ones on the solutions of eqs . ( [ eq : fadd_1w ] ) . however , it is show in ref . @xcite that they are crucial to guarantee the exact separation of the spurious solutions always introduced by the faddeev formalism @xcite if rpa phonons are used . for the same reasons , the last terms in curly brackets of eqs . ( [ eq : vert2p1h ] ) and ( [ eq : vert2h1p ] ) should be dropped whenever tamm - dancoff ( tda ) phonons are propagated . the approach followed in this work for solving eqs . ( [ eq : fadd_1w ] ) is to transform them into a matrix representations @xcite . once this is done , one is left with an eigenvalue problem that depends only on the _ 2p1h _ ( _ 2h1p _ ) configurations ( @xmath117 ) [ ( @xmath118 ) ] . the spurious states are known exactly @xcite and can be projected out analytically to reduce the computational load . in any case , they would give vanishing contributions to eq . ( [ eq : sigma1 ] ) . w. kohn and l. j. sham , phys . 140 , a1133 ( 1965 ) . p. hohenberg and w. kohn , phys . 136 , b864 ( 1964 ) . a. grling and m. levy , phys . a 50 , 196 ( 1994 ) . a. grling , j. chem . 123 , 062203 ( 2005 ) . r. j. bartlett _ et al . _ , j. chem . 122 , 034104 ( 2005 ) ; j. chem . 123 , 062205 ( 2005 ) . p. mori - sanchez , q. wu , and w. t. yang , j. chem . 123 , 062204 ( 2005 ) . d. van neck , s. verdonck , g. bonny , p. w. ayers , and m. waroquier , phys . a 74 , 042501(2006 ) . a. b. migdal , _ theory of finite fermi systems and applications to atomic nuclei _ ( john wiley and sons , new york , 1967 ) . a. l. fetter and j. d. walecka , _ quantum theory of many - particle physics _ ( mcgraw - hill , new york , 1971 ) . w. h. dickhoff and d. van neck , _ many - body theory exposed ! _ ( world scientific , singapore , 2005 ) . d. van neck , k. peirs , and m. waroquier , j. chem . 115 , 15 ( 2001 ) . k. peirs , d. van neck , and m. waroquier , j. chem . 117 , 4095 ( 2002 ) . n. e. dahlen and r. van leeuwen , j. chem . 122 , 164102 ( 2005 ) . s. verdonck , d. van neck , p. w. ayers , m. waroquier , phys . a 74 , 062503 ( 2006 ) . shirley and r.m . martin , phys . b 47 , 15404 ( 1993 ) . dahlen , r. van leeuwen , and u. von barth , phys . a 73 , 012511 ( 2006 ) . n.e . dahlen and u. von barth , phys . rev . b 69 , 195102 ( 2004 ) . a. stan , n.e . dahlen and r. van leeuwen , europhys . 76 , 298 ( 2006 ) . l. hedin , phys . 139 , a796 ( 1965 ) . n. fukuda , f. iwamoto , and k. sawada , phys . 135 , a932 ( 1964 ) . c. barbieri , n. paar , r. roth , and p. papakostantinou , arxiv : nucl - th/0608011v1 . j. schirmer , l.s . cederbaum , and o. walter , phys . a * 28 * , 1237 ( 1983 ) o. walter and j. schirmer , j. phys . b : at . mol . 14 , 3805 ( 1981 ) . c. barbieri and w. h. dickhoff , phys . c 63 , 034313 ( 2001 ) . c. barbieri and w. h. dickhoff , phys . rev . c 65 , 064313 ( 2002 ) . w. h. dickhoff and c. barbieri , prog . 52 , 337 ( 2004 ) . c. barbieri , phys . b 643 , 268 ( 2006 ) . w. j. w. geurts , k. allaart , and w. h. dickhoff , phys . rev . c 50 , 514 ( 1994 ) . l. d. faddeev , zh . . fiz . * 39 * 1459 ( 1961 ) [ sov . jetp * 12 * , 1014 ( 1961 ) ] . t.h . dunning jr . , j. chem . phys . 90 , 1007 ( 1989 ) ; d.e . woon and t.h . dunning jr , j. chem . 98 , 1358 ( 1993 ) a.b . trofimov and j. schirmer , j. chem . 123 , 144115 ( 2005 ) nist atomic spectra database , nist standard reference database # 78 , http://physics.nist.gov/physrefdata/asd/in-dex.html a. thompson _ _ , _ x - ray data booklet _ ( lawrence berkeley national laboratory , berkeley , ca , 2001 ) , and references cited therein . u. von barth and b. holm , phys . b * 54 * , 8411 ( 1996 ) ; b. holm and u. von barth , phys . b * 57 * , 2108 ( 1998 ) ; b. holm , phys . rev . lett . * 83 * , 788 ( 1999 ) . p. garcia - gonzales and r.w . godby , phys . b * 63 * , 075112 ( 2001 ) g. baym and l.p . kadanoff , phys . 124 , 287 ( 1961 ) . g. baym , phys . 127 , 1391 ( 1962 ) . c. j. joachain , _ quantum collision theory _ , ( north - holland , amsterdam , 1975 ) . s. k. adhikari and w. glckle , phys . c 19 , 616 ( 1979 ) . j. w. evans and d. k. hoffman , j. math . * 22 * , 2858 ( 1981 ) .
|
the spectral function of the closed - shell neon atom is computed by expanding the electron self - energy through a set of faddeev equations .
this method describes the coupling of single - particle degrees of freedom with correlated two - electron , two - hole , and electron - hole pairs .
the excitation spectra are obtained using the random phase approximation , rather than the tamm - dancoff framework employed in the third - order algebraic diagrammatic contruction [ adc(3 ) ] method .
the difference between these two approaches is studied , as well as the interplay between ladder and ring diagrams in the self - energy .
satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the faddeev - rpa scheme that is also appropriate for the high - density electron gas .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
fmos ( the fiber multi - object spectrograph ) is a near - infrared ( nir ) fiber - fed spectrograph for the subaru telescope , which is capable of collecting nearly 400 spectra in a @xmath1 deg@xmath2 field - of - view ( @xcite ) available at the prime - focus of the telescope . the 400 fibers are divided into two groups of 200 fibers and connected to the two spectrographs , irs1 ( infra - red spectrograph ) and irs2 , for spectroscopy . a feature of fmos is the suppression of bright oh - airglow emission lines using oh mask mirrors , which reduce the oh - airglow emission lines by more than @xmath11 ( @xcite ) . since these airglow lines are the largest noise source in the nir region , the oh suppression allows fmos to perform nir spectroscopic observations with a substantially reduced background level . nir spectroscopy is a useful probe for a variety of topics in astronomy . the detection of emission lines ( e.g. , h@xmath3@xmath12 , [ nii]@xmath13 , [ oiii]@xmath14 ) ) in galaxies provide us with information on diverse galaxy properties including the redshift , star formation rate , metallicity and ionisation state . a cosmological redshift survey , fastsound , is ongoing , using fmos to measure @xmath15 redshifts of star - forming galaxies at @xmath16 from h@xmath3 lines over a total area of @xmath17 ( @xcite ) . the main science goal is the measurement of redshift space distortion ( rsd ) for the first time in this redshift range , to measure the structure growth rate as a test of general relativity on cosmological scales ( see , e.g. , @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite for recent galaxy redshift surveys whose scientific targets include rsd ) . a key for studies treating many emission line galaxies in a large data set is to automatically search and detect emission lines . in the case of fmos , residual oh airglow lines just outside the oh mask regions and cosmic rays often show spectral shapes similar to those of real emission lines . if such spurious line detections are included into a statistical sample of emission lines , they will cause a systematic error in , e.g. , measurements of galaxy clustering power spectra . therefore , an efficient automatic line detection software with minimized probability of false detection is highly desirable . in this paper , we present the development of an emission line detection software for fmos spectroscopy , field ( fmos software for image - based emission line detection ) . two - dimensional spectral images for each fiber are used in this algorithm in order to detect faint emission line features , while efficiently filtering out unwanted spurious detections . we examine the performance of this routine by applying it to a part of the fastsound project data . we test various parameters about shapes of emission lines in the 2d spectral images , to cut spurious objects efficiently . the contamination fraction of spurious objects is estimated by inverted 2d spectral images , which are obtained by exchanging object frames and sky frames . in section [ sample ] , we describe the data used in this work . in section [ method ] , we present the algorithm for detecting emission line candidates and describe key features that effectively reduce false detections caused by residual oh emission lines and cosmic rays . the results are presented in section [ results ] and we summarize this work in section [ conclusion ] . the targets for fmos spectroscopy were selected using photometric redshift and h@xmath3 flux estimates using @xmath18 data of canada - france - hawaii telescope legacy survey ( cfhtls ) wide ( @xcite ; @xcite ) . observations are done with the normal - beam - switch ( nbs ) mode of fmos , in which the object frame is taken using nearly all @xmath0 fibers for targets , followed by an offset - sky frame with the same fiber configuration . in fastsound , the exposure time for one frame is @xmath19 s , and two frames ( i.e. , total exposure of @xmath20 min ) are taken both for the object and sky frames . we use the high - resolution ( hr ) mode with spectral resolution of @xmath21 and the wavelength range of @xmath22@xmath23 m . the detector size is 2k@xmath242k , and the pixel scale is @xmath25}$ ] . the primary fmos data set used in this paper are eight fmos field - of - views ( fovs ) observed during march 2829 and april 12 of 2012 , as part of the fastsound project . the fastsound field ids are w2_041 , 062 , and 063 from the cfhtls w2 field , and w3_162 , 163 , 164 , 177 , and 192 from w3 . ( see a forthcoming fastsound paper tonegawa , in prep . for details of the fastsound observations . ) fmos fibers were allocated to @xmath26 galaxies in the eight fmos fovs and the emission line detection rate is typically @xmath27 as reported in tonegawa et al . the mean seeing is @xmath28 in the @xmath29 band . these eight fovs are representative of fastsound data in normal observing conditions . in rare cases ( four out of @xmath30 fovs in the fastsound project ) , images taken by irs2 spectrograph just after sunset show unusual dark patterns on the edge of detector , likely because of the instability of detector resulting from overexposures by dome - flat lamp or twilight . these patterns usually appear vertically in the left end ( @xmath22@xmath31 m ) of the image , but sometimes appear horizontally in the top or bottom end . though the fraction of such data is not large in the total fastsound data set , line detection must be affected in these regions . hence we also examine the performance of our software in these four fovs in these bad conditions . these are w2_042 ( taken in 2013 feb . ) , w3_173 ( 2012 apr . ) , w3_206 and 221 ( 2012 jun . ) . the fmos two - dimensional raw data are reduced with the standard data reduction pipeline , _ fibre - pac _ ( iwamuro et al , 2012 ) . in fmos images , the @xmath32-axis is used for the wavelength direction , and @xmath33 pixels are assigned in the @xmath34-axis direction for each fiber . the profile in the @xmath34-axis is just the point spread function ( psf ) of light from a fiber . first , sky subtraction is performed for each frame . in order to trace the time - variance of sky background , the linear combination of two sky frames is used for the subtraction . after sky subtraction , the bias difference between the four detector quadrants , cross talk , bad pixels , distortions , and residual oh lines are corrected . the two images are then combined into the total exposure . the wavelength calibration is applied by making a correlation between @xmath32-axis and physical wavelength using images of a th - ar lamp . _ fibre - pac _ creates not only a science frame from object and sky images , but also square - noise and bad pixel maps for each fov ( figure 1 ) . the square - noise map gives the square of noise level of each pixel , which is measured using @xmath35 pixels around each pixel along the @xmath34-axis with a @xmath36 clipping algorithm iterated ten times . the bad pixel map indicates the quality of each pixel by a value between between @xmath37 ( bad ) and @xmath38 ( normal ) . those pixels on the detector flagged as non - functioning by _ fibre - pac _ and temporally prominent pixels appearing in one exposure ( by cosmic ray events in most cases ) are rejected and replaced in the science frames , by an interpolated value from surrounding pixels ( @xcite ) . these pixels are stored as a bad pixel map for each exposure of @xmath39 min . our exposure time is @xmath20 min , and hence we have two bad pixel maps for each field - of - view . when these two maps are combined , true defects of the detector are presented as the value @xmath37 on the bad pixel map because they appear in both of two bad pixel maps , while cosmic rays have a value of @xmath40 because they affect only one of the two science sub - frames . the information from the square - noise and bad pixel maps are utilized in the line detection algorithm presented below . as well as the normal science frame , we also create an inverted " science frame by exchanging object and sky frame in the _ fibre - pac _ procedures . the line detection algorithm is also applied to this inverted science frame . any candidates detected in this inverted frames will not be genuine features , because real emission lines are negative in the frame and absorption lines are usually below the detection limit of fmos for galaxies beyond @xmath41 . therefore we can reduce the overall false detection rate by tuning the software to minimize the detection rate in the inverted frames . ( 80mm,80mm)figure1a.eps ( 80mm,80mm)figure1b.eps our goal is to automatically detect spectral features corresponding to emission lines from the fmos data with high reliability . to achieve this , we develop an emission line detection algorithm which is based on a convolution of the fmos 2-d spectral images weighted by a detection kernel that is similar to the typical emission line profile ( see , e.g. , @xcite for similar approaches ) . note that we do not use the flux - calibrated 1-d spectra for emission line detection because they do not retain the 2-d shape of the psf , which is useful to filter out false detections . the algorithm uses a flat - field image , square - noise map and bad pixel map as well as the science frame ( see [ sample ] ) . the science frame and square - noise map are both convolved with the kernel , producing an effective signal - to - noise ( @xmath6 ) ratio of a line centered at each pixel along the wavelength direction . the flat - field image and bad pixel map are used to remove the oh mask regions or bad pixels from the calculation of @xmath6 , so that the false detection rate is minimized . an important feature of the fmos spectra in the nir is the hardware suppression of the the oh airglow . inclusion of the masked regions does not improve the line detection efficiency , and the noise level is often particularly high at the border of oh masks , which would have a negative effect for efficient line detection . to remove such regions , we utilize the dome - flat image . the dome - flat image is divided by the detector - flat image to correct for different quantum efficiency between pixels ( @xcite ) . after normalization , the flat - field image have values of @xmath42 at normal pixels , while @xmath43 at oh mask regions . we then make a new square - noise map by dividing the original square - noise map by this flat image . this operation increases the noise level inside and around oh mask regions by a large factor , making such regions ineffective in the line @xmath6 calculation ( see the next subsection ) . this approach is better than , e.g. , simply removing oh mask regions by wavelength information , because the detailed performance of the oh mask depends on the temperature of fmos instrument that changes with time . however , the effect of noisy regions around the border of oh masks can not be completely removed by this operation , and hence we further remove the pixels whose original noise level ( before dividing by the flat image ) is higher than the mean by more than @xmath44 , from the line @xmath6 calculation . in order to decrease the chance of detecting cosmic rays and detector defects , we use of the bad pixel map ( figure [ figure : fmosimage ] ) . since the bad pixel map has values of @xmath42 at good pixels , @xmath43 at defects , and @xmath45 at pixels hit by cosmic rays , we exclude the pixels having values lower than @xmath46 in the bad pixel map from the line @xmath6 calculations . finally , the continuum component of galaxies may affect the detection efficiency of emission lines . therefore the continuum component of each spectrum in the object frame was subtracted by applying fit1d task of iraf with a 5th - order chebyshev polynomial along the wavelength direction . this procedure was adopted separately for each pixel along the vertical direction perpendicular to wavelength . we define the detection kernel by a two - dimensional gaussian that imitates the typical shape of an emission line : @xmath47\ ] ] where @xmath48 and @xmath49 are the typical dispersion of emission lines along @xmath32-axis ( corresponding to wavelength ) and @xmath34-axis ( corresponding to the fiber aperture ) on the detector . although @xmath48 and @xmath49 are adjustable , we fix @xmath50 [ pix ] as the quadratic sum of the spectral resolution of fmos ( @xmath51 fwhm at @xmath52 m in the hr mode ) and the typical velocity dispersion of galaxies ( @xmath53 fwhm ) , and @xmath54 [ pix ] as representative of the fmos fiber profile on the detector . because fmos is a fiber - fed spectrograph , @xmath49 is just an instrumetal spread of light from fibers without spatial information of target objects . for each pixel @xmath55 , we define the signal @xmath56 of a supposed line centered at the pixel as : @xmath57 which is an integration over pixels @xmath58 around @xmath55 confined within the region @xmath59 . this region is an ellipse whose center is @xmath55 and the radii along the major and minor axis are @xmath60 [ pix ] and @xmath61 [ pix ] . the parameter @xmath3 is the best - fit flux normalization of this kernel to the observed count @xmath62 in the pixels in @xmath59 : @xmath63 which can be derived by minimizing @xmath64 , where @xmath65 ^ 2}{n_i^2 } \ , \ ] ] where @xmath66 is @xmath67 noise at @xmath68-th pixel from the noise - square ( @xmath69 ) map . from the expression for @xmath3 , it can be understood that the signal @xmath56 ( equation [ s ] ) is a convolution of the count @xmath62 with the detection kernel , weighted by @xmath70 . we then define the noise @xmath71 for the line ; since @xmath56 is a linear combination of @xmath62 whose noise is @xmath66 , its statistical error can be calculated as : @xmath72 then the signal - to - noise ratio @xmath6 is simply calculated from @xmath56 and @xmath71 . the @xmath6 is calculated along the wavelength direction , with @xmath73 fixed at the central pixel ( @xmath74 ) among the 9 pixels for one fiber . when a certain pixel has a local peak of @xmath6 map , the pixel is likely to be the center of an emission line feature . therefore , we select emission line candidates by extracting pixels with @xmath6 values higher than a threshold and locally greatest within the range of @xmath7530 pixels from @xmath76 along the wavelength direction . the latter condition is introduced to avoid multiple detections of the same line - like feature . although the use of @xmath7530 pixels as the minimum separation is a simplistic approach , it suffices for fastsound , because the closest lines ( h@xmath3@xmath12 and [ nii]@xmath13 ) at @xmath16 are separeted by @xmath77 [ pix ] on the image . more sophisticated methods to discriminate multiple lines , such as connected - pixel approach , will be examined in future work . the selection threshold can be changed by the user : if one increases the threshold , the number of false detections decreases but the number of real lines would also be reduced . the false detection rate and its dependence on the threshold are discussed in [ results ] . the image shape of each candidate line includes important information that allows discrimination between real and false lines . we measure the following shape parameters : \(1 ) center along @xmath34-axis : @xmath78 ( 2 ) dispersion along @xmath34-axis : @xmath79 ( 3 ) fraction of positive pixels : @xmath80 where @xmath81 is a step function ( 1 for @xmath82 and 0 otherwise ) . + ( 4 ) position angle of the major axis of an elliptical fit : @xmath83 ( 5 ) axis ratio of the elliptical fit : @xmath84 where @xmath85 and @xmath86 are semi - major and minor axes , respectively . + ( 6 ) signal - to - noise fluctuation per pixel : @xmath87 where angle brackets denote the mean over the region @xmath59 . we applied our algorithm to an inverted " science frame , obtained by running the fmos image reduction pipeline with the object and sky frames exchanged , as well as the normal " science frame . all the candidates in the inverted frames must be spurious , and their statistical nature should be the same as the spurious features in the normal frames , because the analysis procedures are exactly the same except for swapping the object / sky frames . for example , residual oh emission lines in the science frame can be positive or negative at the same probability depending on the observing conditions , and cosmic rays fall randomly both on the object and sky frames . note that an emission line galaxy can accidentally fall in a fiber during the sky exposures , but we did not take these cases into account , because the possibility should be small . figure [ figure : param ] shows the distribution of the shape parameters defined above , as a function of @xmath6 , both for the normal and inverted frames . it is found that in some regions ( especially for the data in the 4 fovs at the bad condition in the plots of @xmath88 , @xmath89 , @xmath90 , and @xmath91 plots ) the number of inverted - frame objects is relatively large compared with the normal - frame objects , and the numbers in normal / inverted frames are similar . false lines are expected to be dominant over real lines in such regions , and hence we introduced an event cut as shown by the solid curves in the figure . these conditions are expressed as : @xmath92 it should be noted that the region of many inverted - frame objects is also found in the plot of minor / major axis ratio . we did not include a condition on this quantity , because we found that almost all of the false objects in this region are effectively removed by the other three conditions . we also examined the spectral images of the rejected events by eye , and confirmed that false detections do indeed dominate . these events mainly arise from unusual dark patterns appearing in the bad - condition four fovs , when the detector was in an unstable state . examples of emission line candidates detected by the software are shown in figure [ figure : sample ] . ( 75mm,75mm)figure2a.eps ( 75mm,75mm)figure2b.eps ( 75mm,75mm)figure2c.eps ( 75mm,75mm)figure2d.eps ( 75mm,75mm)figure2e.eps ( 75mm,75mm)figure2f.eps ( 80mm,66mm)figure3a.eps ( 80mm,66mm)figure3e.eps ( 80mm,66mm)figure3b.eps ( 80mm,66mm)figure3f.eps ( 80mm,66mm)figure3c.eps ( 80mm,66mm)figure3g.eps ( 80mm,66mm)figure3d.eps ( 80mm,66mm)figure3h.eps .detected line candidate statistics for the eight fovs under normal condition . [ cols="^,^,^,^",options="header " , ] [ table : linestat ] in table [ table : linestat ] , the number of emission line candidates in the eight fovs under normal condition is shown for three different thresholds of @xmath6 , both for the normal and inverted frames . there is only one false - detection in the inverted frame above @xmath93 , indicating that the false detection rate should be less than 1% at @xmath8 . we inspected the seven detections in the inverted frame at @xmath94 by eye , and found that five were close to oh mask region , one at the edge of the detector , and the other one due to a cluster of hot pixels that shows a collective instability because of high dark current . although the line detection algorithm suppresses the contribution to @xmath6 from the oh mask region , strong residual oh emission often remains because of unstable oh airglow in the nir , resulting in a small fraction of false detections . since the events at the border of oh masks are the major source of the false detections , more strict cuts in such regions may further improve the fraction of real lines . the cumulative @xmath6 distribution of emission line candidates are displayed in figure [ figure : snhist ] . the lower limit to the number of real emission lines can be estimated from the difference between the normal and inverted frames , i.e. , red and blue lines . ( they are lower limits because the line detection completeness is not 100% by statistical fluctuation and oh mask effects . see the next paragraph . ) one can see that the number of detections sharply increases with decreasing @xmath6 under @xmath95 , but this is mainly due to the increase of false detections in the inverted frame . the wavelength of the detected lines are plotted against @xmath6 in figure [ figure : zhist ] . again , we see a tendency for detections in the inverted images to be close to the oh mask regions , indicating that residuals from the oh airglow is one of the major reasons for false detections . the completeness of detection , i.e. , the probability of successful detection for a given real emission line is also an important statistic to evaluate the performance of the software . to estimate this we ran a simulation by placing artificial objects and then applying our detection algorithm . the completeness against input brightness ( @xmath6 ) is displayed in figure [ figure : completeness ] , for some different values of detection @xmath6 thresholds . the completeness does not reach 100% but stays lower than 90% at large input @xmath6 , and this can be explained by the effect of oh masks that cover about 20% of the observed wavelength range . indeed , we confirmed that the completeness becomes close to @xmath96 when we carried out the same simulation excluding the oh mask regions . the completeness is about 40% for the input @xmath6 same as the adopted @xmath6 threshold , which is a reasonable result expected from statistical fluctuation and the oh mask effect . it should be noted that we assumed the same gaussian psf as the kernel described in [ subsection : algorithm ] for the artificial objects distributed in the simulation . though the kernel is set to be similar to typical line profiles , the completeness for real emission galaxies is expected to be lower than this . a quantitative estimate of the completeness including this effect would be model dependent about the line profile distribution of a galaxy sample , which is beyond the scope of this work . ( 80mm,80mm)figure4a.eps ( 80mm,80mm)figure4b.eps ( 160mm,80mm)figure5.eps ( 80mm,80mm)figure6.eps in this work we developed an automated emission line detection software for fmos on the subaru telescope , using the data from the fastsound project , targeting h@xmath3 emission line galaxies at @xmath16 . emission line detection is based on @xmath6 obtained by a convolution between the two - dimensional science frame and a line - profile kernel . a unique feature of the software for the fmos data having many oh airglow suppression masks is the amplification of noise level ( and consequent suppression in the @xmath6 calculation ) in the masked region by use of a flat - field image . bad pixels on the detectors and pixels affected by cosmic - rays are removed from the @xmath6 calculation . we also calculated six shape parameters for the detected lines , and these are used for further rejection of false detections . this is particularly useful to remove those spurious sources caused by unusual dark patterns when the detectors are in unstable condition . the efficiency and reliability of the line detection were examined by applying the method to inverted science frames obtained by exchanging the object and sky images in the reduction process . the false - detection rate is @xmath97 , @xmath98 and @xmath99 for @xmath6 above @xmath100 , @xmath101 , and @xmath102 , respectively . the emission line flux corresponding to @xmath103 is about @xmath104}$ ] in normal condition . the software is open to the community , and currently available on request to the authors . blake , c. , 2011 , , 415 , 2876 beutler , f. , 2012 , , 423 , 3430 de la torre , s. , guzzo , l. , peacock , j. a. , 2013 , , 557 , a54 guzzo , l. , 2008 , nature , 451 , 541 gilbank , d. g. , baldry , i. k. , balogh , m. l. , glazebrook , k. , & bower , r. g. 2010 , , 405 , 2594 gwyn , s. d. j 2012 , a&a , 143 , 38 hawkins , e. , 2003 , , 346 , 78 iwamuro , f. , motohara , k. , maihara , t. , hata , r. , & harashima , t. 2001 , , 53 , 355 iwamuro , f. , 2012 , , 64 , 59 kimura , m. , maihara , t. , iwamuro , f. , 2010 , , 62 , 1135 reid , b. a. , 2012 , , 426 , 2719 samushia , l , percival , w. j & raccanelli , a 2013 , , 420 , 2102 tonegawa , m. , totani , t. , akiyama , m. , 2014 , , 66 , 43
|
we describe the development of automated emission line detection software for the fiber multi - object spectrograph ( fmos ) , which is a near - infrared spectrograph fed by @xmath0 fibers from the @xmath1 deg@xmath2 prime focus field of view of the subaru telescope .
the software , field ( fmos software for image - based emission line detection ) , is developed and tested mainly for the fastsound survey , which is targeting h@xmath3 emitting galaxies at @xmath4 to measure the redshift space distortion as a test of general relativity beyond @xmath5 . the basic algorithm is to calculate the line signal - to - noise ratio ( @xmath6 ) along the wavelength direction , given by a 2-d convolution of the spectral image and a detection kernel representing a typical emission line profile .
a unique feature of fmos is its use of oh airglow suppression masks , requiring the use of flat - field images to suppress noise around the mask regions .
bad pixels on the detectors and pixels affected by cosmic - rays are efficiently removed by using the information obtained from the fmos analysis pipeline .
we limit the range of acceptable line - shape parameters for the detected candidates to further improve the reliability of line detection .
the final performance of line detection is tested using a subset of the fastsound data ; the false detection rate of spurious objects is examined by using inverted frames obtained by exchanging object and sky frames .
the false detection rate is @xmath7% at @xmath8 , allowing an efficient and objective emission line search for fmos data at the line flux level of @xmath9[erg / cm@xmath2/s ] .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
laboratory experiments @xcite and numerical simulations @xcite can now systematically explore sustained homogeneous nonequilibrium systems of quite large aspect ratios ( @xmath2 ) which possibly approximate a thermodynamic limit of infinite system size . these advances raise the theoretical question of identifying order parameters for analyzing and classifying spatiotemporal chaotic states so that a quantitative comparison can be made between theory and experiment @xcite . the most appropriate order parameter for a given nonequilibrium system is presently not well understood although numerous possibilities have been studied . some order parameters , such as spatial correlation lengths obtained from exponentially decaying correlation functions , emphasize the average spatial disorder and have been widely used in condensed matter physics @xcite . others such as the metric entropy and the lyapunov fractal dimension @xcite are familiar from nonlinear dynamics and emphasize the average temporal disorder or dynamical complexity arising from the geometric structure of an attractor in phase space . we would then like to know whether these different kinds of order parameters are related and whether there is a need for new order parameters . as an example , does knowledge of an easily measured correlation length give information about the fractal dimension , which is difficult to estimate from experimental time series @xcite ? that a relation between spatial disorder and dynamical complexity may exist is suggested by the prominence in many nonequilibrium states of defects @xcite whose dynamics often determine the average spatial disorder @xcite . examples of defects are amplitude holes in one - dimensional complex ginzburg - landau equation ( abbreviated below as cgle ) @xcite , domain walls and droplets of opposite spin in coupled map lattices ( cmls ) with an ising - like transition @xcite , vortices in the two - dimensional cgle @xcite , and spirals and centers in the recently discovered spiral - defect chaos state in thermal convection @xcite . the nucleation , motion , and annihilation of defects are important features of the chaotic dynamics and so dynamical quantities such as the fractal dimension may be related to their spatial statistics @xcite . complicating this simple picture is the fact that not all fluctuations are associated with defect motion , e.g. , phase fluctuations in the cgle @xcite . the fractal dimension may then be larger than that suggested by defect statistics . in recent work @xcite , egolf and greenside have explored the relation between temporal and spatial disorder for the cgle on a large periodic interval @xcite . observing that sufficiently large chaotic systems become extensive so that a fractal dimension @xmath3 grows linearly with volume size @xmath4 ( where @xmath5 is the system size and @xmath6 is the spatial dimensionality ) @xcite , they calculated the dimension correlation length @xmath0 which is defined @xcite in terms of the intensive dimension density @xmath7 by the equation @xmath8 the length @xmath0 can be interpreted crudely as a characteristic size of dynamically independent subsystems or a characteristic range of chaotic fluctuations . near a transition from phase- to defect - turbulent states @xcite , egolf and greenside found that the length @xmath0 was approximately equal to , and had the same parametric dependence as , the spatial correlation length given by the magnitude fluctuations of the ginzburg - landau field @xcite . over the same parameter range , the correlation length @xmath9 arising from phase fluctuations ( and also of the ginzburg - landau field itself ) was found to increase to quite large values , suggesting that the chaos fluctuations measured by @xmath0 were short - ranged and decoupled from the phase . in this paper , the one - dimensional investigations of egolf and greenside @xcite are extended by examining similar questions of how spatial disorder and dynamical complexity are related for spatial dimensionality @xmath10 and @xmath11 . we study a class of dissipative coupled map lattices @xcite that undergoes an ising - like ferromagnetic - ordering transition as a lattice coupling constant @xmath1 ( defined below in eq . ( [ eq : miller - huse - map ] ) ) is increased through a range of finite positive values , corresponding to a transition from a high - temperature paramagnetic phase to a low - temperature ferromagnetic phase . instead of the point - like space - time defects of the one - dimensional cgle @xcite , long - lived defects occur in the form of domain walls and droplets involving regions of opposite sign . at a critical transition point @xmath12 , the usual two - point correlation length @xmath13 diverges to infinity while the magnetization ( the average of the signs of all lattice variables ) bifurcates from a zero to nonzero value . the main attraction of the miller - huse model @xcite is that a transition with a diverging correlation length occurs from one chaotic state to another . this permits a careful comparison of different length scales near the transition point . by calculating the lyapunov spectrum and related dynamical quantities such as the dimension correlation length @xmath0 , we show that correlations in chaotic fluctuations near the ordering transition have a short range and are decoupled from the diverging long - range order measured by the length @xmath13 . as was the case for the 1d periodic cgle @xcite , the length @xmath0 and the correlation length @xmath14 arising from magnitude fluctuations of the cml variables both turn out to be short here about one lattice spacing but a quantitative relation can not be determined since these lengths do not vary substantially with parameters . in one - space dimension , a more substantial variation in these quantities is obtained by increasing the radius @xmath15 of the coupling from nearest to @xmath15th - nearest lattice neighbors . with increasing radius @xmath15 , the lengths @xmath0 and @xmath16 then increase approximately linearly and with slopes related by a factor of order one . this suggests that the two lengths may be related and that @xmath15 is important in determining the length scale of dynamical fluctuations . several dynamical quantities such as the largest lyapunov exponent @xmath17 , the metric entropy density @xmath18 , and the lyapunov fractal dimension density @xmath19 attain minimum values near but distinctly not at the critical transition point @xmath12 . at first glance , minima in quantities such as @xmath20 or @xmath19 seem counterintuitive since the onset of ferromagnetic ordering should correspond to increased correlations , i.e. , decreased dynamical complexity and a decreased dimension density or entropy density . such a monotonic decrease in the metric entropy is , in fact , observed for a non - dissipative equilibrium cml that undergoes an ising - like transition @xcite . but because the dimension correlation length @xmath0 is quite short in the miller - huse model , we argue below that dynamical quantities are only sensitive to nearest neighbor dynamics . as the coupling constant @xmath1 increases , the discrete laplacian eventually becomes antidiffusive , magnifying rather than reducing short - wavelength structure , and the dimension density and entropy density start to increase . the role of coupling lattice neighbors is demonstrated with calculations on periodic 2d hexagonal and 3d cubic lattices . for these cases , @xmath0 is again about one lattice spacing in size and the extrema in dynamical quantities occur at a value @xmath21 determined by the number @xmath22 of nearest neighbors ( @xmath23 for the square lattice , @xmath24 for the hexagonal and cubic lattices ) . the positions of the extrema do not coincide with the bifurcation of the magnetization and do not seem to depend on the lattice symmetry and dimensionality . the origin of these minima remains to be explained . these results , together with previous work on the 1d cgle and with some unpublished work on cmls with algebraic decay of spatial correlations @xcite , suggest that the dimension correlation length will typically be short so that chaotic fluctuations are decoupled from long - range spatial order as measured by correlation functions . this leads to the qualitative conclusion that defects , although a striking visual feature of spatiotemporal chaos , are not the source of complexity that leads to large fractal dimensions and to small dimension correlation lengths . the physical meaning and utility of the length @xmath0 remains to be understood and further studies on different kinds of systems will be useful . the rest of this paper is organized as follows . in section [ methods ] , we define the coupled map lattice and discuss some details of how its lyapunov spectrum is calculated using a cm-5 parallel computer @xcite . in section [ results ] , we discuss various results of our simulations , especially the dependence of order parameters on the lattice coupling constant and on the symmetry and dimensionality of the lattice . finally , in section [ conclusions ] , we summarize our results and relate them to other recent research . in this section , we define the mathematical models used in our simulations and discuss some details about how the lyapunov spectrum and spatial correlation lengths were calculated numerically . since a cm-5 parallel computer played an important role in our being able to explore large space - time regions for many parameter values , we also discuss some details of how the algorithms were adapted for parallel computation . as easily simulated models of spatiotemporal chaos , we consider homogeneous coupled map lattices ( cmls ) in which the same chaotic map @xmath25 is associated with each point of a finite periodic lattice and for which nearest neighbor maps are coupled linearly by diffusion . cmls have a significant advantage over partial differential equations of being analytically amenable @xcite and easier to simulate on a computer . cmls have the drawback that their solutions can not generally be related quantitatively to experiment and they may not have universal critical properties near transitions @xcite . we study cmls suggested by recent work of miller and huse @xcite , who analyzed the long - wavelength properties of a two - dimensional cml that orders ferromagnetically , in analogy to the equilibrium ising model @xcite . following these authors , we use a lattice map @xmath26 with odd symmetry so that domains of opposite `` spin '' or sign arise . the odd symmetry is a necessary but not sufficient condition @xcite for obtaining an ising - like transition in which the magnetization ( defined below in eq . ( [ eq : magnetization - defn ] ) ) bifurcates to a nonzero value as a coupling constant @xmath1 is varied . following miller and huse , we choose @xmath25 to be a piecewise - linear map with slope of constant magnitude greater than one , for which only chaotic states of constant measure exist in the absence of lattice coupling . if @xmath27 denotes the variable at spatial site @xmath28 at integer time @xmath29 ( with @xmath30 , @xmath31 , @xmath32 ) , then the rule for updating each lattice variable to time @xmath33 is given by @xcite : @xmath34 where the parameter @xmath1 is the spatial coupling constant . the sum goes over indices @xmath35 that denote nearest neighbors sites of site @xmath28 , e.g. , the four nearest neighbors on a square two - dimensional lattice or the six nearest neighbors in a 2d hexagonal lattice or 3d cubic lattice . for most of our calculations , we used the same local map as in ref . @xcite : @xmath36 with a slope of constant magnitude equal to 3 . to understand the competition between local chaos and diffusion ( which decrease and increase spatial correlations respectively ) , we also used a more weakly chaotic map with a slope of smaller constant magnitude equal to 1.1 : @xmath37 once a local map @xmath25 has been chosen , a numerical simulation is specified by the dimensionality of the lattice @xmath6 , the symmetry of the lattice ( e.g. , square , hexagonal , or cubic ) , the size of the lattice @xmath5 ( number of sites along an edge ) , the coupling constant @xmath1 , the initial condition @xmath38 , and the total integration time @xmath39 . in this paper , we used an integer lattice in 1d , square and hexagonal lattices in 2d and a cubic lattice in 3d . initial conditions consisted of assigning a random uniformly - distributed number in the interval @xmath40 $ ] to each site ; results were not dependent on the choice of initial conditions provided the integration time was sufficiently long . typical integration times were @xmath41 iterations for calculations of lyapunov exponents and @xmath42 iterations for calculations of correlation functions . we made a few runs with longer integration times of @xmath43 and @xmath44 to check the convergence of the lyapunov exponents and the correlation functions , respectively . different lattice sizes were used depending on which order parameters were being studied , typically @xmath45 for calculating lyapunov exponents ( which are quite costly to compute ) and @xmath46 for estimating correlation lengths . many runs were repeated using several lattice sizes to verify the absence of significant finite - size effects . for calculations of statistical averages such as the magnetization and correlation functions , the statistics could often be improved substantially by averaging the results of an ensemble of @xmath47 runs each of duration @xmath39 , with each run differing only in the choice of initial conditions . calculations indicate that this ensemble average is ergodically equivalent to a single integration of duration @xmath48 @xcite . for most of the results reported below , we used an ensemble average over @xmath49 runs which could be executed simultaneously and in parallel on the vector units of a 16-node partition of a cm-5 computer . some dynamical order parameters can be constructed by combining in different ways the lyapunov exponents @xmath50 associated with a given attractor @xcite . for the cml given by eq . ( [ eq : the - cml ] ) with a total of @xmath51 lattice sites , there are @xmath47 real - valued lyapunov exponents @xmath50 ( labeled in decreasing order @xmath52 ) that characterize the long - time average - rate - of - separation of nearby orbits in phase space . from the @xmath50 , we can calculate a lyapunov dimension @xmath3 given by the kaplan - yorke formula @xcite : @xmath53 and calculate an entropy defined by the sum of the positive exponents @xcite : @xmath54 the number @xmath55 in eq . ( [ eq : dim - defn ] ) is the largest integer such that the sum @xmath56 of the first @xmath55 exponents is nonnegative ; this sum is positive for @xmath57 if an orbit is chaotic ( @xmath58 ) and is negative for @xmath59 if the dynamics is dissipative and so the sum typically crosses zero at an intermediate index @xmath60 for a chaotic dissipative system . the exponents @xmath50 were calculated numerically by a now - standard numerical method @xcite , in which @xmath61 linearizations of the equations of motion , eq . ( [ eq : the - cml ] ) , are evolved in time . this allows one to follow @xmath55 lyapunov vectors in a tangent space from which local stretching information and the lyapunov exponents can be extracted . together with a particular nonlinear orbit defined by the equations of motion , a total of @xmath62 cmls was evolved to calculate @xmath55 lyapunov exponents . repeated orthonormalizations of lyapunov vectors at time intervals @xmath63 are needed to prevent floating - point overflow from the exponentially growing values and to prevent inaccuracies arising from the loss of linear independence as they fold up along the direction of the fasting growing exponent @xcite . for the maps eqs . ( [ eq : miller - huse - map ] ) and ( [ eq : slope-1.1-map ] ) , we found empirically that values @xmath64 gave reasonable results for all lattices studied with the largest value of @xmath63 depending on the parameters @xmath1 and @xmath5 . smaller renormalization times did not change the values of the lyapunov spectrum ( although the code was more costly to run ) while larger values led to serious errors due to linear dependence of the lyapunov vectors . the orthonormalizations of tangent vectors consumed most of the computing time on a thinking machines cm-5 parallel computer . the orthonormalizations require substantial communication between different processors since each processor evolves independently only a few of the @xmath55 linearized equations in its own local memory ; this communication decreases the efficiency of the code . although the communication inherent in the orthonormalization procedure could not be avoided , in all other portions of the code communication between nodes was eliminated by iterating redundantly an identical copy of the nonlinear cml eq . ( [ eq : the - cml ] ) with identical initial conditions on each processor . in this way , information about the underlying orbit ( needed when iterating the linearized cmls ) did not have to be communicated from one processor to all others at each time step . by monitoring the lyapunov exponents and the lyapunov dimension as a function of time @xmath29 , we found empirically that an integration time @xmath65 time steps gave an acceptable relative accuracy of better than one percent for calculating the dimension @xmath3 and entropy @xmath66 for all lattices studied ( @xmath67 ) . [ fig : dim - vs - time ] shows how the dimension @xmath3 converges over time for lattice size @xmath68 and for @xmath69 . the dimension curve is noisy with fluctuations that diminish slowly over time ( the @xmath50 , not shown , have substantially noiser time series ) . goldhirsch et al @xcite have argued that the amplitude of the exponent fluctuations should decay as @xmath70 where @xmath39 is the total integration time and so one could fit and extrapolate to get a better estimate @xcite . extrapolation was not needed in plots like fig . [ fig : dim - vs - time ] which already give an adequate relative accuracy of better than one percent . by repeating plots such as fig . [ fig : dim - vs - time ] for different system sizes @xmath5 with all other parameters held fixed , we found extensive scaling of the dimension @xmath3 with the volume of the system @xmath71 for a wide range of parameter values @xmath1 , an example of which is given in fig . [ fig : ext - chaos ] . fig . [ fig : ext - chaos](a ) shows that the dimension @xmath3 increases linearly and extensively with @xmath47 beyond a system size of about @xmath72 . the lyapunov dimension density @xmath19 could then be obtained from the slope of a least - squares fitted line in the extensive region . the intercept of a least - squares - fitted line through the four right - most points was @xmath73 which is quite small ( and is also approximately zero for extensively chaotic solutions of the 1d cgle @xcite ) . there is then the possibility that a single dimension calculation for a sufficiently large system may suffice to estimate its dimension density . by comparing the dimension per volume @xmath74 with the dimension density @xmath19 , fig . [ fig : ext - chaos](b ) shows how the extensive regime is approached rapidly and achieved for fairly small system sizes @xmath75 . we finish this subsection with two comments about the meaning of the lyapunov dimension @xmath3 and about why we chose to calculate @xmath3 from lyapunov vectors rather than from time series measurements . the reader should recall that there is an infinity of fractal dimensions @xmath76 ( often called the renyi dimensions ) associated with a strange attractor with the parameter @xmath77 varying over the real numbers @xcite . since each dimension @xmath76 will be extensive for a homogeneous extensively chaotic system , the particular values only reflect the system size and are not interesting themselves . instead , one should define a continuum of corresponding intensive dimension densities @xmath78 to provide a partial characterization of such systems . for large fractal dimensions ( @xmath79 ) , present computers and algorithms only allow the calculation of the lyapunov fractal dimension eq . ( [ eq : dim - defn ] ) and its corresponding density . the lyapunov dimension is conjectured to be the same as the renyi dimension with @xmath80 ( i.e. , the information dimension @xmath81 @xcite ) and it is not known to what extent the corresponding density @xmath82 characterizes the unknown function of densities @xmath83 , e.g. , whether it is close to the mean value of the @xmath83 . there are one - dimensional maps for which the ratio of @xmath81 to @xmath84 ( the two most commonly calculated fractal dimensions ) can be arbitrarily large @xcite , and so the variation of the function @xmath83 around its mean value can be large . a perhaps even more important question is whether the different dimension densities @xmath83 each have a similar dependence on model parameters , e.g. , all increasing or decreasing together . until this issue is resolved , the dimension density @xmath82 and the corresponding length @xmath85 need to be interpreted with caution . the dynamical quantities @xmath3 and @xmath66 are calculated in terms of the lyapunov spectrum @xmath50 , and not in terms of time series @xmath86 at a given lattice site @xmath28 , because of the impractical computational demands of time series algorithms @xcite . while the computational complexity of the method based on the lyapunov spectrum scales algebraically with system volume @xmath87 or dimension @xmath3 @xcite , it is well known that the computational complexity of classical time series algorithms such as that proposed by procaccia and grassberger grows _ exponentially _ with @xmath3 , imposing severe restrictions on the largest dimension that can be estimated from experimental data . although it remains controversial what is the practical upper bound for clean time series of less than a million points ( there are claims from 6 to 20 @xcite ) , existing time series methods can not treat extensively chaotic systems whose fractal dimensions may be in the hundreds ( see fig . [ fig : ext - chaos ] ) . in addition to the dynamical quantities described in the previous section , we quantify the evolution of the cmls by a `` magnetization '' @xmath88 and by length scales measured from two - point and mutual information correlation functions . we describe these briefly to indicate the method and errors involved . following miller and huse @xcite , the average magnetization @xmath88 of the cml is defined by a space - time and ensemble average of the signs @xmath89 of the lattice values @xmath27 , @xmath90 where the index @xmath91 labels a particular cml running on processor @xmath91 . an average over 64 independent cmls running on a 16-node partition of a cm-5 was typically used . for the local maps ( [ eq : miller - huse - map ] ) and eq . ( [ eq : slope-1.1-map ] ) , @xmath88 undergoes a pitchfork bifurcation from a zero to finite value as the lattice coupling @xmath1 is increased from small values . the critical value @xmath92 at which @xmath88 bifurcates to a nonzero value is unchanged if the values of the site variables @xmath86 are used instead of their signs in eq . ( [ eq : magnetization - defn ] ) . the bifurcation of @xmath88 to a nonzero value defines the onset of ferromagnetic order at @xmath12 @xcite as illustrated in fig . [ fig : m - for - square - lattice ] . to characterize the average spatial disorder , we examined two of many possible definitions of spatial correlation lengths , one from an exponentially decaying two - point correlation function , another from an exponentially decaying mutual information function @xcite . ( some other correlation lengths are discussed on pages 945 - 947 of ref . @xcite . ) the two - point correlation function was defined in the usual way : @xmath93 where @xmath94 denotes the position of lattice point @xmath28 and where the brackets @xmath95 denote the averaging process of eq . ( [ eq : magnetization - defn ] ) . given the periodicity of the lattice and the availability of efficient parallel fast fourier transforms ( ffts ) on the cm-5 , we calculated eq . ( [ eq:2-pnt - correlation - fn ] ) via the wiener - khintchin theorem @xcite , first obtaining the time - averaged magnitude squared of the fourier coefficients , from which eq . ( [ eq:2-pnt - correlation - fn ] ) was obtained by an inverse fft . in most cases , there was a substantial region of exponential decay from which the two - point correlation length @xmath13 was obtained by a least - squares fit of the form @xmath96 ; a representative plot is given in fig . [ fig:2-point - corr - fn ] . the correlation functions and corresponding values of @xmath13 do not change if eq . ( [ eq:2-pnt - correlation - fn ] ) is defined in terms of the sign of the variables , @xmath97 . the two - point correlation functions decay more rapidly as shown in fig . [ fig:2-point - corr - mag - fn ] if the signs of the field values @xmath98 in eq . ( [ eq:2-pnt - correlation - fn ] ) are replaced by their magnitudes @xmath99 . the correlation length @xmath100 obtained from the initial rapid exponential decay is approximately one lattice spacing and changes little when the coupling constant @xmath1 is varied over a large range , including near the bifurcation point @xmath12 . for dimensionality @xmath101 , we show below that this short length scale @xmath16 is related to the dimension correlation length @xmath0 and that both vary linearly with the radius @xmath15 of neighboring lattice sites that are coupled together spatially . as a possible alternative for characterizing the spatial disorder of a nonlinear system , we also calculated a correlation length @xmath102 based on the exponential decay of the mutual information function @xmath103 @xcite of the variables @xmath86 . [ fig : mutual - info - decay ] shows the exponential decay of a mutual information function @xmath104 for two - dimensional square lattice of size @xmath105 and for the parameter @xmath106 . again if the magnitudes of the field variables are used when calculating @xmath103 , the exponential decay is much more rapid , with @xmath107 being approximately one lattice spacing for a wide range in @xmath1 . although there is not yet a compelling theoretical reason to prefer @xmath102 over other correlation lengths such as @xmath13 @xcite , the former has the distinction of depending nonlinearly on the dynamics and so may depend on details that are missed by @xmath13 . for this reason , an increasing number of scientists have reported correlation lengths in terms of @xmath102 @xcite . as shown in fig . [ fig : mi - vs-2-point ] for the two - dimensional miller - huse cml on a square lattice , the length scales @xmath13 and @xmath102 are linearly related over a substantial dynamical range near the ferromagnetic transition . at least for the present models , these lengths are equivalent measures of spatial disorder and we report values only for @xmath13 below . in this section , we discuss our calculations of the lyapunov spectrum and of correlation lengths . our goal is to explore how spatial disorder ( as characterized by the two - point correlation length or by the mutual information correlation length ) is related to dynamical complexity ( as measured by the intensive dimension density and by the dimension correlation length eq . ( [ xi - delta - defn ] ) ) and to investigate how these order parameters vary near the nonequilibrium transition point @xmath12 at which the magnetization bifurcates to nonzero values ( fig . [ fig : m - for - square - lattice ] ) . results for the 2d square lattice are given first , followed by results for lattices with different symmetries and dimensionalities . we do not address issues related to critical exponents of these different models which have been discussed by miller and huse @xcite and more recently by marcq and chat @xcite . related interesting results on similar cmls have also recently been reported by boldrighini et al @xcite . for the two - dimensional periodic square lattice with map eq . ( [ eq : miller - huse - map ] ) , fig . [ fig : m - for - square - lattice ] shows that there is a bifurcation at @xmath108 . this bifurcation corresponds to the onset of long - range order of the lattice variables @xmath86 as demonstrated by the divergence of the two - point correlation length @xmath13 as @xmath1 approaches @xmath92 ( fig . [ fig : divergence - of - xi_2](a ) ) . over this same parameter range , the dimension correlation length @xmath0 varies smoothly ( fig . [ fig : divergence - of - xi_2](b ) ) , deviating by less than four percent from a value of one lattice spacing and attaining a maximum value close to where the correlation length diverges . the lyapunov spectrum of exponents also varies smoothly from one side of the transition to the other as shown in fig . [ fig : lyapunov - spectra ] . we conclude that chaotic fluctuations have a short range , are decoupled from the onset of long - range order measured by @xmath13 , and that the spectrum of exponents is at most weakly dependent on the onset of long - range spatial order . to understand further how various dynamical quantities change near the transition point , we have plotted in fig . [ fig : d - h - l1-vs - g-2d - square ] the variation of the lyapunov fractal dimension density @xmath19 , of the metric entropy density @xmath20 , and of the largest lyapunov exponent @xmath17 across the ferromagnetic transition for a lattice of size @xmath68 , which is already extensively chaotic according to fig . [ fig : ext - chaos ] . as was the case for the length @xmath0 in fig . [ fig : divergence - of - xi_2](b ) , these quantities change by only a small amount through the transition ( at most by 20% ) and all go through a minimum close to , but distinct from , the ferromagnetic transition at @xmath109 . this result was surprising to us since one consequence of coupling neighboring maps more strongly ( increasing the parameter @xmath1 ) would intuitively be to increase correlations between their dynamics , which should decrease both @xmath19 and @xmath20 . [ fig : d - h - l1-vs - g-2d - square](a ) indicates that roughly one quarter of the maximum number of degrees of freedom disappear when the lattice attains its minimum dimension density of @xmath110 . ( an upper bound of @xmath111 is set by the integer lattice spacing . ) it is not clear why the fractal dimension density and other dynamical quantities have extrema near @xmath112 . for an equilibrium non - dissipative cml of ising dynamics on a two - dimensional square lattice , sakaguchi @xcite did not find a local minimum in the entropy @xmath66 but instead found a monotonic decrease consistent with the analytical solution of the spin-@xmath113 ising model on a square lattice @xcite . one explanation for the extrema may be that the dissipative linear coupling in eq . ( [ eq : the - cml ] ) becomes antidiffusive for @xmath114 , enhancing rather than damping short - wavelength fluctuations and so decorrelating nearby lattice variables . the issue is somewhat more subtle than this because the existence of the minimum depends also on details of the local map @xmath25 in eq . ( [ eq : the - cml ] ) . for the less chaotic lattice map eq . ( [ eq : slope-1.1-map ] ) with slope of constant magnitude 1.1 , an ising - like transition still occurs as shown by the bifurcation of the magnetization near @xmath115 in fig . [ fig : m - for - square - lattice - map2 ] . fig . [ fig : d - h - l1-for - square - lattice - map2 ] now shows that the lyapunov dimension density @xmath19 and entropy density @xmath20 decrease monotonically as the parameter @xmath1 is increased , with the largest exponent @xmath17 remaining constant . figures [ fig : m - for - square - lattice ] , [ fig : divergence - of - xi_2](a ) , and [ fig : d - h - l1-vs - g-2d - square ] suggest that the extrema of dynamical quantities may be related to the ferromagnetic transition . on the other hand , the short dimension correlation length in fig . [ fig : divergence - of - xi_2](b ) contradicts this by implying that chaotic fluctuations occur over a length scale that is short compared to the ferromagnetic ordering . to understand this further , we have explored cmls of different symmetry and dimensionality and found that the near - proximity of the extrema with the transition is a coincidence for the two - dimensional lattice with square symmetry . more generally , the positions of extrema seem to be determined simply by the number of nearest neighbors @xmath22 , and not by the symmetry or dimensionality of the cml or by the position of the magnetization bifurcation point . [ fig : m - for - hexagonal - lattice - map1 ] summarizes calculations for a two - dimensional periodic hexagonal lattice by plotting the dependence of magnetization @xmath88 and of dimension correlation length @xmath0 on the coupling constant @xmath1 . the magnetization bifurcates to a nonzero value at @xmath116 which is a smaller value than that for the square lattice since the larger number of nearest neighbors ( six versus four ) increases the effective strength of the diffusive coupling . the relative difference between the transition at @xmath12 and the positions of the extrema in @xmath0 and in related dynamical quantities is substantially larger than was the case for the 2d square lattice . for the hexagonal lattice , extrema in quantities like the length @xmath0 occur at a value close to @xmath117 where @xmath24 is the number of nearest neighbors . a similar result is found for the same cml on a 3d cubic lattice , as shown in fig . [ fig : m - for - cubic - lattice - map1 ] . the transition at @xmath118 occurs at a value close to but smaller than the value on the hexagonal lattice . extrema in the dynamical quantities like @xmath0 again occur at a value close to @xmath119 with @xmath24 . that the positions of the extrema of dynamical quantities is dependent primarily on the number of nearest neighbors is a consequence of the nearest - neighbor diffusive coupling in eq . ( [ eq : the - cml ] ) and of the fact that , for the local map eq . ( [ eq : miller - huse - map ] ) , the chaos is sufficiently strong to make the dimension correlation length @xmath0 quite small , about one lattice length . that the positions of the extrema seem to be given quantitatively by the specific formula @xmath120 is more delicate to understand but may be related to the fact that the discrete laplacian operator changes from diffusive to antidiffusive behavior at this value . the value @xmath120 is the value for which the weight of each of the neighbors is equal to the weight of the central lattice site to be updated . for all cmls that we studied , the dimension correlation length @xmath0 was about one lattice spacing in size and this was also the `` radius '' of the diffusive coupling in eq . ( [ eq : the - cml ] ) . this suggests that the length @xmath0 may be determined by the spatial extent of the diffusive coupling , becoming larger as more sites are coupled to a given site . this conjecture was tested in one - space dimension by coupling together , with equal weight @xmath1 , all lattice variables within a radius @xmath15 of a given site @xmath28 : @xmath121 for this one - dimensional periodic cml with the lattice map eq . ( [ eq : miller - huse - map ] ) , the dynamics varies in a complicated way with radius @xmath15 . for most initial conditions , chaos was found for smaller radii ( @xmath122 ) . for larger radii @xmath123 , the transients lasted much longer and the asymptotic dynamics was periodic . as an example , for @xmath124 the dimension as a function of time initially reached a value @xmath125 even after @xmath126 transient iterations were skipped ; however , the dimension then decreased steadily to zero over the next 30,000 iterations . we believe that this asymptotic periodic behavior is a finite - size effect . for a sufficiently large system size @xmath5 , with the crossover length increasing with the radius @xmath15 , the asymptotic state should be chaotic . for dimensionality @xmath101 , the dimension correlation length @xmath0 varies more strongly with increasing radius @xmath15 than with coupling constant @xmath1 , which allows several different length scales to be compared . [ fig:1d - cml - results ] shows that the two - point correlation length @xmath100 obtained using the magnitudes @xmath127 of the field values has approximately the same linear dependence on the coupling radius @xmath15 as the dimension correlation length @xmath0 . in addition , these two length scales are the same order of magnitude . the two - point correlation length @xmath13 obtained using the actual field values is larger than @xmath0 and does not have the same simple linear dependence on @xmath15 . the situation is then similar to results found for spatiotemporal chaotic solutions of the 1d periodic cgle @xcite in that the spatial correlation length of fluctuations in the magnitude of a field provides a way of estimating the length @xmath0 . in this paper , we have extended recent calculations @xcite concerning the relation between spatial disorder and dynamical complexity of a sustained homogeneous nonequilibrium system from dimensionality @xmath101 to dimensionalities @xmath10 and @xmath11 . this was accomplished by choosing a coupled map lattice , eq . ( [ eq : the - cml ] ) , that underwent an ising - like transition with diverging two - point correlation length as a parameter @xmath1 was varied @xcite . by comparing various length scales such as the two - point correlation length @xmath13 , the dimension correlation length @xmath0 , and the two - point correlation length of magnitude fluctuations @xmath14 near the transition point , we were able to show that the lengths @xmath0 and @xmath14 were short , of order one lattice spacing , even as the length @xmath13 diverged to infinity . in agreement with previous work @xcite , the chaotic fluctuations are decoupled from the average long - range spatial order . the correlation length of magnitude fluctuations seems to provide an effective way to estimate the size of the length @xmath0 . our calculations of the lyapunov spectrum and related quantities such as the lyapunov dimension density @xmath19 , metric entropy density @xmath20 , and the largest lyapunov exponent @xmath17 show that the onset of long - range spatial order ( diverging @xmath13 ) does not affect dynamical invariants , which vary smoothly and weakly through the transition point @xmath12 . thus the average spatial disorder ( measured by @xmath13 ) does not determine dynamical complexity ( measured by @xmath0 ) . rather surprisingly , the intensive densities @xmath19 and @xmath20 go through a minimum near the transition point so that the onset of long - range order does not correspond to a decrease in complexity . by examining cmls of different symmetry and of different dimensionality @xmath6 , we showed that the positions @xmath1 of the extrema were determined by the number @xmath22 of neighbors nearest to a given lattice site ( with @xmath128 ) but not by the symmetry or by @xmath6 . this result can be understood as a consequence of the extremely short dimension correlation length @xmath0 , about one lattice size , so that lattice variables are independent except when they are nearest neighbors . we believe that the minima in @xmath19 and in @xmath20 occur approximately when the discrete laplacian in eq . ( [ eq : the - cml ] ) becomes antidiffusive with increasing parameter @xmath1 . short - wavelength fluctuations are then magnified instead of damped , decreasing correlations between neighboring sites . some of our results concerning extrema in dynamical quantities have been independently obtained by boldrighini et al @xcite although these authors worked with extensive , rather than intensive , quantities and they did not determine whether their calculations corresponded to extensively chaotic regimes . boldrighini et al investigated cmls of the form eq . ( [ eq : the - cml ] ) for dimensionalities @xmath101 and @xmath10 but with some new local maps @xmath25 . besides also finding extrema in the lyapunov fractal dimension , boldrighini et al showed that the odd symmetry of the map @xmath25 was not a sufficient condition for the magnetization to bifurcate to a nonzero value . using a strongly chaotic local map with slope of constant magnitude equal to 5 , they also showed that the magnetization @xmath88 did not bifurcate to a nonzero value if the local map were made sufficiently chaotic compared to the ordering caused by diffusion . in section [ other - lattices ] , we used the less - chaotic local map eq . ( [ eq : slope-1.1-map ] ) with slope of constant magnitude 1.1 to show that the dimension density @xmath19 can decrease monotonically without a minimum even when the magnetization @xmath88 bifurcates to a nonzero value . the dependence of these minima on details of the local map is not yet understood and should be pursued with further studies . the small values of @xmath0 in the present cmls , in the 1d cgle @xcite , and in a nonequilibrium cml with algebraic decay of spatial correlations @xcite have several interesting implications . one is that many previous laboratory experiments @xcite and numerical simulations @xcite concerning spatiotemporal chaos are likely already extensive so that it is meaningful to talk about dimension and entropy densities ( see also the earlier paper by bohr @xcite . ) a second implication is that the dimension correlation length @xmath0 may not be a useful order parameter for future studies of spatiotemporal chaos since it depends only weakly on parameters . a third implication is that the short value of @xmath0 suggests the nonexistence of macroscopic chaotic states for dynamics with local interactions , a point already made by several researchers @xcite . finally , we speculate that @xmath0 is the length scale below which one can replace chaotic fluctuations by a white noise source when trying to develop a hydrodynamic ( long wavelength ) description of spatiotemporal chaos ( see the discussion on pages 953 - 954 in ref . @xcite ) . the short values of @xmath0 raise the question of what determines this length scale . our calculations on the 1d cml eq . ( [ eq : radius - r - cml ] ) with a variable radius of coupling @xmath15 suggest that the length @xmath0 is determined partly by the length @xmath14 characterizing magnitude fluctuations although the reason for and the generality of this correspondence is not understood @xcite . the length @xmath0 is also related to the radius @xmath15 over which nearby lattice sites are coupled together ( fig . [ fig:1d - cml - results ] ) . further calculations with different kinds local maps and of diffusive operators and for different values of @xmath15 should provide further insight . it is appropriate to finish with a discussion about the relevance of these results for laboratory experiments . as discussed at the end of section [ exponents ] , it does not seem possible in the near future to calculate the lyapunov spectrum , the fractal dimension , or the fractal dimension density of a high - dimensional extensively chaotic experimental system for which only time series measurements are available @xcite . our success in calculating these quantities was a result of having explicit knowledge of the dynamical equations which could then be integrated numerically on a powerful parallel computer using algorithms whose complexity only grew algebraically with the dimension @xmath3 @xcite . for many laboratory experiments , a quantitative mathematical description is either lacking ( e.g. , for chemical reactions ) or , if known , is too difficult to work with numerically ( e.g. , the five three - dimensional boussinesq equations describing buoyancy - induced convection in a large - aspect - ratio container @xcite ) . our calculations in section [ results ] suggest that one possible way to estimate the dimension correlation length @xmath0 may be to calculate the correlation length of some function of the physical fields , e.g. , the field magnitude . another possibility will be to discover and to validate algorithms that can calculate the intensive dimension density eq . ( [ delta - defn ] ) directly from time - series measurements that are localized in space @xcite , in lieu of calculating a large extensive fractal dimension @xmath3 and then dividing by the extensive system volume @xmath87 . several steps have been taken in this direction @xcite , but a theoretical foundation has not yet been established nor have the numerical algorithms been adequately tested . we thank l. bunimovich , h. chat , p. hohenberg , d. huse , h. riecke , and j. socolar for useful discussions . this work was supported by grants nsf - cda-91 - 23483 and nsf - dms-93 - 07893 of the national science foundation , by grant doe - de - fg05 - 94er25214 of the department of energy and by an allotment of cm-5 computer time at the national center for supercomputing applications . the second author ( d.a.e . ) would like to thank the office of naval research for fellowship support . much of the research reported here was carried out by the first author as part of his undergraduate honors thesis in physics at duke university . j. p. gollub and r. ramshankar , in _ new perspectives in turbulence _ , edited by s. orszag and l. sirovich ( springer - verlag , berlin , 1990 ) , pp.165194 ; s. w. morris , e. bodenschatz , d. s. cannell , and g. ahlers , phys . * 71 * , 2026 ( 1993 ) ; m. assenheimer and v. steinberg , nature * 367 * , 345 ( 1994 ) ; f. t. arecchi _ et al . _ , physica d * 61 * , 25 ( 1992 ) ; q. ouyang and h. l.swinney , chaos * 1 * , 411 ( 1991 ) . b. i. shraiman _ et al . _ , physica d * 57 * , 241 ( 1992 ) ; h .- w . xi , j. d. gunton , and j. vials , phys . lett . * 71 * , 2030 ( 1993 ) ; j. e. pearson , science * 261 * , 189 ( 1993 ) ; d. a. egolf and h. s. greenside , nature * 369 * , 129 ( 1994 ) ; w. decker , w. pesch , and a. weber , phys . lett . , 648 ( 1994 ) ; m. cross , d. meiron , and y. tu , chaos * 4 * , ( 1994 ) . s. w. morris , e. bodenschatz , d. s. cannell , and g. ahlers , phys . * 71 * , 2026 ( 1993 ) ; m. assenheimer and v. steinberg , phys . rev . lett . * 70 * , 3888 ( 1993 ) ; w. decker , w. pesch , and a. weber , phys . lett . * 73 * , 648 ( 1994 ) . h. chat , in _ spatio - temporal patterns in nonequilibrium complex systems _ , vol . xxi of _ santa fe institute studies in the science of complexity _ , santa fe institute , edited by p. e. cladis and p. palffy - muhoray ( addison - wesley , reading , massachusetts , 1995 ) , pp . 3350 . for a numerical dynamical model with @xmath47 independent variables , an optimal integration scheme will advance the variables in one time step with a computational effort that grows at best as slowly as @xmath129 . each linearized equation also involves @xmath47 variables and so the effort needed to calculate @xmath55 lyapunov exponents by the algorithm discussed in section [ methods ] over a fixed time time @xmath39 will grow at best as slowly as @xmath130 . since the number of exponents needed to estimate the lyapunov dimension grows linearly with system volume , as does the number @xmath47 itself , we conclude that the computational effort to calculate the lyapunov dimension for an extensively chaotic system , for an observation time @xmath39 , will grow at best as slowly as @xmath131 , i.e. , algebraically with @xmath47 . it is presently not understood how the total integration time @xmath39 should itself scale with system volume or with @xmath47 and so the exponent of the algebraic scaling with dimension @xmath3 is not yet known @xcite . y. pomeau , acad . paris * 300 * , ser . ii , 239 ( 1985 ) ; p. grassberger phys . scripta * 40 * , 346 ( 1989 ) ; a. torcini , a. politi , g. p. puccioni , and g. dalessendro , physica d * 53 * , 85 ( 1991 ) ; l. s.tsimring , phys . e * 48 * , 3421 ( 1993 ) ; m. bauer , h. heng , and w. martienssen , phys . rev.lett . * 71 * , 521 ( 1993 ) .
|
by simulating a nonequilibrium coupled map lattice that undergoes an ising - like phase transition , we show that the lyapunov spectrum and related dynamical quantities such as the dimension correlation length @xmath0 are insensitive to the onset of long - range ferromagnetic order . as a function of lattice coupling constant @xmath1 and for certain lattice maps , the lyapunov dimension density and other dynamical order parameters go through a minimum .
the occurrence of this minimum as a function of @xmath1 depends on the number of nearest neighbors of a lattice point but not on the lattice symmetry , on the lattice dimensionality or on the position of the ising - like transition . in one - space dimension , the spatial correlation length associated with magnitude fluctuations and the length @xmath0 are approximately equal , with both varying linearly with the radius of the lattice coupling .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
systems of interacting open- and closed - strings play important roles in several aspects of string theory . it has been found out that theories even formulated as pure closed - strings have open - string sectors when they are accompanied by d - branes @xcite . one of the most important features of such open - closed mixed systems is a duality between open- and closed - strings . this duality becomes manifest , for instance , by seeing one - loop diagrams of open - string as tree propagations of closed - string through modular transformations on the string world - sheets . in the systems of d - branes , this duality should be a rationale for correspondence between gauge theory on the world - volumes ( open - string sector ) and gravity theory in the bulk space - time ( closed - string sector ) . ads / cft correspondence @xcite@xcite@xcite may be regarded as one of the most remarkable examples of such bulk - boundary correspondence . this viewpoint was emphasized also in @xcite . nevertheless it becomes very difficult to establish the correspondence between the two . this is essentially because light particles ( ir effects ) in the one sector are realized by summing up whole massive towers ( uv effects ) in the other sector . when a constant @xmath0 field background of closed - string is turned on , however , we expect that the situation could be drastically changed . on this background , the world - volumes of d - branes become non - commutative @xcite@xcite@xcite@xcite@xcite . low - energy effective theories of the open - string sector , therefore , become field theories on the non - commutative world - volumes . it was pointed out @xcite that in such theories there happens a mixing of the uv and the ir . this uv / ir mixing gives @xcite us a chance to capture some effects of light particles in the bulk gravity theories , e.g. gravitons , by investigating non - commutative field theories on the world - volumes . to pursue such possibility , non - planar one - loop amplitudes of open - string were studied on this background @xcite@xcite@xcite@xcite@xcite . higher loops were investigated as well in @xcite . coupling of non - commutative d - branes to closed - string in the bulk has attracted much interest particularly from the viewpoint of bulk - boundary correspondence @xcite@xcite . it was pointed out in @xcite@xcite that the generalized star - products arise in disk amplitudes consisting of a closed - string vertex operator and open - string vertex ones on a constant @xmath0 field background . the generalized star products are also found to appear in straight open wilson lines by the expansions in powers of non - commutative gauge fields @xcite . these observations were combined in @xcite and it was shown there that in the zero - slope limit of @xcite disk amplitudes of a closed - string tachyon and arbitrary numbers of gauge fields on this background give rise to a straight open wilson line . thus it has been clarified that open wilson lines play a role in the correspondence between the bulk gravity and the non - commutative gauge theories . open wilson lines found in @xcite are remarkable gauge invariant objects in non - commutative gauge theories . gauge invariant operators in non - commutative gauge theories can be constructed in the forms of local operators smeared along straight wilson lines @xcite@xcite@xcite . it was also shown in @xcite that coupling of closed - string graviton to non - commutative gauge theories actually is an operator of this form . open wilson lines need not be straight in order to be gauge invariant . taking account of the above role played by straight open wilson lines in the bulk - boundary duality , dhar and kitazawa @xcite conjectured that curved open wilson lines fluctuating around a straight path should be brought about by higher level ( stringy ) states of closed - string . they showed that the coupling of graviton state is in fact the gauge invariant operator which appears in the leading coefficients in harmonic expansions of curved open wilson lines around a straight path . it is well - known that the so - called boundary states @xcite@xcite@xcite provide a description of d - branes in closed - string theory . open- and closed - strings interact on the branes in space - time . therefore it is further expected that boundary states admit to describe these interactions . in this article , receiving the above development and understanding , we study the system of a d - brane in bosonic string theory on a constant @xmath0 field background . in order to obtain further insight into the bulk - boundary duality , we first exploit the boundary state formalism to include states which describe arbitrary numbers of open - string tachyons and gluons . open - string legs of these boundary states , that is , tachyons and gluons , need not to be on - shell . thereby it becomes possible to study the duality . on a constant @xmath0 field background , boundary state without open - string legs has been constructed in @xcite@xcite@xcite . it is briefly reviewed in the next section . let us denote the state by @xmath1 . this state does not lead any couplings of a closed - string to open - string excitations on the world - volume . we may regard @xmath1 as a perturbative vacuum of closed - string in the presence of the brane . it is also possible to interpret the state as a bogolubov transform of the @xmath2-invariant vacuum of closed - string : @xmath3 , where @xmath4 is a suitable generator of the transformation . in section [ sec : tachyon boundary state ] we construct boundary states of open - string tachyons . basic idea of the construction may be explained as follows . we first examine an insertion of a closed - string tachyon into the brane . let us suppose that the closed - string world - sheet is an infinite semi - cylinder @xmath5 with @xmath6 and that the brane or the state @xmath1 is located at @xmath7 . we may describe the insertion by @xmath8 , where @xmath9 is a closed - string tachyon vertex operator . however , there occurs a singularity within @xmath9 at the world - sheet boundary . this originates in correlations between the chiral and the anti - chiral sectors which are caused by the brane . we then regularize such a singularity and define a renormalized tachyon vertex operator @xmath10 . it turns out surprisingly that the renormalized operator becomes an open - string tachyon vertex at the boundary . in our prescription of the renormalization , disk green s function and the transform @xmath11 play essential roles . boundary state of a single open - string tachyon is given by @xmath12 . boundary states of arbitrary numbers of open - string tachyons turn out to be realized by the successive insertions of the above renormalized operators . as for boundary states of gluons , they can be obtained by taking the same steps as illustrated above , except that we need to introduce a suitable local operator of closed - string in place of @xmath9 and renormalize it in a suitable fashion . these are presented in section [ sec : gluon boundary state ] . consistency of the constructions is examined from several aspects at some length . in particular , we compute closed - string tree propagations between these boundary states and make sure that they reproduce the corresponding open - string one - loop amplitudes . these are presented separately in section [ sec : tachyon boundary state ] for the tachyons and in section [ sec : gluon boundary state ] for the gluons . it is also worth noting that the present constructions are relevant on a vanishing @xmath0 field background . low - energy description of world - volume theories of d - branes is obtained by taking a zero - slope limit @xmath13 so that it makes all perturbative stringy states ( @xmath14 ) of open - string infinitely heavy and decouple from the light states . as regards one - loop amplitudes of open - string , this can be achieved by taking the limit with @xmath15 fixed . here @xmath16 is the proper time on the world - sheet of open - string . the fixed parameter @xmath17 becomes the schwinger parameter of one - loop amplitudes of low - energy effective world - volume theories . it is shown @xcite@xcite@xcite@xcite@xcite@xcite that one - loop amplitudes of open - string tachyons on a constant @xmath0 field background reduce to those of a non - commutative ( tachyonic ) scalar field theory on the world - volume . the above zero - slope limit can be interpreted as magnification of the string amplitudes in the vicinity of @xmath18 . in section [ sec : uv nc scalar ] we investigate a possible uv behavior of this non - commutative field theory . in order to know the uv behavior one needs to focus on the region @xmath19 . for the description we take the following route . we introduce a parameter @xmath20 , where @xmath21 is the closed - string proper time . @xmath22 and @xmath23 are related to each other through the modular transformation of the world - sheet by @xmath24 . then we take a zero - slope limit with @xmath23 fixed . existence of @xmath0 field allows us to make the limit slightly different from the standard field theory limit of closed - string ( gravity limit ) . this enables us to capture the world - volume theories . this zero - slope limit is magnification of the string amplitudes in the vicinity of @xmath25 and hence the region @xmath19 . strictly speaking , this is a trans - string scale of the world - volume theories . generating function of one - loop amplitudes of open - string tachyons is found out to be factorized at the limit into two ( analogues of ) straight open wilson lines exchanging closed - string tachyons . in general , one - loop amplitudes of open - string are factorized by a tower of closed - string states . in gravity limit the propagations of closed - string tachyons become dominant . although the present limit is slightly different from gravity limit , these zero - slope limits share the common property . uv behavior of the non - commutative gauge theory is explored in section [ sec : uv nc gauge ] by following the same route . generating function of one - loop amplitudes of gluons is also found out to be factorized at this limit into two straight open wilson lines exchanging closed - string tachyons . these analyses indicate a possibility that field theories on the non - commutative world - volume become topological at such a trans - string scale of the world - volume . the conjecture @xcite leads us to expect that curved open wilson lines somehow factorize generating functions of one - loop amplitudes of gluons or open - string tachyons . the standard factorization made by particle states of closed - string seems to be little use . instead , momentum eigenstates of closed - string are used for the factorization . closed - string momentum @xmath26 is a momentum loop ( a loop in the momentum space ) while closed - string coordinate @xmath27 is a space - time loop . section [ sec : comments on eigenstates ] is devoted to introduction of coordinate and momentum eigenstates of closed - string . we also provide some observations on their relations with boundary states . factorizations made by the momentum eigenstates are examined in section [ sec : open wilson lines ( i ) ] for the tachyons and in section [ sec : open wilson lines ( ii ) ] for the gluons . it turns out that in the zero - slope limit ( the previous uv limit of the world - volume theories ) momentum loops become curves in the non - commutative world - volume and that open wilson lines along these curves appear in the factorizations . the generating functions are factorized by two curved open wilson lines . more precisely , they become integrals on the space of curves with a gaussian distribution around straight lines . as regards fluctuations from the straight lines , width of the distribution becomes so sharp that the integrals reduce to the previous factorizations by straight open wilson lines . we start section [ sec : open wilson lines ( ii ) ] by giving a proof of the conjecture made by dhar and kitazawa . we introduce the closed - string state @xmath28 , where @xmath26 is a momentum loop . this state is not an eigenstate of closed - string momentum but serves as a generating function of ( generally off - shell ) closed - string states . overlap between @xmath29 and boundary states with open - string legs is a generating function of couplings of all the closed - string states to the non - commutative d - brane . the overlap with the boundary states of gluons is shown to become a curved open wilson line in the zero - slope limit . we make an explicit correspondence between the closed - string states and the coefficients of harmonic expansions of the curve . these provide the proof . we also show that in the zero - slope limit the momentum eigenstate is identified with the state @xmath30 after some manipulation . this accounts for the previous factorizations by curved open wilson lines . wilson line is invariant under reparametrizations of the path . the corresponding transformations on the string world - sheet are reparametrizations of the boundary . the reparametrization invariance of boundary states is fulfilled by imposing the ishibashi condition @xcite or equivalently the brst invariance on these states . as regards the boundary states of open - string tachyon and gluon it is shown in sections [ sec : tachyon boundary state ] and [ sec : gluon boundary state ] that their reparametrization invariance is equivalent to the on - shell conditions . on the other hand we do not require any condition on gluons to obtain open wilson lines . this puzzle is solved by seeing that the ishibashi condition or the brst invariance becomes null in the present zero - slope limit . it is observed in section [ sec : discussions ] that all the boundary states constructed so far in this article are eigenstates of the closed - string momentum operators . the eigenvalues are essentially delta functions on the world - sheet boundary . their boundary actions are computed by following the prescription given in @xcite . it turns out that they are the standard boundary actions used in the path - integral formalism of the world - sheet theory . after a speculation based on these observations we finally make a conjecture on the duality between open- and closed- strings . in appendix [ sec : open - string tensors ] world - volume and space - time tensors used in the text are summarized including their relations . in appendix [ sec : formulae ] some formulae of creation and annihilation modes are described . these are necessary for our computations of several string amplitudes in the text . oscillator realizations of coordinate and momentum eigenstates are presented in appendix [ sec : eigen ] . let us consider the system of d@xmath31-brane in bosonic string theory . we take the following closed - string background : @xmath32 here @xmath33 is a flat space - time metric , which we refer to as closed - string metric . two - form gauge field @xmath34 is constant . we divide the space - time directions into two pieces , @xmath35 , where @xmath36 and @xmath37 . the directions @xmath38 are supposed to be parallel to the d@xmath31-brane . the directions @xmath39 are perpendicular to the brane . closed - string may capture the brane . relevant action of a closed - string takes the form : @xmath40=\frac{1}{4\pi\alpha'}\int_{\sigma } d\tau d\sigma \bigl \ { \partial_{a}x^{m } \partial^{a}x^{n}g_{mn } -i2\pi\alpha ' \epsilon^{ab}\partial_{a}x^{\mu } \partial_{b}x^{\nu}b_{\mu\nu } \bigr \}~ , \label{eq : action1}\ ] ] where @xmath41 is the antisymmetric tensor on the world - sheet with @xmath42 . the world - sheet @xmath43 is a disk or an infinite semi - cylinder , and we use the cylinder coordinates @xmath44 ( @xmath6 and @xmath45 ) . closed - string interacts with the brane at the boundary of the world - sheet , that is , at @xmath7 . the second term of the action is an integration of the two - form ( its pull - back ) on @xmath43 . since it is an exact two - form , by applying the stokes theorem we can recast it into a boundary integral : @xmath40 = \frac{1}{4\pi\alpha'}\int_{\sigma } d\tau d\sigma \partial_{a}x^{m } \partial^{a}x^{n}g_{mn } -\frac{i}{2}\int_{\partial\sigma } d\sigma b_{\mu\nu}x^{\mu } \partial_{\sigma}x^{\nu}~. \label{eq : action2}\ ] ] energy - momentum tensor @xmath46 is obtained from the action . since the integration of the two - form is independent of the world - sheet metric , @xmath47 acquires the standard form without the @xmath0-field . closed - string coordinates @xmath48 have mode expansions of the form , @xmath49 and the standard first quantization requires the following commutation relations : @xmath50=ig^{mn}~ , \quad [ \hat{x}_{0}^{m},\hat{x}^{n}_{0 } ] = [ \hat{p}^{m}_{0},\hat{p}^{n}_{0}]=0~,\nonumber\\ & & [ \alpha^{m}_{m},\alpha^{n}_{n}]=mg^{mn}\delta_{m+n}~ , \quad [ \tilde{\alpha}^{m}_{m } , \tilde{\alpha}^{n}_{n } ] = mg^{mn}\delta_{m+n}~ , \quad [ \alpha^{m}_{m } , \tilde{\alpha}^{n}_{n}]=0~. \label{eq : ccr}\end{aligned}\ ] ] the energy - momentum tensor generates reparametrizations of the world - sheet . their infinitesimal forms turn out to be the virasoro algebras with the central charges equal to @xmath51 . the virasoro generators are given by the expansions , @xmath52 @xmath53 and @xmath54 @xmath55 . here we use the complex coordinates @xmath56 instead of the cylinder coordinates . they are related by @xmath57 . @xmath58 and @xmath59 are generators of the chiral and the anti - chiral sectors respectively . these have the following representations in terms of the oscillator modes : @xmath60 where @xmath61 . @xmath62 denotes the standard normal ordering with respect to the @xmath2-invariant vacuum of closed - string @xmath63 : @xmath64 for @xmath65 and the similar prescription for the anti - chiral modes . the above conditions are used to introduce a boundary state in the first quantized picture of closed - string . let us denote it by @xmath77 . it is a state which satisfies the following conditions : @xmath78(\sigma,0 ) = x^{i}_{0 } \label{eq : bstate1}\end{aligned}\ ] ] these are linear constraints on @xmath1 and determine the state modulo its normalization . it turns out that the state satisfies the ishibashi condition @xcite as follows : @xmath79 the above state can be interpreted as a perturbative vacuum of closed - string in the presence of the brane . since there are no correlations between string - coordinates of the neumann and the dirichlet directions , we can factorize the state into a product : @xmath80 where the subscripts @xmath81 and @xmath51 denote the corresponding boundary conditions . we may refer to these two states respectively as the neumann and the dirichlet boundary states for short . they satisfy the following constraints : @xmath82 ( \sigma,0 ) { \bigl |}b_{n } { \bigr\rangle}=0~ , \quad x^{i}(\sigma,0 ) { \bigl |}b_{d } { \bigr\rangle}= x^{i}_{0}{\bigl |}b_{d } { \bigr\rangle}~. \label{eq : bstatend}\ ] ] these constraints may be handled in the oscillator representations . for the neumann boundary state it can be read as follows : @xmath83 where @xmath84 and its transpose @xmath85 are tensors defined as @xmath86 as regards the dirichlet boundary state the constraint can be read as follows : @xmath87 let us describe these boundary states including their normalizations . the above constraints can be solved without any difficulty in the oscillator representations . determinations of their normalization factors need to be cared . we start with the dirichlet boundary state . the dirichlet boundary state turns out to be given by @xmath89 where we put @xmath90 . the state @xmath91 is an eigenstate of the zero modes @xmath92 with eigenvalues @xmath93 . momentum representation of @xmath94 is given by @xmath95 where @xmath96 denotes an eigenstate of the momentum zero modes @xmath97 with eigenvalues @xmath98 defined as @xmath99 the dual state can be obtained by taking the bpz conjugation : @xmath100 it is easy to see that the state ( [ state bd ] ) satisfies the conditions ( [ eq : bstate3 ] ) . since they are linear conditions , normalization factor of the state should be determined by other means . we fixed it to be @xmath101 in the above . it is determined by claiming that closed - string propagations along the dirichlet directions reproduce the vacuum one - loop amplitude of open - string which satisfies the dirichlet - dirichlet ( d - d ) boundary conditions . we parametrize propagations of closed - string by @xmath102 . closed - string evolves by an imaginary time @xmath103 with the hamiltonian @xmath104 . see figure [ cylinder ] . .,height=264 ] closed - string propagations along the dirichlet directions are measured by @xmath105 , where we put @xmath106 . this amplitude can be easily calculated by using the formula in appendix [ sec : formulae ] . it turns out to be @xmath107 where @xmath108 with @xmath109 . contribution of the world - sheet reparametrization ghosts is excluded in eq.([bd - bd ] ) . it will be included in string propagations along the neumann directions . the corresponding amplitude of open - string is given by @xmath110 . open - string propagates by an imaginary time @xmath16 @xmath111 with the hamiltonian @xmath112 . see figure [ strip ] . . two bold horizontal lines are identified.,height=264 ] the trace is taken over the sector of open - string which satisfies the d - d boundary conditions . the amplitude becomes as follows : @xmath113 the standard argument allows us to interpret these open - string amplitudes as the free propagations of closed - string . two imaginary times are related with each other by @xmath114 . this leads us to write eq.([d - d ] ) as @xmath115 where the modular transformation @xmath116 is used . a comparison between eqs.([bd - bd ] ) and ( [ d - d2 ] ) gives the identity : @xmath117 therefore the propagations of closed - string reproduce correctly the corresponding one - loop amplitude of open - string except the factor @xmath118 . this factor turns out to be canceled by a similar one appearing in closed - string propagations along the neumann directions . the neumann boundary state is given @xcite by @xmath120 where @xmath121 is a tensor defined by @xmath122 the dual state is obtained by the bpz conjugation . its explicit form is as follows : @xmath123 the above normalization factor of the state is chosen to reproduce the related one - loop amplitudes of open - string . this will be shown in the next section . as can be seen in eqs.([state bd ] ) and ( [ state bn ] ) these boundary states are the bogolubov transforms of perturbative vacua . generators of the transformations are @xmath124 for the dirichlet boundary and @xmath125 for the neumann boundary . we can expect that information on the boundary conditions are all encoded in these generators . let @xmath126 be a local operator of closed - string . action of this operator on a boundary state @xmath127 , where @xmath4 is a generator , can be written as follows : @xmath128 describes the bogolubov transform of this local operator and ask the physical implication particularly from the viewpoint of boundary conformal field theories . however , story is not so simple . in general , @xmath129 turns out to be singular . more precisely it becomes singular at @xmath7 , where the boundary state resides . this reflects the fact that the system under consideration is actually a system of closed- and open - strings . we wish to make an idea of the bogolubov transformations of these local operators rigorous . for this purpose , we have to perform a regularization by which the above singularity becomes tractable . this leads us to define @xmath130 , which becomes regular at the world - sheet boundary . it is a local operator and interpreted as the adjoint transform of @xmath126 by @xmath131 . we will find out their physical interpretation . in this section we concentrate on tachyon vertex operators . since we are mainly interested in the world - volume theory of @xmath31-brane , we restrict ourselves to the bogolubov transform associated with the neumann boundary state . we denote the generator by @xmath132 , @xmath133 let @xmath134 be closed - string tachyon vertex operator of momentum @xmath135 . momentum @xmath135 is supposed to have only components along the neumann directions . an explicit form is given by @xmath136 \nonumber \\ & & ~~ \times \prod_{n=1}^{\infty } \exp \left [ -\sqrt{\frac{\alpha'}{2 } } \frac{1}{n}k_{\mu } ( \alpha_{n}^{\mu}z^{-n } + \tilde{\alpha}_{n}^{\mu}\bar{z}^{-n } ) \right ] . \label{closed - string tachyon vertex}\end{aligned}\ ] ] to discuss the bogolubov transform it is convenient to write down the transforms of oscillator modes of the string coordinates . these can be read as : @xmath137 for @xmath138 . the modes @xmath139 and @xmath140 for @xmath141 are kept intact . ( we put @xmath142 . ) the above mixture of the creation- and annihilation - modes makes the transform @xmath143 singular at @xmath7 or equivalently , since we put @xmath144 , at @xmath145 . it can be written in the following form for @xmath146 : @xmath147 here we introduce a local operator @xmath148 which we interpret as the bogolubov transform of @xmath9 . it takes the form of @xmath149 where @xmath150 and @xmath151 are operators consisting only of the creation modes : @xmath152 singularity of @xmath153 comes from the factor @xmath154 in eq.([transform of vt ] ) . because of this factor , the transform ( [ transform of vt ] ) becomes singular at @xmath7 where the boundary state @xmath155 is located . as will be seen soon , this factor should be subtracted in our construction of open - string tachyon vertex operator from a closed - string local operator . this factor can therefore be regarded as a renormalization factor of open - string tachyon vertex operator under its interpretation in terms of closed - string . putting this factor aside for a while , let us examine the operators @xmath148 . we first discuss their operator product expansion ( ope ) . it is convenient to recall the ope between closed - string tachyon vertex operators . it can be read from the expansion ( [ closed - string tachyon vertex ] ) as follows : @xmath156 for @xmath157 and @xmath158 . the ope under consideration changes from eq.([ope vt ] ) . the modification comes from operator products between @xmath159 , @xmath160 and @xmath9 . it can be calculated by using the expansions ( [ n and tilded n ] ) . we finally obtain : @xmath161 for @xmath162 and @xmath163 . here @xmath164 is green s function on the unit disk @xmath165 in the presence of a constant @xmath0 field . it is defined by @xmath166 the rhs can be evaluated by using the bogolubov transforms ( [ transforms of massive modes ] ) and written down explicitly as follows : @xmath167 let us recall that a system only of closed - strings admits a holomorphic factorization . particularly there is no correlation between chiral and anti - chiral pieces , @xmath168 and @xmath169 , of string coordinates @xmath170 . the factorized term of @xmath171 in eq.([disk green function ] ) is a sum of the correlations @xmath172 and @xmath173 . the second and third terms are not factorized and they are respectively the correlations @xmath174 and @xmath175 . these correlations are characteristic of open - string theory . green s function provides a nice description of the singular factor in eq.([transform of vt ] ) . using this terminology we can write it as follows : @xmath176 where the first equality follows from eq.([transform of vt ] ) and @xmath177 . taking account of this expression , the transform ( [ transform of vt ] ) indicates that boundary states give rise to extra correlations , i.e. correlations between the chiral and the anti - chiral sectors , by the amount of @xmath178 . in the presence of the world - sheet boundary , or equivalently in the open - closed mixed system , correlations between the chiral and anti - chiral sectors exist even in the closed string sector , and we need to take care of them . as pointed out in @xcite , this is a direct result of the fact that boundary states on the unit circle reflect a vertex operator at @xmath179 making its mirror image at @xmath180 : @xmath181 ( see also @xcite ) . this brings about a short distance singularity at the boundary @xmath182 . since we intend to construct open - string vertex operators in terms of closed - string , we need to carry out a renormalization to manage this type of singularity . the ope ( [ ope renormalized tachyon vertex ] ) itself strongly suggests an interpretation of @xmath148 from the open - string viewpoint . to pursue such a possibility we introduce a renormalized tachyon vertex operator @xmath183 by subtracting the above singular factor : @xmath184 in other words we have @xmath185 we call @xmath186 renormalized _ open - string _ tachyon vertex operator with momentum @xmath135 . the action ( [ kn on lim ren vt ] ) is completely the same as the action of the virasoro algebra on open - string tachyon vertex operator with the same momentum . in addition to the ope ( [ ope renormalized tachyon vertex ] ) this is the reason why we identify @xmath204 with renormalized open - string tachyon vertex operators . we can also make the above consideration in terms of boundary states . let us consider the state @xmath205 . by using the relation ( [ def2 of ren vt ] ) we can write it in a form , @xmath206 @xmath207 . an explicit form of @xmath208 has been given in eq.([normal ordered transformed vt ] ) . thereby we can express the state in the oscillator representation and then take the limit @xmath195 without ambiguity . we call thus obtained state @xmath209 . it turns out to have the following form : @xmath210 g_{n}|k \rangle . \nonumber \\ \label{one tachyon bn}\end{aligned}\ ] ] here we express the normalization factor of @xmath211 in terms of the open - string tensors . this translation is done by using the identity , @xmath212 , which also follows from eq.([def of g and theta ] ) . action of @xmath187 on this state can be obtained from eq.([kn on lim ren vt ] ) . it can be read as @xmath213 for @xmath189 . the ishibashi condition imposed on @xmath214 , i.e. vanishing of the rhs of the above equation for an arbitrary @xmath215 , ( strictly speaking , modulo a total derivative with respect to @xmath216 ) , requires @xmath217 . it is the on - shell condition of open - string tachyon . the action of @xmath187 on the boundary state has an interpretation in terms of string field theory . fundamental ingredient in string field theory is a brst charge @xmath218 . it is a grassmann - odd operator obeying the usual relations , @xmath219 and @xmath220 , where @xmath47 and @xmath221 are respectively total energy - momentum tensor and anti - ghost field of a world - sheet theory . in the case of bosonic closed - string field theory @xcite world - sheet theory consists of closed - string coordinates @xmath222 ( a matter system ) and the world - sheet reparametrization ghosts . the ghost system is described by @xmath223 for the chiral part and @xmath224 for the anti - chiral part . closed - string brst charge @xmath225 is decomposed into @xmath226 . the chiral part @xmath227 has the following form : @xmath228 where @xmath229 are the virasoro generators of the matter and the ghost systems . @xmath230 and @xmath231 are the fourier modes of ghost and anti - ghost fields : @xmath232 and @xmath233 . the anti - chiral part @xmath234 has the same form as the above except replacing the chiral quantities with the anti - chiral ones . these @xmath227 and @xmath234 are nilpotent independently and satisfy the relations , @xmath235 and @xmath236 . here @xmath237 and @xmath238 are the virasoro generators of the total system . let @xmath239 and @xmath240 be respectively the hilbert spaces of matter and ghost systems . we put @xmath241 . closed - string hilbert space @xmath242 consists of vectors @xmath243 of @xmath244 which satisfy the conditions , @xmath245 necessity of the two conditions is explained in @xcite from the perspective of two - dimensional conformal field theories . now we want to interpret the state @xmath246 as a state of the closed - string hilbert space . since it is a boundary state of the neumann directions we first extend it to a vector of @xmath239 by tensoring the boundary state ( [ state bd ] ) of the dirichlet directions . as for the ghost sector an appropriate state is known . it is given @xcite by @xmath247 and satisfies the following boundary conditions : @xmath248 for @xmath189 . it can be seen from these conditions that @xmath249 satisfies the ishibashi condition . further tensoring the ghost boundary state we obtain @xmath250 of @xmath244 . we call it @xmath251 . this state satisfies the first condition in eq.([state condition of closed - string ] ) because of eq.([boundary condition bghost ] ) . as for the second , the @xmath252 case of eq.([kn on one tachyon bn ] ) gives @xmath253 @xmath254 . this means that we need to integrate over @xmath216 to regard the state as a vector of @xmath242 . the above action can be identified with the brst transformation of open - string tachyon vertex operator . @xmath259 in eq.([qc on one tachyon b ] ) correspond to the same modes of ghost field appearing in open - string field theory . actually eq.([qc on one tachyon b ] ) is an example of the formula given in @xcite . it states that the closed - string brst charge @xmath225 acts on boundary states as the generator of the brst transformation of open - string field . it is used there to show a macroscopic description of the cubic open - string field theory is given by a boundary open - string field theory . we wish to interpret these states as boundary states with off - shell open - string tachyons . this is verified by comparing closed - string tree propagations between these states with the corresponding open - string one - loop amplitudes and testing their coincidence . these will be done in the next subsection . it is worth mentioning that the present construction is also applicable in a vanishing @xmath0 field background and in particular the boundary states ( [ m open - string tachyon boundary state ] ) become available just by putting @xmath285 . to describe the closed - string propagations we need the dual boundary states . although they are given by the bpz conjugation , we apply the previous formalism to their construction as well in order to gain some insight into them . instead of @xmath132 we use the bpz dual : @xmath286 the transform of tachyon vertex operator is given by @xmath287 . it can be written in the following form for @xmath288 : @xmath289 here we introduce a local operator @xmath290 , which we interpret as the ( dual ) bogolubov transform of @xmath291 . it takes the form of @xmath292 where @xmath293 and @xmath294 are operators consisting only of the annihilation modes : @xmath295 singularity of @xmath287 comes from @xmath296 in eq . ( [ dual transform of vt ] ) . it becomes the same as the singular factor appearing in eq.([transform of vt ] ) after changing the coordinate @xmath179 to @xmath297 . similar calculation to what is made to obtain eq.([ope renormalized tachyon vertex ] ) leads to the ope : @xmath298 for @xmath162 and @xmath163 . here @xmath299 is green s function on the unit disk @xmath162 . it is defined by @xmath300 and written down explicitly as follows : @xmath301 similarly to eq.([eq : singular factor as self - contraction of vt ] ) , the singular factor appearing in eq.([dual transform of vt ] ) is expressed as the self - contraction of @xmath302 between its chiral and anti - chiral parts . this may be seen as follows . let @xmath303 be the chiral anti - chiral correlation of green s function @xmath304 : @xmath305 . using this quantity , we can write the singular factor as @xmath306 dual of the renormalized open - string tachyon operator , which we call @xmath307 , is given by subtracting the singular factor appearing in eq.([dual transform of vt ] ) as follows : @xmath308 we remark that the @xmath187 action on the dual vertex operator reduces to eq.([kn on lim ren vt ] ) after we take the limit @xmath309 . let us describe the dual boundary state . it is obtained by following the same route as taken for eq.([m open - string tachyon boundary state ] ) . let @xmath310 ( @xmath262 ) be the dual vertex operators . let @xmath311 be distinct points on the infinite semi - cylinder satisfying the condition , @xmath312 . we consider the state @xmath313 and take the limit @xmath314 . thus obtained state , which we call @xmath315 , is the dual . explicitly it is given by @xmath316 . \nonumber \\ \label{m open - string tachyon dual boundary state } \end{aligned}\ ] ] the hamiltonian operator of closed - string is @xmath317 . we can conveniently parametrize propagations of closed - string by @xmath102 . the evolution by an imaginary time @xmath103 is given by the operator @xmath318 , where we put @xmath106 . in this subsection we only consider the propagations along the neumann directions . propagations of closed - string between the boundary states ( [ m open - string tachyon boundary state ] ) are measured by the following amplitudes : @xmath319 \nonumber \\ \label{def of f } \end{aligned}\ ] ] further evaluations of the infinite products are carried out in appendix [ sec : formulae ] . we just quote the result obtained there . the contributions from the massive modes turn out to be as follows : @xmath326^{2\alpha'k_{\mu}^{(r)}g_{\mu \nu}k_{\nu}^{(s ) } } \nonumber \\ & & ~~ \times \prod_{m+1 \leq r < s \leq m+n } \left [ \frac { \prod_{n=1}^{\infty } \left ( 1-e^{i(\sigma_r-\sigma_s)}q_c^n \right ) \left ( 1-e^{-i(\sigma_r-\sigma_s)}q_c^n \right ) } { \prod_{n=1}^{\infty } \left ( 1-q_c^n \right)^2 } \right]^{2\alpha'k_{\mu}^{(r)}g_{\mu \nu}k_{\nu}^{(s ) } } \nonumber \\ & & ~~ \times \prod_{r=1}^m \prod_{s = m+1}^{m+n } \left [ \frac { \prod_{n=0}^{\infty } \left ( 1-e^{i(\sigma_r-\sigma_s)}q_c^{n+1/2}\right ) \left ( 1-e^{-i(\sigma_r-\sigma_s)}q_c^{n+1/2}\right ) } { \prod_{n=1}^{\infty } \left ( 1-q_c^n\right)^2 } \right]^{2\alpha'k_{\mu}^{(r)}g_{\mu \nu}k_{\nu}^{(s)}}. \nonumber \\ \label{result on f}\end{aligned}\ ] ] for the comparison with open - string one - loop calculation it is convenient to write down the amplitudes ( [ pre tachyon amplitude by boundary state ] ) by using the elliptic @xmath327-functions . appropriate @xmath327-functions are @xmath328 . they have the following representations : @xmath329 where we put @xmath330 . the amplitudes ( [ tachyon amplitude by boundary state ] ) , combined with closed - string propagations along the dirichlet directions , should be compared with open - string tachyon one - loop amplitudes . we begin this subsection with a brief exposition on the open - string calculation in a constant @xmath0 field background of closed - string . related with a prescription of normal orderings relative to the @xmath362-invariant vacuum of open - string , it is convenient to introduce @xcite commutative zero - modes @xmath363 instead of the non - commutative ones . they satisfy the standard canonical relations ; @xmath364= [ \hat{p}_0^{\mu},\hat{p}_0^{\nu}]=0 $ ] and @xmath365=i g^{\mu \nu } $ ] . in what follows we adopt the standard normal ordering of @xmath366 . it is also denoted by @xmath367 . the virasoro generators @xmath368 can be obtained from the energy - momentum tensor @xmath369 by @xmath370 . they turn out to have the following forms : @xmath371 where we put @xmath372 . among these generators open - string propagator is given by @xmath373 . evolution of open - string on the upper half - plane is described by @xmath374 due to an integral representation @xmath375 . it is useful to observe the open - string coordinates , particularly @xmath376 , at the boundary of the world - sheet . we may use the radial coordinates @xmath377 , where @xmath378 , as a parametrization of the upper half - plane . they can be read as follows : @xmath379 at both boundaries @xmath380 and @xmath381 , effect of the non - commutativity of the world - volume ( the @xmath327-dependence ) is encoded only in the zero modes while the massive part acquires the standard mode expansion . open - string amplitudes are correlation functions between local operators inserted at the boundaries . it follows from eq.([open - string at boundary ] ) that @xmath327-dependence of local operators such as tachyon- and gluon - vertices are encoded only in their zero modes . one - loop amplitudes can be obtained in the operator formalism simply by taking a trace of these operators with the propagators inserting among them . the integral representation of the propagator may be used . the zero mode dependent part decouples from the others in @xmath374 . therefore , for the particular cases of tachyons and gluons , only the zero - modes of open - string are influenced by the non - commutativity of the world - volume . let us consider one - loop amplitudes of tachyon vertex operators . we introduce open - string tachyon vertex operator of momentum @xmath135 by @xmath382 we consider the scattering process of @xmath383 tachyons with momenta @xmath384 . it is worth noting that the momenta @xmath384 have only components along the neumann directions . it is because the propagation of open - string is restricted along the d - brane world - volume . diagram describing the one - loop scattering process can be drawn on the upper half - plane as depicted in figure [ upper - half - plane ] . an open - string evolves along the radial direction from the outer semi - circle to the inner one and interacts with the tachyon vertices at its ends . two semi - circles are identified with each other and thereby the diagrams is interpreted as an open - string one - loop . the corresponding tachyon amplitude , which we call @xmath385 , is given by a sum of traces of their products arranged in cyclically distinct orders with keeping their partial cyclic orderings at the both ends : @xmath386 we can evaluate the amplitude in the standard manner . let us recall that the tachyon vertex operator enjoys the properties : @xmath387 combined with the integral form of @xmath388 , this enables us to write the amplitude as follows : @xmath389 the coordinates @xmath390 , which appear in the rhs as insertion points of the tachyon vertices , can be thought to provide a parametrization of the diagram . another parametrization may be obtained by mapping the diagram to a cylinder with width @xmath391 . see figure [ open - string diagram ] . two bold vertical lines are identified with each other.,height=529 ] correspondingly @xmath390 are mapped to @xmath392 by @xmath393 . we put @xmath394 . we may use @xmath16 instead of @xmath395 . the evolution of open - string becomes manifest in this parameterization . in the figure an open - string with width @xmath391 evolves along the real axis from the origin to @xmath396 , interacting with the tachyon vertices inserted at @xmath392 or @xmath397 . to compare the open - string one - loop amplitude ( [ tachyon one - loop by open - string parameters ] ) with eq.([tachyon amplitude by boundary state ] ) we map the cylinder drawn on the @xmath402-plane ( figure [ open - string diagram ] ) to a cylinder with width @xmath403 on the @xmath404-plane by the conformal transformation @xmath405 . see figure [ closed - string diagram ] . two bold vertical lines are identified with each other.,height=529 ] @xmath392 are mapped to @xmath406 by @xmath407 . we put @xmath408 . these provide a new parametrization of the diagram , which makes an evolution of closed - string manifest . a closed - string with circumference @xmath409 evolves along the imaginary axis , starting from the real axis where it interacts with @xmath260 open - string tachyons and ending on @xmath410 where it interacts with @xmath81 open - string tachyons . we note that the insertion point of the @xmath411-th open - string tachyon is fixed at @xmath412 . low - energy description of world - volume theories of @xmath31-brane can be obtained by taking a zero - slope limit @xmath13 so that it makes all perturbative stringy states ( @xmath14 ) of open - string infinitely heavy and decouple from the light states . low - energy effective theory relevant to open - string tachyons in the presence of a constant @xmath0 field is a scalar field theory on the non - commutative world - volume . in general , quantum field theories on a non - commutative space suffer the uv - ir mixing @xcite which originates in the non - commutativity . it causes @xcite a serious problem on the renormalization prescription of these theories . in this section , based on the results obtained so far , we investigate the uv behavior of the non - commutative scalar field theory . for this sake it is convenient to start with a brief description of the above zero - slope limit of the one - loop amplitudes of @xmath383 tachyons . our study in this section is restricted to the case of @xmath421-brane in the critical dimensions . eq.([open - string ftl of one - loop amplitude ] ) can be identified with one - loop amplitude of open - string tachyon @xmath430 ( a scalar particle of @xmath431 ) living on the non - commutative world - volume . it is the amplitude obtained from the corresponding one - loop feynman diagram consisting of @xmath383 trivalent vertices . each of the vertices represents the following cubic interaction : @xmath432 here we introduce the moyal product ( an associative non - commutative product ) by @xmath433 the feynman rule becomes a little complicated due to the above moyal products . but the feynman integral can be evaluated by using the standard technique @xcite and translated to eq.([open - string ftl of one - loop amplitude ] ) . we can find @xmath434 in the integral of eq.([open - string ftl of one - loop amplitude ] ) . the first term comes from the schwinger representation of the tachyon propagator , @xmath435 . this is the standard term which appears in ordinary field theory one - loop amplitudes . the second term can be understood @xcite as a uv regularization ( a regularization of the integration near @xmath436 ) . it is a curious regularization since it depends on the external momentum , @xmath437 . this feature can be thought of as a characteristic of quantum field theories on a non - commutative space and makes field theoretical description of physics at high energy scale difficult . it should be also noted that integrations in eq.([open - string ftl of one - loop amplitude ] ) have potential singularities at @xmath438 ( @xmath439 ) . these singularities can be already seen in the integral ( [ tachyon one - loop by open - string parameters ] ) . although we do not describe their regularization here , prescription used @xcite for the cubic open - string field theory will be effective . both parameters @xmath440 and @xmath441 are regarded as coordinates of the moduli space of conformal classes of cylinder with @xmath383 punctures at the boundaries . for each values of the closed - string parameters we obtain a graph as depicted in figure [ closed - string diagram ] . set of these graphs is the above moduli space . strictly speaking , these graphs are representatives of the conformal classes and thus it is possible to make different choices . for each values of the open - string parameters we obtain a graph as depicted in figure [ open - string diagram ] . it can be identified in the standard manner with a metrized trivalent one - loop ribbon graph . metric of the graph is given by the open - string parameters . these metrized ribbon graphs can be chosen as the representatives . therefore the open - string parameters give another set of coordinates of the moduli space . the above moduli space has several ends . typically we have two ends at @xmath25 and @xmath18 . the previous zero - slope limit of open - string is related with the end at the infinity as can be seen from eq.([schwinger parameters of scalar field theory ] ) . actual procedure to obtain the zero - slope limit ( [ open - string ftl of one - loop amplitude ] ) shows that the field theory amplitude is a suitable magnification of the integral ( [ tachyon one - loop by open - string parameters ] ) on an infinitesimal neighbourhood around this end . we want to know the uv behavior of the world - volume theory . for such a purpose we need to focus on the region @xmath442 . one possible resolution may be obtained by taking a zero - slope limit such that it magnifies an infinitesimal neighbourhood around another end located at @xmath25 . near this end use of the closed - string parameters will be effective as the open - string parameters near the infinity . let us examine a zero - slope limit which will be taken by fixing the following parameters : @xmath443 in the course of taking the limit we also fix open - string tensors @xmath200 and @xmath201 in order to capture the world - volume theory . it follows from eq.([equalities2 and 3 ] ) that this makes the limit different from the standard field theory limit of closed - string . transformation between the open- and the closed - string parameters leads @xmath444 and @xmath445 . for @xmath23 and @xmath406 to remain finite the original field theory parameters need to satisfy @xmath446 and @xmath447 . we can not neglect effects of all the perturbative stringy states of open - string at such a trans - string scale . hence the limit in question includes their effects . it is important to compare the above zero - slope limit with the previous one . in eq.([closed - string ftl of one - loop amplitude ] ) the schwinger propagator @xmath459 is integrated and gives rise to the propagator of _ closed - string tachyon _ of momentum @xmath460 , while the counterpart in eq.([open - string ftl of one - loop amplitude ] ) consists of two parts one of which is standard in field theories and the other is a curious regularization factor originated in the non - commutativity . the factors @xmath461^{k_{\mu}^{(r)}g^{\mu \nu}k_{\nu}^{(s ) } } $ ] found in eq.([open - string ftl of one - loop amplitude ] ) describe correlations between two open - string tachyons inserted at @xmath423 and @xmath462 in the schwinger time . these correlations are made by propagations of open - string tachyon between them and particularly have their origin in kinetic energy @xmath463 of the propagating open - string tachyon . these correlations are lost in eq.([closed - string ftl of one - loop amplitude ] ) . this indicates that open - string tachyon becomes topological at the trans - string scale and that it is described by a one - dimensional topological field theory . vanishing of the correlations originates in the modular transforms ( [ modular trans of tachyon correlations ] ) . so this is a stringy effect . we now consider generating function of amplitudes ( [ def of jt ] ) and examine the zero - slope limit . let @xmath464 be the fourier modes of open - string tachyon field : @xmath465 since it is a real scalar field , the fourier modes satisfy @xmath466 , where the overline denotes complex conjugation . we examine the zero - slope limit of the following generating function : @xmath467 a factorization of one - loop amplitudes of non - commutative scalar field theory is considered in @xcite@xcite and claimed there to be realized at the low - energy of this field theory by using the path - ordered exponentials . the procedure of the zero - slope limit adopted in those papers are quite different form that of the present paper . our limiting procedure is not the conventional one to obtain the field theory limit of open - string . in fact the present limit would reduce to the field theory limit of closed - string if the @xmath0 field were vanishing . we would like to emphasize that the existence of the @xmath0 field makes the limit different from the conventional field theory limit of closed - string . we make a digression from the previous discussion on the world - volume theory . in this section we introduce coordinate and momentum eigenstates of closed - string and make some observation on their relation with the boundary states . these eigenstates will play important roles in the subsequent discussions . we begin with a simple remark on closed - string momentum currents . these currents are introduced by taking functional derivatives of the action : @xmath484 } { \delta \partial_{\tau } x^{m}(\sigma,\tau ) } $ ] . we have two expressions of the action , ( [ eq : action1 ] ) and ( [ eq : action2 ] ) . they provide two momentum currents , which coincide for the dirichlet directions but become slightly different from each other for the neumann directions . when the action ( [ eq : action2 ] ) is used one obtains @xmath485 while the action ( [ eq : action1 ] ) leads to @xmath486 their conserved charges become the same , that is , momentum @xmath483 , since these are integrals of the currents over the circle . whichever momentum current one adopts as the conjugate variable of @xmath487 , canonical quantization of closed - string leads to the same commutation relations ( [ eq : ccr ] ) . let @xmath488 be closed - string coordinate operators . they are simply given by @xmath489 . the conjugate momentum operators @xmath490 are those operators satisfying @xmath491 @xmath492 $ ] @xmath493 . ] ] these are given by the above momentum currents . due to the existence of two different currents we have two operators , @xmath494 and @xmath495 . all these operators are periodic with respect to @xmath216 . we may provide their mode expansions similar to eq.([eq : modex ] ) . for the later discussion these turn out to be inconvenient . instead we use the following mode expansions : @xmath496~ , \nonumber\\ \hat{p}^{(b)}_{m}(\sigma ) & = & \frac{1}{2\pi } \left [ \hat{p}_{0 m } + \frac{1}{\sqrt{2 } } \sum_{n=1}^{\infty } \left ( \hat{\varrho}_{nm}e^{-in\sigma } + \hat{\varrho}^{\dagger}_{nm}e^{in\sigma } \right ) \right]~ , \label{modes of coordinates and momntum operators}\end{aligned}\ ] ] where @xmath497 , @xmath498 and @xmath499 are the hermitian conjugates of the corresponding ones . their commutation relations can be read as follows : @xmath500=i\delta^{m}_{n}~ , \quad [ \hat{\chi}^{m}_{m},\hat{\psi}^{\dagger}_{nn } ] = [ \hat{\chi}^{\dagger m}_{m},\hat{\psi}_{n n } ] = 2i\delta^{m}_{n}\delta_{m , n}~ , \nonumber\\ & & [ \hat{\chi}^{m}_{m},\hat{\varrho}^{\dagger}_{n n } ] = [ \hat{\chi}^{\dagger m}_{m},\hat{\varrho}_{n n } ] = 2i\delta^{m}_{n}\delta_{m , n}~ , \quad \mbox{otherwise$=0$}~.\end{aligned}\ ] ] these oscillator modes are linear sums of the standard oscillator modes of @xmath501 . for the reader s convenience we attach below the dictionary : @xmath502 where the @xmath503-components of @xmath504 are understood as @xmath505 . therefore we have @xmath506 and @xmath507 . coordinate and momentum eigenstates are respectively eigenstates of @xmath508 and @xmath509 or @xmath510 . we describe them in some detail . these are closed - string extensions of those obtained @xcite for open - string . we first describe the coordinate eigenstates . they are characterized by two conditions ; ( i ) they are eigenstates of the string coordinate operators , @xmath511 where we abbreviate the superscript @xmath260 in the rhs . dual states @xmath512 are defined by the hermitian conjugation . it is done by simultaneous operations of the bpz and complex conjugations . the above conditions determine the eigenstates . we factorize them into products of the neumann and the dirichlet sectors : @xmath513 ) we parametrize the eigenvalues in terms of complex variables @xmath514 ( @xmath515 ) besides the zero - modes @xmath516 . we put @xmath517 the eigenstates are given by the following infinite products : @xmath518 here @xmath91 ( resp . @xmath162 ) are the normalized eigenstates of the zero modes @xmath519 ( resp . @xmath520 ) with eigenvalues @xmath521 ( resp . @xmath522 ) , and their normalization conditions are @xmath523 @xmath524 ( resp . @xmath525 ) are the normalized eigenstates of @xmath526 ( resp . @xmath527 ) with eigenvalues @xmath528 ( resp . @xmath529 ) . their normalization conditions are chosen as @xmath530 the orthonormality ( [ eq : orthonormality of x bs ] ) follows from the above conditions imposed on each eigenstates : @xmath531 where we put @xmath162 @xmath532 and @xmath533 @xmath532 . the steps to obtain oscillator realizations of eigenstates @xmath162 and @xmath534 are presented in appendix [ sec : eigen ] . ] ] they turn out to be as follows : @xmath535 |0 \rangle~ , \label{eq : chi - chi n}\\ \lefteqn { \left|\chi_{n},\bar{\chi}_{n}\right\rangle_{d } = \left(\frac{n}{\pi\alpha'}\right)^{\frac{d}{2 } } \sqrt{\det g_{ij } } } \nonumber\\ & & \times \exp\left[\frac{1}{n}\alpha^{i}_{-n}g_{ij } \tilde{\alpha}^{\nu}_{j } -\frac{i}{\sqrt{\alpha ' } } \left(\chi^{i}_{n}g_{ij}\alpha^{i}_{-n } + \bar{\chi}^{i}_{n } g_{ij } \tilde{\alpha}^{j}_{-n } \right ) -\frac{n}{2\alpha ' } \bar{\chi}^{i}_{n}g_{ij } \chi^{j}_{n } \right ] |0\rangle~. \label{eq : chi - chi d}\end{aligned}\ ] ] the coordinate eigenstates provide a complete basis of the hilbert space @xmath536 , where @xmath537 and @xmath538 stand for the neumann and the dirichlet sectors . we have the following completeness relation : @xmath539 \;|x\rangle \langle x| \nonumber\\ & = & \int d^{d}x_{0 } \prod_{n=1}^{\infty } \left(\prod_{m=0}^{d-1 } \frac{d\bar{\chi}^{m}_{n}d\chi^{m}_{n}}{2i}\right ) \left|\chi_{n},\bar{\chi}_{n}\right\rangle |x_{0}\rangle \langle x_{0}| \left\langle \chi_{n},\bar{\chi}_{n}\right|~.\end{aligned}\ ] ] we start with eigenstates of @xmath509 . they are characterized by two conditions ; ( i ) they are eigenstates of @xmath509 , @xmath540 and ( ii ) they satisfy the orthonormality condition , @xmath541 these conditions determine the eigenstates . we factorize them into products of the neumann and the dirichlet sectors . @xmath542 ( @xmath515 ) besides the zero - modes @xmath543 . we put @xmath544~.\end{aligned}\ ] ] the eigenstates are given by the following infinite products : @xmath545 where @xmath162 denote the normalized eigenstates of @xmath546 with eigenvalues @xmath547 , and the normalization conditions are taken as @xmath548 these conditions ensure the orthonormality of @xmath549 . oscillator realizations of @xmath550 are given in appendix [ sec : eigen ] . they turn out to be as follows : @xmath551 |0 \rangle~. \label{eq : psi - psi}\end{aligned}\ ] ] similarly to @xmath552 , momentum eigenstates @xmath549 provide another complete basis of @xmath537 . the completeness relation is @xmath553\ ; |p_{n}\rangle\langle p_{n}| \nonumber\\ & = & \int d^{p+1}p_{0 } \prod_{n=1}^{\infty } \left ( \prod_{\mu=0}^{p } \frac{d\bar{\psi}_{n\mu } d\psi_{n\mu}}{2i}\right ) \left|\psi_{n } , \bar{\psi}_{n}\right\rangle_n |p_{0n}\rangle \langle p_{0n}| { } _ n\!\left\langle \psi_{n},\bar{\psi}_{n}\right|~. \label{partition unity by pn}\end{aligned}\ ] ] next we describe eigenstates of @xmath510 . they are determined by similar conditions to those imposed on @xmath509 and are factorized into products of the neumann and the dirichlet sectors : @xmath554 by complex variables @xmath555 ( @xmath515 ) and the zero - modes @xmath543 as @xmath556 . \label{eq : pbn - eigenvalue}\end{aligned}\ ] ] the eigenstates are given by the following infinite products : @xmath557 where @xmath558 are the normalized eigenstates of @xmath559 with eigenvalues @xmath560 . we normalize them by @xmath561 the orthonormality of @xmath562 follows from these conditions . oscillator realizations of these states are given in the appendix . they turn out to be as follows : @xmath563 |0\rangle~. \label{eq : varrho - varrho}\end{aligned}\ ] ] momentum eigenstates @xmath564 provide a complete basis of @xmath537 as well . the completeness relation reads as follows : @xmath565\ ; \left|p_n^{(b)}\right\rangle \left\langle p_n^{(b ) } \right| \nonumber\\ & = & \int d^{p+1}p_{0 } \prod_{n=1}^{\infty } \left ( \prod_{\mu=0}^{p } \frac{d\bar{\varrho}_{n\mu } d\varrho_{n\mu}}{2i}\right ) \left|\varrho_{n},\bar{\varrho}_{n}\right\rangle_n |p_{0n}\rangle \langle p_{0n}| { } _ n\!\left\langle \varrho_{n},\bar{\varrho}_{n}\right|~.\end{aligned}\ ] ] the previous constructions give all the eigenstates in terms of infinite products of the correctly normalized eigenstates of each massive modes . for instance the eigenstates @xmath552 and @xmath566 in ( [ eq : es - x ] ) are given by the infinite products of @xmath567 and @xmath568 . one can find in eqs.([eq : chi - chi n ] ) and ( [ eq : chi - chi d ] ) that the normalization factors are respectively @xmath569 and @xmath570 . infinite products of these constants become the normalization factors of @xmath552 and @xmath566 . let us denote them by @xmath571 @xmath572 $ ] and @xmath573 @xmath574 $ ] . the eigenstates can be written as follows : @xmath575 |x_{0n}\rangle~ , \nonumber\\ & & \nonumber\\ & & = \mathcal{c}^{(d)}_{x } \prod_{n=1}^{\infty } \exp \left [ \frac{1}{n}\alpha^{i}_{-n } g_{ij}\tilde{\alpha}^{j}_{-n } -\frac{i}{\sqrt{\alpha ' } } \left ( \chi^{i}_{n}g_{ij } \alpha^{j}_{-n } + \bar{\chi}^{i}_{n}g_{ij } \tilde{\alpha}^{j}_{-n } \right ) -\frac{n}{2\alpha ' } \bar{\chi}^{i}_{n}g_{ij}\chi^{j}_{n } \right ] \nonumber \\ \label{coordinates 2}\end{aligned}\ ] ] as is performed in @xcite , one may evaluate @xmath576 by using the zeta - function regularization . this regularization scheme leads the following identities : ; @xmath577 , @xmath578 etc . ] @xmath579 and thus we obtain @xmath580 it is important to recall that the boundary state @xmath581 is an eigenstate of @xmath582 with eigenvalues @xmath583 . we have @xmath584 . if one uses the zeta - function regularization and adopts the above @xmath585 as the normalization constant , these two states become precisely identical : @xmath586 let us also recall that in quantum mechanics of a single particle , coordinate eigenstates @xmath587 are described by @xmath588 , where @xmath589 is the momentum operator . one may find an analogous realization for the string coordinate eigenstates . taking account of eq.([bd as xd ] ) one can infer the following one : @xmath590 where @xmath591 is the state ( [ bd as xd ] ) with @xmath592 . this turns out to be the case . using the previous parametrization of the eigenvalues one can find @xmath593 it is easy to see that the normal ordered exponential in the rhs of eq.([eq : x - shift ] ) reproduces @xmath566 . the formula ( [ eq : ee ] ) may be useful . the above discussion is also applicable to the string momentum eigenstates . we can write these states as follows : @xmath594 |p_{0n}\rangle~,~~ \nonumber \\ & & \left| p_n^{(b)}\right\rangle \nonumber\\ & & = \mathcal{c}_{p_n^{(b ) } } \prod_{n=1}^{\infty } \exp \left [ -\frac{1}{n}\alpha^{\mu}_{-n}n_{\mu\nu}\tilde{\alpha}^{\nu}_{-n } + \frac{\sqrt{\alpha'}}{n}\left\ { \varrho_{n\mu } { \left(\frac{1}{e^{t}}g\right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } + \bar{\varrho}_{n\mu } { \left(\frac{1}{e}g\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n}\right\ } \right . \nonumber\\ & & \hspace{8em } \left . -\frac{\alpha'}{2n}\bar{\varrho}_{n\mu } \left(\frac{1}{e^{t}}g\frac{1}{e}\right)^{\mu\nu } \varrho_{n\nu}\right ] |p_{0n}\rangle~ , \label{eq : pn - state}\end{aligned}\ ] ] where the normalization factors @xmath595 and @xmath596 are originally given by the infinite products as follow from eqs.([eq : psi - psi ] ) and ( [ eq : varrho - varrho ] ) . they are regularized to @xmath597 boundary state @xmath155 is an eigenstate of @xmath598 with vanishing eigenvalue . we have @xmath599 . if we use the above @xmath600 as the normalization constant , these two states become identical including their normalizations : @xmath601 momentum eigenstates can be realized in the quantum mechanics by @xmath602 , where @xmath603 is the position operator . we can find an analogous realization of the string momentum eigenstates . it turns out to be as follows : @xmath604\nonumber\\ & & \times : \exp\left(i\int^{2\pi}_{0}d\sigma p^{(b)}_{\mu}(\sigma ) \hat{x}^{\mu}(\sigma)\right ) : { \bigl |}b_{n}{\bigr\rangle}~. \label{eq : pnb - shift}\end{aligned}\ ] ] for the later convenience we provide a similar observation for the eigenstates @xmath605 as well . let @xmath606 be the state which is obtained from the neumann boundary state @xmath607 by putting @xmath285 : @xmath608 this is the neumann boundary state in a vanishing @xmath0 field background . one can readily find that this is an eigenstate of @xmath609 with zero eigenvalue : @xmath610 and hence @xmath611 . comparing definitions ( [ eq : pn - state ] ) and ( [ eq : bn with zero b ] ) , we find that these states coincide with each other including their normalizations , if we adopt @xmath612 in eq.([cpb ] ) as the normalization of the momentum eigenstates . concerning the state @xmath605 , a similar formula to eq.([eq : pnb - shift ] ) becomes @xmath613 \ , : \exp \left(i \int^{2\pi}_{0 } d\sigma p_{\mu}(\sigma ) \hat{x}^{\mu}(\sigma ) \right ) : \left| b_{n } \right\rangle_{b=0}~. \label{eq : pn - shift}\ ] ] this has the same form as eq.([eq : pnb - shift ] ) with putting @xmath285 . at the zero - slope limit , which is introduced in eq.([field theory parameters of closed - string ] ) and its below in order to capture the uv behavior of the world - volume theory , the generating function of one - loop amplitudes of open - string tachyons is shown to exhibit the factorized form ( [ closed - string ftl by straight open wilson lines ] ) . it is expressed as a sum of products of two wilson lines ( strictly speaking , their analogues ) along the same straight lines @xmath470 , multiplying the propagators of closed - string tachyon . this shows that closed - string tachyon @xmath614 has a tadpole interaction with the open wilson line . it can be written in a form , @xmath615 $ ] . its origin in string theory can be found in eq.([def of jt ] ) . the interaction simply comes from the closed - string tachyon modes of boundary states , @xmath616 @xmath617 @xmath618 . actually these boundary states have all the components of perturbative closed - string states . at the level of string amplitudes all of them propagate between the boundary states and contribute to the amplitudes . it is shown in @xcite that the straight open wilson line can couple with the on - shell graviton and , as will be seen in the later section , it can be generalized to the off - shell . this indicates that all the perturbative closed - string states have tadpole interactions with the open wilson line . therefore we may unfasten the zero - slope limit ( [ closed - string ftl by straight open wilson lines ] ) so that propagations of all these states are made manifest . in this section we pursue such a possibility . we also restrict to the case of @xmath421-brane in the critical dimensions . let us provide a general perspective on this issue before we start calculations . first of all , it can be expected @xcite that closed - string propagations including gravitons fluctuate the straight line appearing in eq.([closed - string ftl by moyal products ] ) and transform it into curved ones . in other words we can expect that there are correlations between their deviations from the straight line and the propagations of closed - string states . these curves will appear as the corresponding wilson lines , and we may factorize the generating function into a sum of these products at the zero - slope limit . analogously to the standard factorization of closed - string amplitudes the sum must be taken originally over the perturbative closed - string states . the above correspondence between curves and states will enable us to translate the sum as an integral over the space of curves . this integral may be suppressed by a suitable weight as the straight line is suppressed in eq.([closed - string ftl by straight open wilson lines ] ) by the closed - string tachyon propagator . it is amazing that one can interpret the tachyon propagator as a propagator of the straight open wilson line by notifying @xmath619 . the factorization ( [ closed - string ftl by straight open wilson lines ] ) may be obtained from the aforementioned integral by integrating out the fluctuations . we may say that the straight open wilson line is the average . in order to justify the above perspective let us first factorize the string amplitudes by an insertion of a partition of unity . use of that constructed from closed - string momentum eigenstates turns out to be relevant . it is given in eq.([partition unity by pn ] ) as @xmath620~ \langle p_n | , $ ] where the eigenvalue @xmath621 is parametrized by @xmath622 $ ] . ^{2\alpha'k_{\mu}^{(r)}g^{\mu \nu}k_{\nu}^{(s ) } } \times \prod_{m+1 \leq r < s \leq m+n } the above factorization may be compared with the previous expression of the amplitudes . terms in the first four lines can be found exactly in eq.([pre tachyon amplitude by boundary state ] ) . the other terms describe a factorization of @xmath324 in the same equation . if we integrate out @xmath628 and @xmath629 they provide eq.([result on f ] ) . these variables describe fluctuations of @xmath621 . corresponding degrees of freedom of closed - string are given by the massive modes . the factorization of @xmath324 in terms of those complex variables are plausible since @xmath324 is the sum of contributions of these massive modes . we examine the zero - slope limit of the factorization ( [ momentum factorization of tachyon amplitude ] ) . the limit we discuss is same as that investigated previously to capture the uv behavior of the world - volume theory . it is taken by fixing parameters @xmath630 and @xmath406 besides the open - string tensors . we also keep @xmath625 and @xmath631 intact . they do not scale under the limit . it should be noticed that the complex variables used originally to parametrize @xmath621 do scale under the limit . this is because @xmath625 and @xmath631 in ( [ momentum factorization of tachyon amplitude ] ) are the rescaled ones introduced by multiplying the original variables by @xmath632 . we first focus on the integral over @xmath625 and @xmath631 in eq.([momentum factorization of tachyon amplitude ] ) . let us start by considering the first exponential in the integral . scaling part in the exponent is expressed by means of the closed - string tensors and @xmath633 . it has the form : @xmath634 dominant contribution at the limit clearly comes from the first term : @xmath635 where we use equalities ( [ equalities2 and 3 ] ) . we turn to the determinant factors , which is also described by the closed - string tensors . their contributions can be read as follows : @xmath636 \approx \left ( \frac{1}{2\pi } \right)^{26 } \mbox{det } \left[- \frac{\theta g \theta } { 8\pi^2nq_c^{\frac{n}{2}}\alpha ' } \right].\end{aligned}\ ] ] the other two exponentials in the integral have similar forms . it is enough to know the behavior of @xmath637 in their exponents . it can be read by using eq.([def of g and theta ] ) as follows : @xmath638 as we studied previously , contributions of the first four lines in the factorization ( [ momentum factorization of tachyon amplitude ] ) are to give rise to the straight open wilson lines . the last pieces we need to estimate are the exponentials whose exponents are bilinear of @xmath639 with the weights , @xmath640 . it is enough to know the behavior of these weights . again using eqs . ( [ def of g and theta ] ) and ( [ equalities2 and 3 ] ) it can be read as @xmath641 @xmath642 . thus we can neglect these pieces in the zero - slope limit . the integration of @xmath662 are suppressed exponentially compared with @xmath470 . it can be accomplished by replacing the integration variables with their mean values and becomes unity at the zero - slope limit . then the above factorization reduces to that by the straight open wilson lines . our study of gauge theory starts from this section . an analogue of gluon vertex operator of open - string is introduced in closed - string theory . investigation of its bogolubov transformation leads a renormalization of this operator . it will be shown that the renormalized operator enjoys the standard properties of ( open - string ) gluon vertex operator , including the action of the virasoro algebra . these operators , acting on the neumann boundary state , give rise to boundary states which turn out to be identified with the boundary states of ( open - string ) off - shell gluons . in particular we will show that closed - string tree amplitudes between these states coincide with the corresponding gluon one - loop amplitudes of open - string . our discussion in this section goes almost parallel to section [ sec : tachyon boundary state ] where the boundary states of off - shell open - string tachyons are constructed . in closed - string theory an analogue of gluon - vertex operator may be taken as @xmath667 where @xmath668-vectors @xmath669 and @xmath670 are the momentum and the polarization vectors . the polarization vector is the fourier transform of @xmath479 gauge field @xmath671 : @xmath672 the gauge field takes value in @xmath673 . this yields @xmath674 . while we concentrate on @xmath479 gauge group in this paper , it can be straightforwardly generalized to @xmath675 by assigning the chan - paton indices . let us express the above operator in an auxiliary form . this often makes subsequent calculation facile . let @xmath676 be an auxiliary parameter . we write the operator in an exponential form : @xmath677 with @xmath678 \nonumber\\ & & \quad \times \prod_{n=1}^{\infty } \exp \left[-\sqrt{\frac{\alpha'}{2}}\frac{1}{n } \left\ { \bigg(k_{\mu}+ina a_{\mu}(k)\bigg ) \alpha_{n}^{\mu}z^{-n } + \bigg(k_{\mu}-ina a_{\mu}(k)\bigg ) \tilde{\alpha}_{n}^{\mu}\bar{z}^{-n } \right\}\right]~.\end{aligned}\ ] ] the relation ( [ vg and hvg ] ) implies that the terms proportional to higher powers of @xmath679 become irrelevant to the amplitudes . we consider the bogolubov transformation for @xmath680 generated by @xmath681 given in eq.([gn ] ) . using ( [ transforms of massive modes ] ) we can write down the transform as follows : @xmath682 where @xmath683~. \label{r}\end{aligned}\ ] ] the operator @xmath684 is defined to be @xmath685 where @xmath686 and @xmath687 consist of the creation modes alone and take the forms of @xmath688 in the transform ( [ transform of hvg ] ) we can find the same singularity structure as that of the tachyon . it is due to the factor @xmath689 , which is responsible for the singularity of the transform ( [ transform of hvg ] ) at the world - sheet boundary , @xmath690 . this factor contains an exponential in addition to the same factor which appeared in the transform @xmath691 . let us make a few comments about this extra factor . the first term of the exponent is proportional to @xmath679 . the closed - string tensor @xmath692 used there is translated into @xmath693 . the second term becomes irrelevant since it is proportional to @xmath694 . therefore the net effect of this extra factor is to give a term proportional to @xmath327 . particularly in the absence of a constant @xmath0 field , @xmath695 reduces to that of the tachyon . the singular factor which appears in the transform of the tachyon vertex operator has been expressed in eq.([eq : singular factor as self - contraction of vt ] ) as the self - contraction between the chiral and the anti - chiral parts of @xmath9 . in the present case as well , we can think of the singular factor as such a self - contraction of @xmath696 . in fact , we can express @xmath689 by using the chiral anti - chiral correlation @xmath697 of green s function @xmath698 as follows : @xmath699~. \label{r in terms of green function}\end{aligned}\ ] ] as mentioned in the analysis of tachyon , boundary states induce correlations between the chiral and the anti - chiral sectors , and this feature is characteristic of systems of interacting closed- and open - strings . we wish to find out a relation between @xmath700 and open - string gluon vertex operators . previous discussions to establish the relation between @xmath701 and open - string tachyon vertex operators consist of : i ) description of the ope between @xmath701 , which turns out to be given in ( [ ope renormalized tachyon vertex ] ) by using green s function @xmath698 and allows us to introduce the renormalized open - string tachyon vertex operators @xmath183 , ii ) description of the diff@xmath702 action on @xmath183 , which turns out to reduce at the boundary to the standard virasoro action on open - string tachyon vertex operators , and iii ) reproduction of open - string one - loop tachyon amplitudes by the boundary state formalism . we repeat these analyses as for gluons in the following . let us examine the ope between @xmath703 . it is convenient to start with the ope between the auxiliary operators . the standard calculation leads to @xmath704 \nonumber\\ & & \quad \ ; \times : { \hat{v}_{a}}\left(\sigma_{1},\tau_{1};k^{(1)};a_{1}\right ) { \hat{v}_{a}}\left(\sigma_{2},\tau_{2};k^{(2)};a_{2}\right):~,\end{aligned}\ ] ] for @xmath705 and @xmath706 . since @xmath703 have the form given in ( [ normal ordered transformed hvg ] ) , we can obtain their ope from the above by taking account of the opes between @xmath707 and @xmath708 . we find that @xmath709 \nonumber\\ & & \quad \ ; \times : \mbox{ad}_{g_{n}^{-1 } } { \hat{v}_{a}}\left(\sigma_{1},\tau_{1};k^{(1)};a_{1 } \right)\ ; \mbox{ad}_{g_{n}^{-1 } } { \hat{v}_{a}}\left(\sigma_{2},\tau_{2};k^{(2)};a_{2 } \right ) : ~ , \label{ope renormalized gluon}\end{aligned}\ ] ] for @xmath705 , @xmath710 and @xmath706 . this is a direct extension of the tachyon case ( [ ope renormalized tachyon vertex ] ) and suggests an interpretation of @xmath703 in terms of the gluon vertex operator . let us introduce the renormalized operator @xmath711 by the following subtraction : @xmath712 in other words , @xmath713 in addition to the subtraction of the singular factor @xmath689 , a finite subtraction is made in the above by the multiplication of @xmath714 . although it vanishes in the absence of @xmath0 field , this factor becomes surely necessary in general . without this subtraction we can not reproduce even the standard virasoro action which we present below . the renormalized operator ( [ def of renormalized hvg ] ) gives rise to @xmath715 by the relation ( [ vg and hvg ] ) . we refer to @xmath716 as renormalized gluon vertex operator with momentum @xmath669 and polarization @xmath670 . it can be written as follows : @xmath717 where @xmath302 @xmath718 is the closed - string tachyon vertex operator . next we will examine the action of diff@xmath702 on the above renormalized operators . as a shortcut we use the description in terms of boundary states . relevant states are introduced as @xmath719 ; a;(\sigma , k)\bigr\rangle$ ] and @xmath720 ; ( \sigma , k ) \bigr\rangle$ ] . the first states are defined by using @xmath721 as follows : @xmath722;a ; ( \sigma , k ) \bigr\rangle \equiv \lim_{\tau\rightarrow 0 + } { \hat{v}_{a}}^{ren } ( \sigma,\tau;k;a ) \left| b_{n } \right\rangle } \nonumber \\ & & = \left ( \frac{\left(\det e_{\mu\nu}\right)^{2 } } { \left(2\alpha'\right)^{p+1}(-\det g_{\mu\nu } ) } \right)^{\frac{1}{4 } } \lim_{\tau\rightarrow 0 + } g_{n } \left(g_{n}^{-1}{\hat{v}_{a}}^{ren}(\sigma,\tau;k;a ) g_{n}\right ) |\mathbf{0}\rangle \nonumber\\ & & = \left ( \frac{-\det g_{\mu\nu } } { \left(2\alpha'\right)^{p+1 } } \right)^{\frac{1}{4 } } e^{\frac{i}{2\pi } aa_{\mu}(k)\theta^{\mu\nu}k_{\nu}}\,g_{n } \lim_{\tau\rightarrow 0 + } \mbox{ad}_{g^{-1}_{n}}{\hat{v}_{a}}(\sigma,\tau;k;a ) |\mathbf{0}\rangle \nonumber\\ & & = \left ( \frac{-\det g_{\mu\nu } } { \left(2\alpha'\right)^{p+1 } } \right)^{\frac{1}{4 } } e^{\frac{i}{2\pi}aa_{\mu}(k)\theta^{\mu\nu}k_{\nu } } \nonumber\\ & & \hspace{1.5em } \times \prod_{n=1}^{\infty } \exp \left [ \frac{\sqrt{2\alpha'}}{n } \left\ { \left(k_{\mu}-ina a_{\mu}(k)\right ) { \left(\frac{1}{e^{t } } g\right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } e^{in\sigma } \right . \nonumber\\ & & \hspace{10em}\left.\left . + \left(k_{\mu}+ina a_{\mu}(k)\right ) { \left(\frac{1}{e}g\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n } e^{-in\sigma } \right\}\right]g_{n } |k_{n}\rangle~ , \label{aux 1 gluon bs}\end{aligned}\ ] ] where eqs.([normal ordered transformed hvg ] ) and ( [ renormalized operators ] ) are utilized to obtain the oscillator representation . the second states are defined similarly by using @xmath715 as @xmath723 ; ( \sigma , k ) \bigr\rangle & \equiv & \lim_{\tau\rightarrow 0 + } { v_{a}}^{ren } ( \sigma,\tau;k ) \left| b_{n } \right\rangle i \frac{\partial}{\partial a } \bigl| \hat{b}_{n}[a];a ; ( \sigma , k ) \bigr\rangle \right|_{a=0}~.\end{aligned}\ ] ] the oscillator representation can be obtained from ( [ aux 1 gluon bs ] ) . this yields @xmath724 ; ( \sigma , k)\bigr\rangle = \left . i \frac{\partial}{\partial a } \bigl| \hat{b}_{n}[a ] ; a ; ( \sigma , k)\bigr\rangle \right|_{a=0 } } \nonumber\\ & & = \left[-\frac{1}{2\pi}a_{\mu}(k)\theta^{\mu\nu}k_{\nu } + \sum_{n=1}^{\infty}\sqrt{2\alpha'}a_{\mu}(k ) \left\ { { \left(\frac{1}{e^{t}}g\right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } e^{in\sigma } - { \left(\frac{1}{e}g\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n } e^{-in\sigma } \right\}\right]\nonumber\\ & & \hspace{1.5em}\times { \bigl |}b_{n } ; ( \sigma , k){\bigr\rangle}~ , \label{1 gluon bs}\end{aligned}\ ] ] where @xmath725 is the boundary state with a single open - string tachyon given in ( [ one tachyon bn ] ) . let us recall that generators of diff@xmath702 are @xmath726 @xmath727 . their actions on the states @xmath728;a ; ( \sigma , k)\bigr\rangle$ ] turn out to be as follows : @xmath729;a ; ( \sigma , k ) \bigr\rangle } \nonumber\\ & & = e^{im\sigma } \left [ \sqrt{2\alpha'}\sum_{n=1}^{\infty}\left\ { k_{\mu}-i(n+m)a a_{\mu}(k ) \right\ } { \left(\frac{1}{e^{t}}g\right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } e^{in\sigma } \right . \nonumber\\ & & \hspace{4em } -\sqrt{2\alpha ' } \sum_{n=1 } \left\{k_{\mu}+i(n - m)a a_{\mu}(k ) \right\ } { \left(\frac{1}{e}g\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n}e^{-in\sigma } \nonumber\\ & & \hspace{4em } + \alpha ' mk_{\mu}g^{\mu\nu}k_{\nu } + iaa_{\mu}(k ) \left(\frac{m}{2\pi}\theta^{\mu\nu } -\alpha ' m^{2 } g^{\mu\nu}\right ) k_{\nu } \nonumber\\ & & \hspace{4em } -a^{2 } \alpha ' \frac{1}{6}m(m-1)(m+1 ) a_{\mu}(k)\,g^{\mu\nu}a_{\nu}(k ) \bigg ] \bigl|\hat{b}_{n}[a];a ; ( \sigma , k)\bigr\rangle~,\end{aligned}\ ] ] for @xmath730 . as regards the states @xmath731 ; ( \sigma , k)\bigr\rangle$ ] , the actions of @xmath188 can be read from the above by differentiating it with respect to @xmath679 and then setting @xmath732 . we obtain @xmath733 ; ( \sigma , k)\bigr\rangle & = & e^{im\sigma } \left\ { -i\frac{\partial}{\partial \sigma } + m\left(\alpha'k_{\mu}g^{\mu\nu}k_{\nu}+1\right ) \right\ } \bigl| b_{n}[a ] ; ( \sigma , k)\bigr\rangle \nonumber\\ & & + e^{im\sigma}\alpha ' m^{2}k_{\mu}g^{\mu\nu}a_{\nu}(k)\ ; { \bigl |}b_{n};(\sigma , k ) { \bigr\rangle}~. \label{eq : ishibashi - virasoro}\end{aligned}\ ] ] the rhs can be compared with the standard virasoro action on ( open - string ) gluon vertex operators of the same momenta and polarizations . we find that they are identical with each other . the ishibashi condition imposed on @xmath734 ; ( \sigma , k)\bigr \rangle$ ] , namely the vanishing of the rhs of eq.([eq : ishibashi - virasoro ] ) , modulo total derivative with respect to @xmath216 , requires the well - known physical state condition of the gauge field : @xmath735 by the same argument presented for the tachyon the above action is translated in closed - string field theory to the action of the brst charge @xmath225 . the brst invariance of these boundary states becomes precisely the on - shell condition of the gauge field . previous constructions ( [ aux 1 gluon bs ] ) and ( [ 1 gluon bs ] ) are generalized to the cases of @xmath260 off - shell gluons . let us start with the states @xmath736 @xmath737 of the auxiliary renormalized operators @xmath711 are distinct with each other , satisfying the condition @xmath738 . the oscillator representation can be obtained by taking account of the relation ( [ renormalized operators ] ) and then using the ope ( [ ope renormalized gluon ] ) . it turns out to be @xmath739 \nonumber\\ & & \hspace{2em } \times \prod_{n=1}^{\infty } \exp \bigg [ \sqrt{\frac{\alpha'}{2}}\frac{1}{n } \sum_{r=1}^{m } \bigg\ { \left(k^{(r)}_{\mu}-ina_{r } a_{\mu}(k^{(r)})\right ) \left(z_{r}^{n}\delta^{\mu}_{\nu}+\bar{z}_{r}^{-n } { \left(g^{-1}n^{t}\right)^{\mu}}_{\nu } \right ) \alpha^{\nu}_{-n } \nonumber\\ & & \hspace{11.5em } + \left(k^{(r)}_{\mu}+ina_{r } a_{\mu}(k^{(r)})\right ) \left ( \bar{z}_{r}\delta^{\mu}_{\nu } + z_{r}^{-n}{\left(g^{-1}n\right)^{\mu}}_{\nu } \right ) \tilde{\alpha}^{\nu}_{-n } \bigg\ } \bigg]\nonumber\\ & & \hspace{2em}\times g_{n } \left|\sum_{r=1}^{m}k_{n}^{(r)}\right\rangle~. \label{m gluons + bn}\end{aligned}\ ] ] limits @xmath740 of the above states may be interpreted as auxiliary boundary states of @xmath260 off - shell gluons . they are auxiliary since the parameters @xmath741 are still included . this limiting process corresponds to sending the operators onto the world - sheet boundary . we then obtain the following states : @xmath722;\{a_{r}\ } ; ( \sigma_{1},k^{(1)}),(\sigma_{2},k^{(2 ) } ) , \cdots,(\sigma_{m},k^{(m)})\bigr\rangle } \nonumber\\ & & \equiv \lim_{\forall \tau_{r}\rightarrow 0 + } { \hat{v}_{a}}^{ren}\left(\sigma_{1},\tau_{1};k^{(1)};a_{1}\right ) \cdots { \hat{v}_{a}}^{ren}\left(\sigma_{m},\tau_{m};k^{(m)};a_{m}\right ) \left| b_{n}\right\rangle \nonumber\\ & & = \left(\frac{-\det g_{\mu\nu}}{\left(2\alpha'\right)^{p+1 } } \right)^{\frac{1}{4 } } e^ { \frac{i}{2\pi } \sum_{r=1}^{m } \left ( a_{r}a_{\mu}(k^{(r ) } ) -\sigma_{r } k_{\mu}^{(r ) } \right ) \theta^{\mu\nu } \sum_{s=1}^{m}k_{\nu}^{(s ) } } \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \hspace{2em } \times \prod_{r < s}^{m } \exp \left[2\alpha ' g^{\mu\nu } \left ( k^{(r)}_{\mu}-a_{r } a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r}}\right ) \left ( k^{(s)}_{\nu}-a_{s } a_{\nu}(k^{(s ) } ) \partial_{\sigma_{s}}\right ) \ln \left|e^{i\sigma_{r } } -e^{i\sigma_{s } } \right| \right ] \nonumber\\ & & \hspace{2em } \times \prod_{n=1}^{\infty } \exp\left [ \frac{\sqrt{2\alpha'}}{n } \sum_{r=1}^{m } \left\ { \left(k_{\mu}^{(r)}-ina_{r } a_{\mu}(k^{(r)})\right ) { \left(\frac{1}{e^{t}}g\right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } e^{in\sigma_{r } } \right . \nonumber\\ & & \hspace{12em } \left.\left . + \left(k^{(r)}_{\nu}+ina_{r } a_{\mu}(k^{(r)})\right ) { \left(\frac{1}{e}g\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n } e^{-in\sigma_{r}}\right\}\right]\ , g_{n } \left|\sum_{r=1}^{m}k^{(r)}_{n}\right\rangle~. \nonumber\\ \label{aux m gluon bs}\end{aligned}\ ] ] we note that the boundary states ( [ m gluon bs ] ) become available in the absence of @xmath0 field by putting @xmath285 . the dual boundary states are required in order to obtain the closed - string tree amplitudes . we construct these states by taking the same route as the tachyon case . we begin by considering the bogolubov transform of @xmath747 generated by @xmath748 which is the bpz dual of @xmath681 given in eq.([gn dagger ] ) . we have @xmath749 here we introduce a regular operator @xmath750 defined as @xmath751 with @xmath752~,\nonumber\\ \tilde{\mathcal{m}}_{a \infty}(z;k;a ) & = & \prod_{n=1}^{\infty } \exp \left[-\sqrt{\frac{\alpha'}{2 } } \frac{1}{n } \left(k_{\mu}-ina a_{\mu}(k)\right ) { \left(g^{-1}n\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{n}z^{n } \right]~.\end{aligned}\ ] ] @xmath753 in the above denotes the singular factor . it takes the same form as @xmath689 with @xmath179 and @xmath754 replaced by @xmath297 and @xmath755 respectively : @xmath756~. \end{aligned}\ ] ] this factor represents the self - contraction between the chiral and the anti - chiral pieces of @xmath747 . it can be written in terms of the chiral anti - chiral correlation @xmath757 of green s function @xmath304 : @xmath758~.\end{aligned}\ ] ] dual of the auxiliary renormalized gluon vertex operator , @xmath759 is introduced by the following subtractions : @xmath760 where the finite subtraction has been made as required previously . dual of the renormalized gluon operator , which we call @xmath761 , is obtained from the above by using the relation ( [ vg and hvg ] ) : @xmath762 we now come to computations of closed - string tree amplitudes between the boundary states ( [ def m gluon bs ] ) . it is a straightforward generalization of what we did for the boundary states of off - shell open - string tachyons but becomes much complicated . we can factorize the amplitudes into products of two kinds of contributions from the zero - modes and the massive modes of closed - string . by using the oscillator representations ( [ aux m gluon bs ] ) and ( [ dual aux m gluon bs ] ) these become as follows : @xmath773;\{a_{r}\ } ; ( \sigma_{m+1},k^{(m+1)}),\cdots , ( \sigma_{m+n},k^{(m+n ) } ) \right| \nonumber\\ & & \hspace{5em } \times q_{c}^{\frac{1}{2 } \left(l_{0}+\tilde{l}_{0}-2 \right ) } \left| \hat{b}_{n}[a];\{a_{r}\};(\sigma_{1},k^{(1 ) } ) , \cdots , ( \sigma_{m},k^{(m ) } ) \right \rangle \nonumber\\ & & = \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1 } } \right)^{\frac{1}{2}}\ ; \delta^{(p+1 ) } \left(\sum_{r=1}^{m+n } k^{(r ) } \right)\;\ ; q_{c}^{-1-\frac{\alpha'}{4}\sum_{r=1}^{m}\sum_{s = m+1}^{m+n } k_{\mu}^{(r ) } g^{\mu\nu } k_{\nu}^{(s ) } } \nonumber\\ & & ~~ \times \prod_{1\leq r < s \leq m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r } - \sigma_{s } ) } \prod_{m+1 \leq r < s \leq m+n } e^{-\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r } - \sigma_{s } ) } \nonumber\\ & & ~~\times e^{\frac{i}{2\pi } \sum_{r , s=1}^{m}\left(a_{r}a_{\mu}(k^{(r ) } ) -\sigma_{r } k_{\mu}^{(r)}\right ) \theta^{\mu\nu}k_{\nu}^{(s ) } } \ ; e^{- \frac{i}{2\pi } \sum_{r , s = m+1}^{m+n}\left(a_{r}a_{\mu}(k^{(r ) } ) -\sigma_{r } k_{\mu}^{(r)}\right ) \theta^{\mu\nu}k_{\nu}^{(s ) } } \nonumber\\ & & ~~\times \prod_{1\leq r < s \leq m } \exp \left [ 2 \alpha ' g^{\mu\nu } \left(k_{\mu}^{(r)}-a_{r } a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r } } \right ) \left(k_{\mu}^{(s)}-a_{s } a_{\mu}(k^{(s ) } ) \partial_{\sigma_{s } } \right ) \ln \left| e^{i\sigma_{r } } - e^{i\sigma_{s } } \right| \right ] \nonumber\\ & & ~~\times \prod_{m+1\leq r < s \leq m+n } \exp \left [ 2 \alpha ' g^{\mu\nu } \left(k_{\mu}^{(r)}-a_{r}a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r } } \right ) \left(k_{\mu}^{(s)}-a_{s } a_{\mu}(k^{(s ) } ) \partial_{\sigma_{s } } \right ) \ln \left| e^{i\sigma_{r } } - e^{i\sigma_{s } } \right| \right ] \nonumber\\ & & ~~\times f_{a}\left ( q_{c } , \left\ { \sigma_{r}\right\ } , \left\ { k^{(r ) } \right\ } ; \left\{a_{r}\right\ } \right)~. \label{pre gluon amplitude by boundary state}\end{aligned}\ ] ] here @xmath774 represents the sum of contributions from the massive modes of closed - string . it is given by the following infinite products : @xmath775 \nonumber\\ & & \hspace{3em } \times \exp \left [ -\frac{1}{n}\alpha^{\mu}_{-n}n_{\mu\nu } \tilde{\alpha}^{\nu}_{-n } \right . \nonumber\\ & & \hspace{6.5em } + \frac{\sqrt{2\alpha'}}{n } \sum_{r=1}^{m}\left\ { \left ( k_{\mu}^{(r ) } -ina_{r } a_{\mu}(k^{(r ) } ) \right ) { \left(\frac{1}{e^{t}}g\right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } e^{in\sigma_{r } } \right.\nonumber\\ & & \hspace{12.5em } \left.\left . + \left(k_{\mu}^{(r)}+ina_{r } a_{\mu}(k^{(r)})\right ) { \left(\frac{1}{e}g\right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n } e^{-in\sigma_{r}}\right\}\right ] |0\rangle~. \nonumber \\ \label{def of fa}\end{aligned}\ ] ] we need to evaluate the above infinite products . these are carried out in appendix [ sec : formulae ] . we just quote the result obtained there . these turn out to be as follows : @xmath776 \nonumber\\ & & \quad \times \prod_{m+1\leq r < s \leq m+n } \exp \bigg [ 2\alpha ' g^{\mu\nu } \left(k_{\mu}^{(r ) } -a_{r } a_{\mu}(k^{(r)})\partial_{\sigma_{r}}\right ) \left(k_{\nu}^{(s ) } -a_{s } a_{\mu}(k^{(s ) } ) \partial_{\sigma_{s}}\right ) \nonumber\\ & & \hspace{12em } \times \ln \left\ { \frac { \prod_{n=1}^{\infty } \left(1-e^{i(\sigma_{r}-\sigma_{s})}q_{c}^{n}\right ) \left(1-e^{-i(\sigma_{r}-\sigma_{s } ) } q_{c}^{n } \right ) } { \prod_{n=1}^{\infty } \left(1-q_{c}^{n}\right)^{2 } } \right\ } \bigg ] \nonumber\\ & & \quad \times \prod_{r=1}^{m } \prod_{s = m+1}^{m+n } \exp \bigg [ 2\alpha ' g^{\mu\nu } \left(k_{\mu}^{(r ) } -a_{r } a_{\mu}(k^{(r)})\partial_{\sigma_{r}}\right ) \left(k_{\nu}^{(s ) } -a_{s } a_{\mu}(k^{(s ) } ) \partial_{\sigma_{s}}\right ) \nonumber\\ & & \hspace{12em } \times \ln \left\ { \frac { \prod_{m=0}^{\infty } \left(1-e^{i(\sigma_{r}-\sigma_{s})}q_{c}^{m+\frac{1}{2 } } \right ) \left(1-e^{-i(\sigma_{r}-\sigma_{s } ) } q_{c}^{m+\frac{1}{2 } } \right ) } { \prod_{n=1}^{\infty } \left(1-q_{c}^{n}\right)^{2 } } \right\ } \bigg ] \nonumber\\ & & \quad \times \exp \left [ 2\alpha'g^{\mu\nu } \sum_{r=1}^{m+n } \left(a_{r}\right)^{2 } a_{\mu}(k^{(r ) } ) a_{\nu}(k^{(r ) } ) \ ; \ln \left\{\prod_{n=1}^{\infty } \left(1-q_{c}^{n}\right)\right\ } \right]~. \label{result on fa}\end{aligned}\ ] ] we have used the total momentum conservation , @xmath777 to obtain the above expression . the last exponential will be ignored since the exponent is proportional to @xmath778 and this term brings about nothing when @xmath779 operate on the amplitudes . the amplitudes which are obtained by plugging eq.([result on fa ] ) into eq.([pre gluon amplitude by boundary state ] ) may be written down by using the elliptic @xmath327-functions . with a similar manipulation which leads eq.([tachyon amplitude by boundary state ] ) we can recast the amplitudes into the following forms : @xmath780;\{a_{r}\ } ; ( \sigma_{m+1},k^{(m+1 ) } ) , \cdots , ( \sigma_{m+n},k^{(m+n ) } ) \right| } \nonumber\\ & & \hspace{4em } \times q_{c}^{\frac{1}{2 } \left(l_{0}+\tilde{l}_{0}-2 \right ) } \left| \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{1},k^{(1 ) } ) , \cdots , ( \sigma_{m},k^{(m ) } ) \right \rangle \nonumber\\ & & = \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1 } } \right)^{\frac{1}{2}}\ ; q_{c}^{\frac{p-25}{24 } } \eta \left(\tau^{(c)}\right)^{-p+1 } \;\delta^{(p+1 ) } \left(\sum_{r=1}^{m+n } k^{(r ) } \right ) \nonumber\\ & & \quad \times \prod_{1\leq r < s \leq m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r } - \sigma_{s } ) } \prod_{m+1 \leq r < s \leq m+n } e^{-\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r } - \sigma_{s } ) } \nonumber\\ & & \quad \times q_{c}^{-\frac{1}{16\pi^{2 } \alpha ' } k_{\mu}(\theta g \theta)^{\mu\nu}k_{\nu } } \ ; \prod_{r=1}^{m+n } e^{\frac{i}{2\pi } \left ( a_{r}a_{\mu}(k^{(r ) } ) -\sigma_{r } k_{\mu}^{(r ) } \right ) \theta^{\mu\nu}k_{\nu } } \nonumber\\ & & \quad \times \prod_{1\leq r < s \leq m } \exp \bigg [ 2\alpha ' g^{\mu\nu } \left(k_{\mu}^{(r ) } -a_{r } a_{\mu}(k^{(r)})\partial_{\sigma_{r } } \right ) \nonumber\\ & & \hspace{10em } \times \left(k_{\mu}^{(s)}-a_{s } a_{\mu}(k^{(s)})\partial_{\sigma_{s } } \right ) \ln \left\{\frac{\theta_{1}\left.\left ( \frac{|\sigma_{r}-\sigma_{s}|}{2\pi } \right| \tau^{(c)}\right ) } { \eta\left(\tau^{(c)}\right)^{3 } } \right\ } \bigg ] \nonumber\\ & & \quad \times \prod_{m+1\leq r < s \leq m+n } \exp \bigg [ 2\alpha ' g^{\mu\nu } \left(k_{\mu}^{(r ) } -a_{r}a_{\mu}(k^{(r)})\partial_{\sigma_{r } } \right ) \nonumber\\ & & \hspace{12em } \times \left(k_{\mu}^{(s ) } -a_{s } a_{\mu}(k^{(s)})\partial_{\sigma_{s } } \right ) \ln \left\{\frac{\theta_{1}\left.\left ( \frac{|\sigma_{r}-\sigma_{s}|}{2\pi } \right| \tau^{(c)}\right ) } { \eta\left(\tau^{(c)}\right)^{3 } } \right\ } \bigg ] \nonumber\\ & & \quad \times \prod_{r=1}^{m } \prod_{s = m+1}^{m+n } \exp \bigg [ 2\alpha ' g^{\mu\nu } \left(k_{\mu}^{(r ) } -a_{r } a_{\mu}(k^{(r)})\partial_{\sigma_{r } } \right ) \nonumber\\ & & \hspace{12em } \times \left(k_{\mu}^{(s ) } -a_{s } a_{\mu}(k^{(s)})\partial_{\sigma_{s } } \right ) \ln \left\{\frac{\theta_{4}\left.\left ( \frac{\sigma_{r}-\sigma_{s}}{2\pi } \right| \tau^{(c)}\right ) } { \eta\left(\tau^{(c)}\right)^{3 } } \right\ } \bigg]~ , \nonumber \\ \label{aux gluon amplitude by boundary state}\end{aligned}\ ] ] where we put @xmath781 . in the above expression we have included the contribution of the ghosts , @xmath782 . we compare the amplitudes ( [ aux gluon amplitude by boundary state ] ) , taking account of closed - string propagations along the dirichlet directions , with open - string gluon one - loop amplitudes . the discussion goes parallel to the case of the tachyon . we first describe the open - string calculation . we use the same conventions as in subsection [ sec : open - string tachyon one - loop ] . open - string gluon vertex operator of momentum @xmath135 is given by @xmath783 where @xmath784 and @xmath377 are respectively the complex and the radial coordinates of the upper half - plane ( open - string world sheet ) . @xmath670 are the polarization vectors or the fourier modes of the @xmath479 gauge field @xmath785 . we consider the scattering process of @xmath383 gluons with momenta @xmath384 . diagram relevant to the one - loop scattering process can be drawn on the upper half - plane as depicted in figure [ upper - half - plane ] . the corresponding gluon amplitude , which we call @xmath786 , is given by a sum of traces of their products arranged in cyclically distinct orders with keeping their partial orderings at the each end : @xmath787 in order to obtain the above amplitude it becomes convenient to introduce the gluon vertex operator in an exponential form . let @xmath676 be an auxiliary parameter . we put @xcite @xmath788 this operator is related with the vertex operator ( [ def of open - string gluon vertex ] ) by @xmath789 the corresponding amplitude constructed from @xmath790 instead of @xmath791 will be called @xmath792 . it is given by @xmath793 the gluon scattering amplitude @xmath794 can be obtained from @xmath792 by differentiating it with respect to @xmath795 and making them vanish : @xmath796 we can evaluate the amplitude @xmath792 in the standard manner . the open - string propagators in ( [ def of open - string aux gluon amplitude ] ) may be replaced by the integral forms . the above virasoro action implies : @xmath801 utilizing these properties we can write the amplitude in the following form : @xmath802 the coordinates @xmath390 in the rhs are insertion points of the auxiliary gluon vertices and provide a parametrization of the diagram . another parametrization can be obtained by mapping the diagram to the cylinder with width @xmath391 as depicted in figure [ open - string diagram ] . correspondingly @xmath390 are mapped to @xmath392 by @xmath393 . we put @xmath394 . these are the open - string parameters . low - energy world - volume theory of open - string gluons in the presence of a constant @xmath0 field is a @xmath479 gauge theory on the non - commutative world - volume . in this section , taking the same route as the previous study of the non - commutative scalar field theory , we investigate the uv behavior of the non - commutative gauge theory . our study in this section is restricted to the case of @xmath421-brane in the critical dimensions . we examine two zero - slope limits of the one - loop amplitudes of gluons . one is based on the open - string parameters and the other is on the closed - string parameters . these two limits , as we explained in the study of the non - commutative scalar field theory , are complementary . we first take a zero - slope limit based on the open - string parameters @xmath440 . eq.([aux gluon one - loop by open - string parameters ] ) may be used as an integral form of the amplitude . strictly speaking , the amplitude is obtained from this integral by using the relation ( [ ia by hatia ] ) . the zero - slope limit will be a gauge theory one - loop amplitude , particularly written in terms of the schwinger parameters , @xmath22 and @xmath423 . these parameters are related with @xmath16 and @xmath392 by the relations ( [ schwinger parameters of scalar field theory ] ) . the auxiliary parameters @xmath795 are also used in the amplitude ( [ aux gluon one - loop by open - string parameters ] ) . they are introduced in order to describe the gluon vertex operators ( [ def of open - string gluon vertex ] ) in the auxiliary forms ( [ def of open - string aux gluon vertex ] ) and to make the loop calculation tractable . at the zero - slope limit the gluon vertex operator become @xmath810 , where @xmath811 is a world - line parametrized by the schwinger parameter @xmath812 . the auxiliary vertex operator is expected to be @xmath813 in the world - line description . here @xmath814 is an auxiliary parameter ( a counterpart of @xmath679 in the field theory ) . a simple dimensional analysis shows that @xmath814 is dimensionful and proportional to @xmath815 . therefore the zero - slope limit must be taken by fixing the following field theory parameters in the amplitude : @xmath816 simultaneously we also need to fix open - string tensors @xmath200 and @xmath201 . we rewrite the amplitude ( [ aux gluon one - loop by open - string parameters ] ) in terms of the above parameters and then pick up the dominant contribution of the @xmath424-expansion . these are parallel to what we did in the previous section to obtain the zero - slope limit ( [ open - string ftl of one - loop amplitude ] ) of the tachyon amplitude . by using the relation ( [ ia by hatia ] ) , we find that the following integral turns out to be the zero - slope limit : @xmath817 , \label{open - string ftl of gluon one - loop amplitude}\end{aligned}\ ] ] where @xmath818 take values @xmath819 such that @xmath820 for @xmath262 and @xmath821 for @xmath822 , and the integral is performed over the region ( [ moduli in open - string ftl ] ) . if we neglect @xmath823 besides their differentiations the above zero - slope limit reduces to eq.([open - string ftl of one - loop amplitude ] ) , modulo the factor @xmath824 . this power has a simple origin in the scaling relations , @xmath825 . it was discussed previously that @xmath826 in the integral comes from the schwinger representation of the open - string tachyon propagator , @xmath827 . presently the same term may be interpreted as an ir regularization of the amplitude by an analytic continuation of @xmath828 to a small positive @xmath829 . the above zero - slope limit can be identified with the corresponding one - loop amplitude of the non - commutative @xmath479 gauge theory . some related calculation in the gauge theory may be found in @xcite@xcite . ) may be compared with eq.(4.9 ) of @xcite . the integration measure used in that paper is different from ours . ] similarly to the scalar field theory , @xmath830 in the above integral is understood as a uv regularization @xcite which depends on the external momentum @xmath460 . in the gauge theory as well , this causes the problem of uv - ir mixing and makes the field theory description at high energy scale difficult . as explained in section [ sec : uv nc scalar ] , the open- and closed - string parameters used in the descriptions of the amplitude are two kinds of coordinates of the moduli space of conformal classes of cylinder with @xmath383 punctures at the boundaries . two ends of the moduli space which are located at @xmath25 and @xmath18 play important roles in the zero - slope limits . the field theory amplitude ( [ open - string ftl of gluon one - loop amplitude ] ) is obtained by a suitable magnification of the integral ( [ aux gluon one - loop by open - string parameters ] ) on an infinitesimal neighbourhood around @xmath18 . we need to focus on the region @xmath442 to know the uv behavior of the gauge theory . as is the case of the scalar field theory , a possible resolution is to take a zero - slope limit such that an infinitesimal neighbourhood around @xmath25 is magnified . it becomes effective to use the closed - string parameters near this end . the zero - slope limit which we examine is essentially the same as that was examined for the open - string tachyons . explicitly we take the limit by fixing the following parameters : @xmath831 in the limiting process we fix open - string tensors @xmath200 and @xmath201 as well to capture the world - volume theory . the auxiliary parameters in the above are the rescaled ones used in the integral ( [ aux gluon one - loop by closed - string parameters ] ) . in contrast with the previous scaling we keep them intact . this is because the present limit is not expected to allow a naive world - line description . as can be observed in the expression ( [ equivalence with boundary state formalism for gluon ] ) these parameters are nothing but the auxiliary parameters used for the description of gluon vertex operators in the boundary state formalism . as was pointed out in the previous study of the scalar field theory , the zero - slope limit based on the closed - string parameters describes physics at the trans - string scale of the world - volume theory . all the perturbative stringy states of open - string contribute to the limit . they bring about a striking contrast between the two limits . we can find the propagator of closed - string tachyon of momentum @xmath460 in the limit ( [ closed - string ftl of one - loop amplitude for gluon ] ) while the counterpart in ( [ open - string ftl of gluon one - loop amplitude ] ) is the curious regularization factor . in that limit there appear terms describing correlations between two gluons inserted at @xmath423 and @xmath462 . these correlations are caused by the kinetic energies of the propagating gluons . but they are lost in ( [ closed - string ftl of one - loop amplitude for gluon ] ) and gluons become topological in this limit . the disappearance originates in the modular transforms ( [ modular trans of tachyon correlations ] ) . it was shown in @xcite disk amplitudes of a closed - string tachyon scattering with arbitrary number of gluons turn out to generate a _ straight _ open wilson line in seiberg - witten s zero - slope limit . the path is a straight line connecting @xmath38 and @xmath843 , where @xmath645 is the momentum of the tachyon . the displacement @xmath844 is required by the gauge invariance @xcite . in this section we first reproduce the above result by using the boundary states constructed in section [ sec : gluon boundary state ] . computations provided below become also helpful for our subsequent investigations of generically curved open wilson lines . closed - string tachyon state with momentum @xmath845 and its bpz dual state are given by @xmath846 disk amplitude of the closed - string tachyon scattering with @xmath260 gluons is obtained in the boundary state formalism by integrating the overlap @xmath847 ; ( \sigma_{1},k^{(1)}),\cdots , ( \sigma_{m},k^{(m ) } ) \right\rangle\right.$ ] on its moduli space . to evaluate this overlap , we start by computing the overlap with the auxiliary boundary states . this becomes @xmath848;\{a_r\ } ; ( \sigma_{1},k^{(1)}),\cdots , ( \sigma_{m},k^{(m ) } ) \right\rangle\right . } \nonumber\\ & & = \left(\frac{-\det g_{\mu\nu } } { ( 2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \ , \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left(p_{0}+\sum_{r=1}^{m}k^{(r ) } \right ) \nonumber\\ & & \quad \times \prod_{r < s}^{m } \exp \left [ 2\alpha ' g^{\mu\nu } \left(k_{\mu}^{(r)}-a_r a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r } } \right ) \left(k_{\nu}^{(s)}-a_s a_{\nu}(k^{(s ) } ) \partial_{\sigma_{s } } \right ) \ln \left|e^{i\sigma_{r}}-e^{i\sigma_{s } } \right| \right]\nonumber\\ & & \quad \times \exp \left[-\frac{i}{2\pi}\sum_{r=1}^{m } \left(a_ra_{\mu}(k^{(r ) } ) -\sigma_{r } k_{\mu}^{(r)}\right ) \theta^{\mu\nu}p_{0\nu}\right]~. \label{eq : overlap with tachyon 1}\end{aligned}\ ] ] the zero - slope limit can be read from the rhs as follows : @xmath849;\{a_r\ } ; ( \sigma_{1},k^{(1 ) } ) , \cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \right . } \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu } } { ( 2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \ , \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left(p_{0}+\sum_{r=1}^{m}k^{(r ) } \right ) \nonumber\\ & & \quad \times \exp \left[-\frac{i}{2\pi } \sum_{r=1}^{m}\left(a_ra_{\mu}(k^{(r ) } ) -\sigma_{r } k_{\mu}^{(r)}\right ) \theta^{\mu\nu}p_{0\nu}\right]~. \label{eq : overlap with tachyon 2}\end{aligned}\ ] ] this gives @xmath850;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \right.\nonumber\\ & & \quad = \left . \prod_{r=1}^{m } \left ( i \frac{\partial}{\partial a_r}\right ) \right|_{a_r=0 } \left\langle -p_{0n}\left| \hat{b}_{n}[a];\ { a_r\};(\sigma_{1},k^{(1 ) } ) , \cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \right . \nonumber\\ & & \quad \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1 } } \right)^{\frac{1}{4 } } \ , \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left(p_{0}+\sum_{r=1}^{m}k^{(r ) } \right ) \nonumber\\ & & \qquad \quad \times \prod_{r=1}^{m } \left ( \frac{dy_{0}^{\mu } ( \sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) e^{ik_{\nu}^{(r ) } y^{\nu}_{0 } ( \sigma_{r } ) } \right ) \nonumber\\ & & \quad = \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \ , \int \frac{d^{p+1}x}{(2\pi)^{p+1 } } e^{ip_{0\mu}x^{\mu } } e^{-\frac{i}{2}(k_{\mu}^{(1)}+\cdots + k^{(m)}_{\mu } ) \theta^{\mu\nu}p_{0\nu } } \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \qquad \quad \times \prod_{r=1}^{m } \left\ { \frac{dy_{0}^{\mu } ( \sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) e^{ik_{\nu}^{(r)}\left(x^{\nu}+ y^{\nu}_{0 } ( \sigma_{r})\right ) } \right\}~ , \label{eq : overlap with tachyon 3}\end{aligned}\ ] ] where @xmath851 @xmath852 is a straight line defined as @xmath853 in the above we have used the following equations to obtain the last equality : @xmath854 the relevant moduli parameters are @xmath406 with @xmath855 @xmath856 @xmath857 . their integration will give us the amplitude . eq.([eq : overlap with tachyon 3 ] ) yields @xmath858;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \right . } \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1 } } \right)^{\frac{1}{4 } } \int \frac{d^{p+1}x}{(2\pi)^{p+1 } } \int_{0\leq \sigma_{1}\leq \sigma_{2 } \leq \cdots \leq \sigma_{m } \leq 2\pi } \prod_{r=1}^{m } d\sigma_{r } \nonumber\\ & & \qquad \left\ { \frac{dy_{0}^{\mu } ( \sigma_{1})}{d\sigma_{r } } a_{\mu}(k^{(1 ) } ) e^{ik_{\nu}^{(1 ) } \left(x^{\nu}+ y^{\nu}_{0 } ( \sigma_{1})\right ) } \right\ } \star \left\ { \frac{dy_{0}^{\mu } ( \sigma_{2})}{d\sigma_{r } } a_{\mu}(k^{(2 ) } ) e^{ik_{\nu}^{(2 ) } \left(x^{\nu}+ y^{\nu}_{0 } ( \sigma_{2})\right ) } \right\ } \star \cdots\nonumber\\ & & \qquad \qquad \qquad \cdots \star \left\ { \frac{dy_{0}^{\mu } ( \sigma_{m})}{d\sigma_{r } } a_{\mu}(k^{(m ) } ) e^{ik_{\nu}^{(m ) } \left(x^{\nu}+ y^{\nu}_{0 } ( \sigma_{m})\right ) } \right\ } \star e^{ip_{0\mu}x^{\mu}}~ , \label{eq : overlap with tachyon 4}\end{aligned}\ ] ] where the moyal products are taken with respect to @xmath473 . the straight open wilson line is obtained from the above equation by summing up with respect to @xmath260 as follows : @xmath859;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \right . \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1 } } \right)^{\frac{1}{4 } } \int \frac{d^{p+1}x}{(2\pi)^{p+1 } } \left [ \mathcal{p}_{\star } \exp \left ( i \int_{0}^{2\pi } d\sigma \frac{dy^{\mu}_{0}(\sigma)}{d\sigma } \mathcal{a}_{\mu } \left(x+y_{0}(\sigma)\right ) \right ) \right ] \star e^{ip_{0\mu}\hat{x}^{\mu}}~ , \label{eq : straight open wilson line}\end{aligned}\ ] ] where the path is taken along the straight line @xmath860 . taking account of the fact that closed - string tachyons give rise to straight open wilson lines , dhar and kitazawa suggested in @xcite that fluctuations of open wilson lines should originate in the massive states of closed - string . they conjectured a possible correspondence between the perturbative massive states of closed - string and the gauge theory operators obtained as the coefficients in a perturbative expansion of open wilson line ( the harmonic expansion at the straight line ) . in this subsection we prove their conjecture . we show , in a self - contained fashion , how one can obtain curved open wilson lines and present an explicit correspondence between their fluctuations and the closed - string states . let @xmath621 be a loop in the momentum space . the harmonic expansion is given by @xmath861 @xmath862 $ ] . we first introduce the following _ out_-state of closed - string : @xmath863 one may think of this state as a stringy extension , ( or a generalization to include the massive modes ) , of the closed - string tachyon state . the oscillator representation can be read as @xmath864~. \label{eq : oscillator expression of momentum loop state}\ ] ] this tells us that we can write the state in terms of the coherent state given in appendix [ sec : formulae ] as follows : @xmath865 where @xmath866 are the coherent states @xmath867 of the @xmath215-th levels defined in eq.([eq : dual of coherent state ] ) with setting the complex variables @xmath868 and @xmath869 . one can readily find that for @xmath65 @xmath870 recalling @xmath631 is the complex conjugate of @xmath625 , one can think of the state @xmath871 as a real section of the coherent state of closed - string . eq.([eq : oscillator expression of momentum loop state ] ) also yields the following equalities : @xmath872 where @xmath873 and @xmath874 are integers greater than or equal to zero . this implies that the state @xmath871 is a generating function of the closed - string states which are off - shell in general . now we wish to compute overlaps of the above state , instead of the tachyon , with the boundary states . since it is a generating function of the closed - string states , these overlaps give us a generating function of amplitudes between closed - string states and gluons . we will show in the below that the zero - slope limit of this generating function is nothing but an open wilson line taken along a curve parametrized by @xmath875 and @xmath876 . by using eq.([eq : momentum loop and coherent state 2 ] ) , we obtain @xmath877;\{a_r\};(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . } \nonumber\\ & & = \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left(p_{0 } + \sum_{r=1}^{m}k^{(r)}\right ) \nonumber\\ & & \quad \times \prod_{r < s}^{m } \exp \left[2\alpha ' g^{\mu\nu } \left ( k_{\mu}^{(r)}-a_r a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r } } \right ) \left ( k_{\nu}^{(s)}-a_s a_{\nu}(k^{(s ) } ) \partial_{\sigma_{s } } \right ) \ln \left|e^{i\sigma_{r}}-e^{i\sigma_{s}}\right| \right ] \nonumber\\ & & \quad \times \prod_{n=1}^{\infty } \exp \left [ -\frac{\alpha'}{4n}\bar{\psi}_{n\mu } \left(g^{-1}ng^{-1}\right)^{\mu\nu}\psi_{n\nu } \right ] \nonumber\\ & & \quad \times \exp \left[-i \sum_{r=1}^{m } a_ra_{\mu}(k^{(r ) } ) \right.\nonumber\\ & & \hspace{8em } \left . \times \left\ { \frac{\theta^{\mu\nu}}{2\pi}p_{0\nu}+ \frac{\alpha'}{\sqrt{2 } } \sum_{n=1}^{\infty}\left ( \left(\frac{1}{e}\right)^{\mu\nu}\psi_{n\nu}e^{-in\sigma_{r } } -\left(\frac{1}{e^{t}}\right)^{\mu\nu } \bar{\psi}_{n\nu } e^{in\sigma_{r}}\right)\right\}\right ] \nonumber\\ & & \quad \times \exp\left [ i\sum_{r=1}^{m}k_{\mu}^{(r ) } \left\{\frac{\theta^{\mu\nu}}{2\pi}p_{0\nu}\sigma_{r}+ \frac{\alpha'}{\sqrt{2 } } \sum_{n=1}^{\infty } \frac{i}{n } \left ( \left(\frac{1}{e}\right)^{\mu\nu}\psi_{n\nu } e^{-in\sigma_{r } } + \left(\frac{1}{e^{t}}\right)^{\mu\nu}\bar{\psi}_{n\nu } e^{in\sigma_{r}}\right ) \right\}\right]~.\nonumber\\ \label{eq : overlap with momentum loop 1}\end{aligned}\ ] ] the zero - slope limit can be read from the rhs . in the present limiting procedure , we have @xmath878 this enables us to find out that @xmath879;\{a_r\};(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left(p_{0 } + \sum_{r=1}^{m}k^{(r)}\right ) \nonumber\\ & & \quad \times\prod_{n=1}^{\infty}\exp \left [ -\frac{\bar{\psi}_{n\mu}\left(\theta g \theta\right)^{\mu\nu } \psi_{n\nu } } { 16\pi^{2}n\alpha ' } \right ] \exp \left [ -i\sum_{r=1}^{m}\left\ { a_r \frac{dy^{\mu}(\sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) -k^{(r)}_{\mu } y^{\mu}(\sigma_{r})\right\ } \right]~. \nonumber \\ \label{eq : overlap with momentum loop 2}\end{aligned}\ ] ] here we introduce @xmath880 with @xmath881 where @xmath882 it is a curve deviating from the straight line @xmath883 and the deviation is denoted by @xmath884 . it follows from eq.([eq : overlap with momentum loop 2 ] ) that @xmath885;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . } \nonumber\\ & & = \left.\prod_{r=1}^{m } \left(i \frac{\partial}{\partial a_r}\right ) \right |_{a_r=0 } \left\langle \omega ( \psi_{n},\bar{\psi}_{n};p_{0})\left| \hat{b}_{n}[a];\{a_r\}(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1 } } \right)^{\frac{1}{4 } } \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left(p_{0 } + \sum_{r=1}^{m}k^{(r)}\right ) \nonumber\\ & & \quad \times\prod_{n=1}^{\infty}\exp \left [ -\frac{\bar{\psi}_{n\mu}\left(\theta g \theta\right)^{\mu\nu } \psi_{n\nu } } { 16\pi^{2}n\alpha ' } \right ] \prod_{r=1}^{m } \left\ { \frac{dy^{\mu}(\sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) e^{ik^{(r)}_{\nu } y^{\nu}(\sigma_{r } ) } \right\}~. \label{eq : overlap with momentum loop 3}\end{aligned}\ ] ] this takes essentially the same form as eq.([eq : overlap with tachyon 3 ] ) with the straight line @xmath883 replaced by the curved one @xmath880 . therefore , performing the same rearrangement as carried out in the last subsection , we obtain @xmath886;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{n=1}^{\infty}\exp \left [ -\frac{\bar{\psi}_{n\mu}\left(\theta g \theta\right)^{\mu\nu } \psi_{n\nu } } { 16\pi^{2}n\alpha ' } \right ] \nonumber\\ & & \qquad \quad \times \int \frac{d^{p+1}x}{(2\pi)^{p+1 } } \left[\mathcal{p}_{\star } \exp \left ( i\int_{0}^{2\pi } d\sigma \frac{dy^{\mu}(\sigma)}{d\sigma } \mathcal{a}_{\mu}(x+y(\sigma ) \right)\right ] \star e^{ip_{0\mu}x^{\mu}}~. \label{eq : overlap with momentum loop 4}\end{aligned}\ ] ] this is the open wilson line taken along the curve @xmath887 . thus we have shown that the zero - slope limit of the generating function of the amplitudes between closed - string states and gluons becomes the open wilson line multiplied by a gaussian weight . the path is curved by @xmath625 and @xmath631 . as can be seen in eq.([eq : generating function ] ) , these variables originally measure condensations of the @xmath215-th massive modes of closed - string . this completes the proof of the conjecture . as has been mentioned , the overlap @xmath888;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right.$ ] serves as a generating function of the amplitudes between closed - string states and gluons . as an illustration , let us consider the case of gravitons . the graviton states can be written as follows : @xmath889 where @xmath890 denotes the polarization tensor of graviton which is symmetric and traceless . using eq.([eq : overlap with momentum loop 3 ] ) , we obtain @xmath891;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle } \nonumber \\ & & = \left . \frac{4}{\alpha ' } \frac{\partial^{2}}{\partial \bar{\psi}_{1\mu}\partial \psi_{1\nu } } \left\langle \omega \left(\psi_{n},\bar{\psi}_{n};p_{0}\right ) \left| b_{n};(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . \right|_{\psi,\bar{\psi}=0 } \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{r=1}^{m } e^{ik_{\mu}^{(r)}\theta^{\mu\nu}p_{0\nu } \frac{\sigma_{r}}{2\pi } } \prod_{r < s}^{m}e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \delta^{(p+1 ) } \left ( p_{0\mu}+\sum_{r=1}^{m}k_{\mu}^{(r)}\right ) \nonumber\\ & & \quad \times \frac{2}{\alpha ' } \left(-\frac{1}{2\pi}\right)^{m } \left[-\frac{\left(\theta g\theta \right)^{\mu\nu}}{8\pi^{2}\alpha ' } \prod_{u=1}^{m}\left(p_{0}\theta a(k^{(u ) } ) \right ) \right.\nonumber\\ & & \hspace{2em } + \sum_{r=1}^{m } \left\ { \left(\theta a ( k^{(r ) } ) \right)^{\mu } \frac{\left(\theta k^{(r ) } \right)^{\nu}}{2\pi } -\frac{\left(\theta k^{(r)}\right)^{\mu}}{2\pi } \left(\theta a(k^{(r)})\right)^{\nu } \right.\nonumber\\ & & \hspace{8em } \left . + i\frac{\left(\theta k^{(r)}\right)^{\mu}}{2\pi } i\frac{\left(\theta k^{(r)}\right)^{\nu}}{2\pi } \left(p_{0}\theta a(k^{(r)})\right ) \right\ } \prod_{u\neq r}\left(p_{0}\theta a(k^{(u ) } ) \right)\nonumber\\ & & \hspace{2em } + \sum_{r\neq s } \left\ { \left(i\frac{\left(\theta k^{(r)}\right)^{\mu } \left(p_{0}\theta a(k^{(r)})\right ) } { 2\pi } -i\left(\theta a(k^{(r)})\right)^{\mu}\right ) e^{i\sigma_{r } } \right . \nonumber\\ & & \hspace{4.5em } \left . \times \left(i\frac{\left(\theta k^{(s)}\right)^{\nu } \left(p_{0}\theta a(k^{(s)})\right ) } { 2\pi } + i\left(\theta a(k^{(s)})\right)^{\nu}\right ) e^{-i\sigma_{s}}\right\ } \prod_{u\neq r , s}\left(p_{0}\theta a(k^{(u)})\right ) \right]~. \nonumber \\ \label{eq : overlap with graviton 2}\end{aligned}\ ] ] this reproduces the result obtained in appendix a of @xcite . in fact , combining the above equation with eq.([eq : graviton out - state ] ) , we obtain @xmath892;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle } \nonumber \\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{r=1}^{m } e^{ik_{\mu}^{(r)}\theta^{\mu\nu}p_{0\nu } \frac{\sigma_{r}}{2\pi } } \prod_{r < s}^{m}e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \prod_{\mu=0}^{p } \delta \left ( p_{0\mu}+\sum_{r=1}^{m}k_{\mu}^{(r)}\right ) \nonumber\\ & & \quad \times \frac{2}{\alpha ' } \left(-\frac{1}{2\pi}\right)^{m } \left [ \sum_{r=1}^{m } \frac{1}{(2\pi)^{2 } } k_{\mu}^{(r)}\left(\theta h(p_{0})\theta\right)^{\mu\nu } k_{\nu}^{(r ) } \prod_{u=1}^{m}\left(p_{0}\theta a(k^{(u ) } ) \right ) \right.\nonumber\\ & & \hspace{3.5em } + \sum_{r\neq s}\left\ { \left ( \frac{\left(p_{0}\theta a(k^{(r)})\right)}{2\pi } k_{\mu}^{(r ) } -a_{\mu}(k^{(r ) } ) \right ) \left(\theta h(p_{0})\theta\right)^{\mu\nu } \left(\frac{\left(p_{0}\theta a(k^{(s)})\right)}{2\pi } k_{\nu}^{(s)}+a_{\nu}(k^{(s)})\right ) \right . \nonumber\\ & & \hspace{8em } \left.\left.\times e^{i(\sigma_{r}-\sigma_{s } ) } \prod_{u\neq r , s}\left(p_{0}\theta a(k^{(u)})\right ) \right\}\right]~ , \label{overlap with graviton 3}\end{aligned}\ ] ] where the properties of @xmath890 are used in the following manner : @xmath893 it is worth emphasizing that eq.([overlap with graviton 3 ] ) is derived without using the on - shell conditions of graviton , i.e. neither @xmath894 or @xmath895 . wilson lines in gauge theories are invariant under reparametrizations of the paths . transformations analogous to the reparametrizations are generated by @xmath896 in closed - string theory . in fact , @xmath896 can be identified with vector fields @xmath897 on the world - sheet . ] and at @xmath7 , where the boundary states reside , these vector fields have the forms of @xmath898 . we have observed that the action of diff@xmath702 on the boundary states is identified with the action of the closed - string brst charge . the boundary states are in general not brst - closed . hence the reparametrization invariance of open wilson lines indicates that the action of diff@xmath702 or the brst charge @xmath225 becomes null at the zero - slope limit . let us verify the above observation . by making use of eqs.([ishibashi condition ] ) and ( [ eq : ishibashi - virasoro ] ) , we obtain for @xmath899 @xmath900;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \nonumber\\ & & \quad = \sum_{r=1}^{m } e^{in\sigma_{r } } \left\{-i\frac{\partial}{\partial\sigma_{r } } + n\left(\alpha ' k^{(r)}_{\mu}g^{\mu\nu}k_{\nu}^{(r ) } + 1\right)\right\ } \left|b_{n}[a];(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \nonumber\\ & & \qquad + \sum_{r=1}^{m } e^{in\sigma_{r}}\alpha ' k_{\mu}^{(r)}g^{\mu\nu } a_{\nu}(k^{(r ) } ) \left|b_{n}[a]_{\check{r } } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle~ , \label{eq : ishibashi - virasoro 2}\end{aligned}\ ] ] where @xmath901_{\check{r } } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle$ ] denotes @xmath902 combining eq.([eq : ishibashi - virasoro 2 ] ) with eq.([eq : overlap with momentum loop 4 ] ) , we find that in the zero - slope limit @xmath903;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \nonumber\\ & & \approx \int \prod_{r=1}^{m } \frac{d^{p+1}k^{(r)}}{(2\pi)^{\frac{p+1}{2 } } } \int_{0\leq\sigma_{1 } \leq \cdots \leq \sigma_{m } \leq 2\pi } \prod_{s=1}^{m } d\sigma_{s } \nonumber\\ & & \hspace{3em } \times \sum_{r=1}^{m } \frac{\partial}{\partial \sigma_{r } } \epsilon_{n}e^{in\sigma_{r } } \left\langle \omega ( \psi_{n},\bar{\psi}_{n};p_{0})\left| b_{n}[a];(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle\right . \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{n=1}^{\infty } \exp \left [ -\frac{\bar{\psi}_{n\mu } \left(\theta g\theta\right)^{\mu\nu } \psi_{n\nu}}{16\pi^{2}n\alpha ' } \right ] \int \frac{d^{p+1}x}{(2\pi)^{p+1 } } \int_{0\leq\sigma_{1 } \leq \cdots \leq \sigma_{m } \leq 2\pi } \prod_{s=1}^{m } d\sigma_{s } \nonumber\\ & & \hspace{1.3em}\sum_{r=1}^{m } \frac{\partial}{\partial\sigma_{r } } \epsilon_{n}e^{in\sigma_{r } } \left\ { \frac{dy^{\mu}(\sigma_{1})}{d\sigma_{1 } } \mathcal{a}_{\mu}(x+y(\sigma_{1 } ) ) \right\}\star \cdots\star \left\ { \frac{dy^{\mu}(\sigma_{m})}{d\sigma_{m } } \mathcal{a}_{\mu}(x+y(\sigma_{m } ) ) \right\ } \star e^{ip_{0\mu}x^{\mu}}~,\nonumber\\ \label{eq : reparametrization 1}\end{aligned}\ ] ] where @xmath904 denotes an infinitesimal parameter . this shows that at the zero - slope limit @xmath896 give rise to infinitesimal reparametrizations @xmath216 @xmath905 @xmath906 , where @xmath907 . this can be checked by recalling that the pull - back @xmath908 on the path transforms under the infinitesimal reparametrizations as @xmath909 , where @xmath910 from this , we can write the integrand in the rhs of eq.([eq : reparametrization 1 ] ) as follows : @xmath911 the above integrand is a total derivative with respect to each @xmath912 . the @xmath912 integrations in ( [ eq : reparametrization 1 ] ) give rise to surface terms . let us show that these surface terms actually cancel out . taking account of the integration region of @xmath912 being @xmath913 $ ] for @xmath914 , the following equality holds for arbitrary @xmath915 : @xmath916 this leads us to find @xmath917~. \nonumber \\ \label{eq : moduli - sekibun}\end{aligned}\ ] ] here we have used the cyclic property of the moyal product inside the integration : @xmath918 . let us recall that @xmath919 plays a role of a translation generator on the non - commutative space - time : @xmath920 combined with the relations @xmath921 and @xmath922 , this tells us that eq.([eq : moduli - sekibun ] ) is vanishing and so is eq.([eq : reparametrization 1 ] ) . therefore we have shown that the action of diff@xmath702 on the boundary states reduces to infinitesimal reparametrizations of paths of open wilson lines and becomes null at the zero - slope limit . the situation is summarized as follows : @xmath923;(\sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{p+1}}\right)^{\frac{1}{4 } } \prod_{n=1}^{\infty } \exp \left [ -\frac{\bar{\psi}_{n\mu } \left(\theta g \theta\right)^{\mu\nu } \psi_{n\nu } } { 16 \pi^{2}n\alpha ' } \right ] \nonumber\\ & & \qquad \delta^{(n)}_{\epsilon } \int \frac{d^{p+1}x}{(2\pi)^{p+1 } } \mathcal{p}_{\star } \left [ \exp \left ( i\int^{2\pi}_{0 } d\sigma \frac{y^{\mu}(\sigma)}{d\sigma } \mathcal{a}_{\mu}(x+y(\sigma))\right)\right ] \star e^{ip_{0\mu}x^{\mu}}\nonumber\\ & & = 0~.\end{aligned}\ ] ] we have seen in section [ sec : open wilson lines ( i ) ] that analogues of open wilson line , @xmath924 $ ] , factorize the generating function of one - loop amplitudes of open - string tachyons at the zero - slope limit . decompositions of the string amplitudes made by insertions of the unity , @xmath925 |p_{n}\rangle \langle p_{n}|$ ] , play an important role to gain the factorization ( [ closed - string ftl by open wilson lines ] ) . with the same manipulation one can expect that a similar factorization is also obtainable for the amplitudes of gluons . prior to the actual computations , let us explain briefly why the decompositions via the momentum eigenstates give us open wilson lines or their analogues . to make the discussion transparent we start with the momentum eigenstate @xmath926 . as described in eq.([eq : pn - shift ] ) , we can write the state as @xmath927 \times { } _ { b=0}\!\left\langle b_{n}\right| : \exp\left(i\int_{0}^{2\pi } d\sigma p_{\mu}(\sigma ) \hat{x}^{\mu}(\sigma)\right):~.\ ] ] taking account of its use in the decompositions of the amplitudes we should consider the state @xmath928 rather than @xmath926 . let us also recall that the above @xmath625 and @xmath631 are rescaled appropriately in order to obtain the factorization ( [ closed - string ftl by open wilson lines ] ) and that the rescaled variables are kept intact under taking the limit . to be explicit , we denote the rescaled @xmath875 and @xmath876 by the following @xmath929 and @xmath930 : @xmath931 oscillator representation of the state @xmath928 can be read as follows by using these rescaled variables : @xmath932~.\end{aligned}\ ] ] now it is clear that this state reduces to the state @xmath933 at the zero - slop limit : @xmath934 \nonumber\\ & & = e^{-\frac{\pi s^{(c)}}{4 } \left(p_{0\mu } g^{\mu\nu}p_{0\nu}-\frac{4}{\alpha'}\right ) } \prod_{n=1}^{\infty } \exp\left\{\frac{\psi'_{n\mu}\left(\theta g\theta\right)^{\mu\nu } \bar{\psi}'_{n\nu } } { 32\pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right\ } \left\langle \omega ( \psi'_{n},\bar{\psi}'_{n};p_{0})\right|~.\end{aligned}\ ] ] as we have seen in subsection [ sec : curved owl ] the above state is a generating function of the closed - string states and its overlaps with the boundary states lead the open wilson line taken along the corresponding path . we will compute the factorization of the generating function of one - loop amplitudes of gluons . we will restrict ourselves to @xmath421-brane in the critical dimensions . computations become parallel to those given in section [ sec : open wilson lines ( i ) ] but much complicated . the factorization at the zero - slope limit by open wilson lines are given in eq.([eq : factorization of gluons 6 ] ) . let us study factorizations of the following amplitudes : @xmath935 ; ( \sigma_{m+1},k^{(m+1)}),\cdots , ( \sigma_{m+n},k^{(m+n ) } ) \right| q_c^{\frac{1}{2}(l_0+\tilde{l}_0 - 2 ) } \nonumber\\ & & \hspace{4em } \times \left| b_{n}[a ] ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle~. \label{amplitude 1}\end{aligned}\ ] ] oscillator representations of the boundary states of gluons are unsuitable for computations of the amplitudes . we instead consider the corresponding amplitudes in auxiliary forms . let us decompose the string amplitudes ( [ def of aux gluon amplitude by bs ] ) by using the momentum eigenstates as follows : @xmath936;\{a_{r}\ } ; ( \sigma_{m+1},k^{(m+1)}),\cdots,(\sigma_{m+n},k^{(m+n ) } ) \right| \nonumber\\ & & \hspace{5em}\times q_{c}^{\frac{1}{2}\left(l_{0}+\tilde{l}_{0}-2\right ) } \left| \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \nonumber\\ & & = \int \left[dp_{n}\right ] \bigl\langle \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{m+1},k^{(m+1)}),\cdots,(\sigma_{m+n},k^{(m+n ) } ) \bigr| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \bigl|-p_{n}\bigr\rangle \nonumber\\ & & \hspace{4em } \times \bigl\langle -p_{n } \bigr| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \bigl| \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \bigr\rangle~. \label{eq : factorization of gluons 1}\end{aligned}\ ] ] here we have changed the integration variables from @xmath621 to @xmath937 for the later convenience . the amplitudes ( [ amplitude 1 ] ) are obtained from the above by the operation @xmath938 . each factor in the above decomposition can be evaluated by using the oscillator representations . these are given in eq.([aux m gluon bs ] ) ( and eq.([dual aux m gluon bs ] ) ) for the boundary states , and eq.([state pn ] ) ( and its hermitian conjugate ) for the momentum eigenstates . we then need to compute matrix elements similar to eq.([difficult matrix element ] ) . eq.([eq : vevee ] ) enables us to calculate them . after the rescaling ( [ eq : rescaled psi ] ) of @xmath625 and @xmath631 we obtain the following expressions : @xmath939 \bigl\langle \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{m+1},k^{(m+1)}),\cdots,(\sigma_{m+n},k^{(m+n ) } ) \bigr| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \bigl|-p_{n } \bigr\rangle\ \nonumber\\ & & \hspace{4em } \times \left\langle -p_{n } \left| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \left| \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \right.\right . \nonumber\\ & & = \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{26}}\right)^{\frac{1}{2 } } \int d^{26}p_{0}\ \delta^{26 } \left(p_{0}+\sum_{r=1}^{m}k^{(r)}\right ) \delta^{26 } \left ( p_{0}-\sum_{r = m+1}^{m+n}k^{(r)}\right ) \ , q_{c}^{\frac{\alpha'}{4}g^{\mu\nu}p_{0\mu}p_{0\nu } -1 } \nonumber\\ & & \quad \times \prod_{1\leq r < s \leq m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \prod_{m+1 \leq r < s \leq m+m } e^{-\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \quad \times \prod_{1\leq r < s \leq m } \exp \left [ \alpha ' g^{\mu\nu } \left(k_{\mu}^{(r)}-a_{r } a_{\mu}(k^{(r)})\partial_{\sigma_{r } } \right ) \left(k_{\nu}^{(s)}-a_{s } a_{\nu}(k^{(s)})\partial_{\sigma_{s } } \right ) \ln \left|e^{i\sigma_{r } } -e^{i\sigma_{s}}\right|^{2 } \right ] \nonumber\\ & & \quad \times \prod_{m+1\leq < s \leq m+n } \exp \left [ \alpha ' g^{\mu\nu } \left(k_{\mu}^{(r)}-a_{r } a_{\mu}(k^{(r)})\partial_{\sigma_{r } } \right ) \left(k_{\nu}^{(s)}-a_{s } a_{\nu}(k^{(s)})\partial_{\sigma_{s } } \right ) \ln \left|e^{i\sigma_{r } } -e^{i\sigma_{s}}\right|^{2 } \right ] \nonumber\\ & & \quad \times \prod_{n=1}^{\infty } \exp \left [ -\frac{2\alpha'q_{c}^{\frac{n}{2}}}{n } \left(\frac{1}{e^{t}-q_{c}^{\frac{n}{2}}e } g \frac{1}{e^{t } } \right)^{\mu\nu } \right.\nonumber\\ & & \hspace{6.5em } \times \left\ { \sum_{r , s=1}^{m } \left(k_{\mu}^{(r)}-ina_{r } a_{\mu}(k^{(r)})\right ) \left(k_{\nu}+ina_{s } a_{\nu}(k^{(s ) } ) \right ) e^{in(\sigma_{r}-\sigma_{s } ) } \right.\nonumber\\ & & \hspace{8em } + \left.\left . \sum_{r , s = m+1}^{m+n } \left(k_{\mu}^{(r)}+ina_{r } a_{\mu}(k^{(r)})\right ) \left(k_{\nu}-ina_{s } a_{\nu}(k^{(s ) } ) \right ) e^{-in(\sigma_{r}-\sigma_{s } ) } \right\ } \right]\nonumber\\ & & \quad \times \int \prod_{n=1}^{\infty } \left [ \frac{d^{26}\bar{\psi}_{n } d^{26}\psi_{n}}{(2i)^{26 } } \left(\frac{\alpha'}{4n\pi q_{c}^{\frac{n}{2 } } } \right)^{26 } \frac{-\det g_{\mu\nu } } { \det^{2 } \left(g - q_{c}^{\frac{n}{2}}n\right)_{\mu\nu } } \right ] \nonumber\\ & & \quad \times \exp \left [ -\sum_{n=1}^{\infty } \frac{\alpha'}{4n } \bar{\psi}_{n\mu}\left ( \frac{1}{q_{c}^{\frac{n}{2}}g } + \frac{1}{g - q_{c}^{\frac{n}{2}}n^{t } } n^{t } \frac{1}{g } + \frac{1}{g}n \frac{1}{g - q_{c}^{\frac{n}{2}}n } \right)^{\mu\nu}\psi_{n\nu}\right ] \nonumber\\ & & \quad \times \exp \left [ -i \sum_{r=1}^{m } a_{r } a_{\mu}(k^{(r)})\left [ \frac{\theta^{\mu\nu}}{2\pi}p_{0\nu } + \frac{\alpha'}{\sqrt{2 } } \sum_{n=1}^{\infty } \left\ { \left(\frac{1}{e - q_{c}^{\frac{n}{4}}e^{t}}\right)^{\mu\nu } \psi_{n\nu}e^{-in\sigma_{r } } \right.\right.\right.\nonumber\\ & & \hspace{21em } \left.\left.\left . -\left(\frac{1}{e^{t}-q_{c}^{\frac{n}{4}}e}\right)^{\mu\nu } \bar{\psi}_{n\nu } e^{in\sigma_{r}}\right\ } \right]\right]\nonumber\\ & & \quad \times \exp \left[i\sum_{r=1}^{m}k_{\mu}^{(r)}\left [ \frac{\theta^{\mu\nu}}{2\pi}p_{0\nu}\sigma_{r } + \frac{\alpha'}{\sqrt{2}}\sum_{n=1}^{\infty}\frac{i}{n } \left\ { \left(\frac{1}{e - q_{c}^{\frac{n}{4}}e^{t}}\right)^{\mu\nu } \psi_{n\nu}e^{-in\sigma_{r } } \right.\right.\right.\nonumber\\ & & \hspace{20em } \left.\left.\left . + \left(\frac{1}{e^{t}-q_{c}^{\frac{n}{4}}e}\right)^{\mu\nu } \bar{\psi}_{n\nu } e^{in\sigma_{r}}\right\ } \right]\right]\nonumber\\ & & \quad \times \exp \left [ -i \sum_{r = m+1}^{m+n } a_{r } a_{\mu}(k^{(r)})\left [ \frac{\theta^{\mu\nu}}{2\pi}p_{0\nu } -\frac{\alpha'}{\sqrt{2 } } \sum_{n=1}^{\infty } \left\ { \left(\frac{1}{e^{t}-q_{c}^{\frac{n}{4}}e}\right)^{\mu\nu } \psi_{n\nu}e^{-in\sigma_{r } } \right.\right.\right.\nonumber\\ & & \hspace{22em } \left.\left.\left . -\left(\frac{1}{e - q_{c}^{\frac{n}{4}}e^{t}}\right)^{\mu\nu } \bar{\psi}_{n\nu } e^{in\sigma_{r}}\right\ } \right]\right]\nonumber\\ & & \quad \times \exp \left[i\sum_{r = m+1}^{m+n}k_{\mu}^{(r)}\left [ \frac{\theta^{\mu\nu}}{2\pi}p_{0\nu}\sigma_{r } -\frac{\alpha'}{\sqrt{2}}\sum_{n=1}^{\infty}\frac{i}{n } \left\ { \left(\frac{1}{e^{t}-q_{c}^{\frac{n}{4}}e}\right)^{\mu\nu } \psi_{n\nu}e^{-in\sigma_{r } } \right.\right.\right.\nonumber\\ & & \hspace{20em } \left.\left.\left . + \left(\frac{1}{e - q_{c}^{\frac{n}{4}}e^{t}}\right)^{\mu\nu } \bar{\psi}_{n\nu } e^{in\sigma_{r}}\right\ } \right]\right]~ , \label{eq : factorization of gluons 2}\end{aligned}\ ] ] where we have newly written @xmath940 as @xmath547 . we examine the zero - slope limit of the above expression . the limiting procedure we consider is the same that was investigated in section [ sec : uv nc gauge ] to capture the uv behavior of the non - commutative gauge theory . it is taken by fixing parameters @xmath630 , @xmath406 and @xmath795 besides the open - string tensors . @xmath625 and @xmath631 in eq.([eq : factorization of gluons 2 ] ) are also left intact . this yields @xmath939 \bigl\langle \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{m+1},k^{(m+1)}),\cdots,(\sigma_{m+n},k^{(m+n ) } ) \bigr| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \bigl|-p_{n } \bigr\rangle \nonumber\\ & & \hspace{4em } \times \bigl\langle -p_{n } \bigr| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \bigl| \hat{b}_{n}[a];\{a_{r}\ } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \bigr\rangle \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{26 } } \right)^{\frac{1}{2 } } \int d^{26}p_{0 } \exp\left\ { -\frac{\pi s^{(c)}}{2 } \left(p_{0\mu}g^{\mu\nu}p_{0\nu}-\frac{4}{\alpha'}\right)\right\ } \nonumber\\ & & \quad \times \int \prod_{n=1}^{\infty } \left [ \frac{d^{26}\bar{\psi}_{n } d^{26}\psi_{n}}{(2i)^{26 } } \left(\frac{1}{\pi } \right)^{26 } \left\ { -\det \left ( - \frac{\left(\theta g\theta\right)^{\mu\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right\ } \exp\left(\frac{\bar{\psi}_{n\mu}\left(\theta g\theta\right)^{\mu\nu } \psi_{n\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right ] \nonumber\\ & & \qquad \times \delta^{26}\left(p_{0}+\sum_{r=1}^{m}k^{(r ) } \right ) \prod_{1\leq r < s \leq m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \hspace{6em } \times \exp\left[-i\sum_{r=1}^{m}\left\ { a_{r } \frac{dy^{\mu}(\sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) -k_{\mu}^{(r ) } y^{\mu}(\sigma_{r } ) \right\}\right ] \nonumber\\ & & \qquad \times \delta^{26}\left(p_{0}-\sum_{r = m+1}^{m+n}k^{(r ) } \right ) \prod_{m+1\leq r < s \leq m+n } e^{-\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \hspace{6em } \times \exp\left[-i\sum_{r = m+1}^{m+n}\left\ { a_{r } \frac{dy^{\mu}(\sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) -k_{\mu}^{(r ) } y^{\mu}(\sigma_{r } ) \right\}\right]~ , \label{eq : factorization of gluons 3}\end{aligned}\ ] ] where @xmath880 is the curve given by eq.([eq : curved path ] ) . the zero - slope limit of the amplitudes ( [ amplitude 1 ] ) can be obtained from the above by the operations @xmath941 , which are carried out without difficulty . the zero - slope limits turn out to be as follows : @xmath939 \left\langle b_{n}[a ] ; ( \sigma_{m+1},k^{(m+1)}),\cdots , ( \sigma_{m+n},k^{(m+n ) } ) \right| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \left|-p_{n}\right\rangle \nonumber\\ & & \hspace{4em } \times \left\langle -p_{n}\right| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \left| b_{n}[a ] ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \right\rangle \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{26 } } \right)^{\frac{1}{2 } } \int d^{26}p_{0 } \exp \left\ { -\frac{\pi s^{(c)}}{2 } \left(p_{0\mu}g^{\mu\nu}p_{0\nu}-\frac{4}{\alpha'}\right ) \right\ } \nonumber\\ & & \quad \times \int \prod_{n=1}^{\infty } \left [ \frac{d^{26}\bar{\psi}_{n } d^{26}\psi_{n}}{(2i)^{26 } } \left(\frac{1}{\pi } \right)^{26 } \left\ { -\det \left(-\frac{\left(\theta g\theta\right)^{\mu\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right\ } \exp\left(\frac{\bar{\psi}_{n\mu}\left(\theta g\theta\right)^{\mu\nu } \psi_{n\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right ] \nonumber\\ & & \quad\qquad \times \delta^{26}\left(p_{0}+\sum_{r=1}^{m}k^{(r ) } \right ) \prod_{1\leq r < s \leq m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \hspace{7em } \times \prod_{r=1}^{m } \left\ { \frac{dy^{\mu}(\sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) e^{ik_{\nu}^{(r)}y^{\nu}(\sigma_{r})}\right\ } \nonumber\\ & & \quad\qquad \times \delta^{26}\left(p_{0}-\sum_{r = m+1}^{m+n}k^{(r ) } \right ) \prod_{m+1\leq r < s \leq m+n } e^{-\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \hspace{7em } \times \prod_{r = m+1}^{m+n } \left\ { \frac{dy^{\mu}(\sigma_{r})}{d\sigma_{r } } a_{\mu}(k^{(r ) } ) e^{ik_{\nu}^{(r)}y^{\nu}(\sigma_{r})}\right\}~. \label{eq : factorization of gluons 4}\end{aligned}\ ] ] factorized form of the @xmath383 gluon amplitude is obtained by integrating the above amplitude over the moduli ( [ moduli in closed - string ftl ] ) , where we set @xmath412 in order to fix the @xmath479 symmetry . this gives rise to the asymmetric term of the factorization ( [ closed - string ftl for gluon by straight open wilson lines ] ) at the zero - slope limit . in order to avoid complexity of expressions we ignore this gauge fixing in the below and integrate @xmath450 over @xmath942 . the integrations over @xmath406 turn out to be written by using the moyal products : @xmath943 \nonumber\\ & & \quad \left\langle b_{n}[a];(\sigma_{m+1},k^{(m+1 ) } ) , \cdots , ( \sigma_{m+n},k^{(m+n)})\right| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \left|-p_{n}\right\rangle\nonumber\\ & & \hspace{3em}\times \left\langle -p_{n}\right| q_{c}^{\frac{1}{4}\left(l_{0}+\tilde{l}_{0}-2\right ) } \left| b_{n}[a];(\sigma_{1},k^{(1)}),\cdots , ( \sigma_{m},k^{(m)})\right\rangle \nonumber\\ & & \approx \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{26 } } \right)^{\frac{1}{2 } } \int d^{26}p_{0 } \exp \left\ { -\frac{\pi s^{(c)}}{2 } \left(p_{0\mu}g^{\mu\nu}p_{0\nu}-\frac{4}{\alpha'}\right ) \right\ } \nonumber\\ & & \quad \times \int \prod_{n=1}^{\infty } \left [ \frac{d^{26}\bar{\psi}_{n } d^{26}\psi_{n}}{(2i)^{26 } } \left(\frac{1}{\pi } \right)^{26 } \left\ { -\det \left(-\frac{\left(\theta g\theta\right)^{\mu\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right\ } \exp\left(\frac{\bar{\psi}_{n\mu}\left(\theta g\theta\right)^{\mu\nu } \psi_{n\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right ] \nonumber\\ & & \hspace{2em } \times\int \frac{d^{26}x}{(2\pi)^{26 } } \left\{\frac{dy^{\mu}(\sigma_{1})}{d\sigma_{1 } } a_{\mu}(k^{(1 ) } ) e^{ik^{(1)}_{\nu}\left(x^{\nu}+y^{\nu}(\sigma_{1})\right ) } \right\ } \nonumber\\ & & \hspace{8em } \star \cdots \star \left\{\frac{dy^{\mu}(\sigma_{m})}{d\sigma_{m } } a_{\mu}(k^{(m ) } ) e^{ik^{(m)}_{\nu}\left(x^{\nu}+y^{\nu}(\sigma_{m})\right ) } \right\ } \star e^{ip_{0\mu}x^{\mu } } \nonumber\\ & & \hspace{2em } \times \int \frac{d^{26}\tilde{x}}{(2\pi)^{26 } } \ ; e^{-ip_{0\mu}\tilde{x}^{\mu}}\star \left\{\frac{dy^{\mu}(\sigma_{m+n})}{d\sigma_{1 } } a_{\mu}(k^{(m+n ) } ) e^{ik^{(m+n)}_{\nu } \left(\tilde{x}^{\nu}+y^{\nu}(\sigma_{m+n})\right ) } \right\ } \nonumber\\ & & \hspace{8em } \star \cdots \star \left\{\frac{dy^{\mu}(\sigma_{m+1})}{d\sigma_{m+1 } } a_{\mu}(k^{(m+1 ) } ) e^{ik^{(m+1)}_{\nu}\left(\tilde{x}^{\nu}+y^{\nu } ( \sigma_{m+1})\right ) } \right\}~. \label{eq : factorization of gluons 5}\end{aligned}\ ] ] factorization of the generating function of the amplitudes ( [ amplitude 1 ] ) at the zero - slope limit can be obtained from the above by integrating out the gluon momenta @xmath944 and then summing up with respect to @xmath260 and @xmath81 . it turns out to have the following form : @xmath945;(\sigma_{m+1},-k^{(m+1 ) } ) , \cdots , ( \sigma_{m+n},-k^{(m+n)})\bigr| \nonumber\\ & & \hspace{15em } \times q_{c}^{\frac{1}{2}\left(l_{0}+\tilde{l}_{0}-2\right ) } \bigl| b_{n}[a];(\sigma_{1},k^{(1)}),\cdots , ( \sigma_{m},k^{(m)})\bigr\rangle \nonumber\\ & & \approx \int d^{26}p_0 \left [ \frac{2\pi}{\alpha ' } \int_0^{+\infty}ds^{(c ) } \exp \left\ { -\frac{\pi s^{(c)}}{2 } \left(p_{0\mu}g^{\mu\nu}p_{0\nu}-\frac{4}{\alpha'}\right ) \right\ } \right ] \nonumber\\ & & \quad \times \int \prod_{n=1}^{\infty } \left [ \frac{d^{26}\bar{\psi}_{n } d^{26}\psi_{n}}{(2\pi i)^{26 } } \left\ { -\det \left ( -\frac{\theta g\theta } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right\ } \exp \left ( \frac{\bar{\psi}_{n\mu}\left(\theta g\theta\right)^{\mu\nu } \psi_{n\nu } } { 16 \pi^{2}nq_{c}^{\frac{n}{2}}\alpha ' } \right ) \right ] \nonumber\\ & & \hspace{2em } \times \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{26 } } \right)^{\frac{1}{4 } } \int \frac{d^{26}x}{(2\pi)^{26 } } \left [ \mathcal{p}_{\star } \exp \left(i\int^{2\pi}_{0 } d\sigma \frac{dy^{\mu}(\sigma)}{d\sigma } \mathcal{a}_{\mu}\left(x+y(\sigma)\right ) \right ) \right]\star e^{ip_{0\mu}x^{\mu } } \nonumber\\ & & \hspace{2em } \times \left(\frac{-\det g_{\mu\nu}}{(2\alpha')^{26 } } \right)^{\frac{1}{4 } } \overline { \int \frac{d^{26}x}{(2\pi)^{26 } } \left [ \mathcal{p}_{\star } \exp \left(i\int^{2\pi}_{0 } d\sigma \frac{dy^{\mu}(\sigma)}{d\sigma } \mathcal{a}_{\mu}\left(x+y(\sigma)\right ) \right ) \right]\star e^{ip_{0\mu}x^{\mu } } } ~. \nonumber \\ \label{eq : factorization of gluons 6}\end{aligned}\ ] ] so far , momentum eigenstates of closed - string play a crucial role in our study . their overlaps with the boundary states provide open wilson lines at the zero - slope limit . the eigenvalues , that is , loops in the momentum space become paths of open wilson lines after suitable rescalings . in this section we wish to deliver the other side of the story . the following discussions include much speculation and therefore they are incomplete . let us first observe that the auxiliary boundary states of gluons are eigenstates of the momentum operator @xmath598 . the boundary state @xmath946;\{a_{r}\};(\sigma_{1},k^{(1 ) } ) , \bigr . $ ] @xmath947 @xmath948 has the eigenvalue equal to @xmath949 @xmath950 @xmath951 . therefore this state is proportional to the eigenstate @xmath952 of the corresponding eigenvalues . this can be seen by a comparison between oscillator representations ( [ aux m gluon bs ] ) and ( [ eq : pn - state ] ) of the boundary state and the momentum eigenstate . in terms of the parametrization ( [ eq : pbn - eigenvalue ] ) the above eigenvalue corresponds to @xmath953 by using these values of @xmath954 and @xmath955 , the precise relation between the two states can be written as follows : @xmath956;\{a_{r}\ } ; ( \sigma_{1},k^{(1)}),\cdots,(\sigma_{m},k^{(m ) } ) \bigr\rangle } \nonumber\\ & & = \prod_{r=1}^{m } \exp \left[\alpha ' g^{\mu\nu } \sum_{n=1}^{\infty } \left ( \frac{1}{n } k^{(r)}_{\mu}k^{(r)}_{\nu } + ( a_{r})^{2 } n a_{\mu}(k^{(r)})a_{\nu}(k^{(r ) } ) \right)\right ] \nonumber\\ & & \quad \times e^{\frac{i}{2\pi}\sum_{r=1}^{m } \left ( a_{r}a_{\mu}(k^{(r)})-\sigma_{r}k_{\mu}^{(r)}\right ) \theta^{\mu\nu}\sum_{s=1}^{m}k^{(s)}_{\nu } } \prod_{r < s}^{m } e^{\frac{i}{2}k_{\mu}^{(r)}\theta^{\mu\nu}k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \quad \times \left| p^{(b)}_{\mu}(\sigma)=\sum_{r=1}^{m } \left(k_{\mu}^{(r ) } -a_{r } a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r}}\right ) \delta ( \sigma -\sigma_{r } ) \right\rangle~ , \label{eq : aux state as eigenstate}\end{aligned}\ ] ] where we have used @xmath957 in ( [ cpb ] ) as the normalization constant of the momentum eigenstate and represented it in terms of open - string tensors . let us comment on the multiplicative factors appearing in the above relation . as can be seen from eq.([boundary limit of disk green function ] ) , the second line on the rhs comes from the terms proportional to @xmath327 of disk green s functions at the world - sheet boundary ( the boundary circle ) . as regard the exponential in the first line on the rhs , we might think of it as being related to short distance singularities between the gluons on the boundary circle . in fact , by recasting the exponent into the following form , @xmath958 we find a singularity similar to that appearing in the ope between auxiliary gluon vertex operators . in the same way , concerning the dual boundary states , we obtain the following relation as well : @xmath959 \nonumber\\ & & \quad \times e^{-\frac{i}{2}\sum_{r=1 } \left(a_{r}a_{\mu}(k^{(r)})-\sigma_{r}k_{\mu}^{(r)}\right ) \theta^{\mu\nu}\sum_{s=1}^{m}k_{\nu}^{(s ) } } \prod_{r < s}^{m } e^{-\frac{i}{2}k_{\mu}^{(r ) } \theta^{\mu\nu } k_{\nu}^{(s ) } \epsilon ( \sigma_{r}-\sigma_{s } ) } \nonumber\\ & & \quad \times \left\langle p^{(b)}_{\mu}(\sigma)= - \sum_{r=1}^{m } \left(k_{\mu}^{(r ) } -a_{r } a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r}}\right ) \delta ( \sigma -\sigma_{r } ) \right|~ , \label{eq : dual gluon bs as eigenstate}\end{aligned}\ ] ] where @xmath960 denotes the hermitian conjugate of @xmath961 . boundary states of open - string tachyons are obtained from auxiliary boundary states of gluons by letting their auxiliary parameters vanish . hence these boundary states are also eigenstates of @xmath962 . the boundary state @xmath963 has the eigenvalue @xmath964 and we can write the state as follows : @xmath965 we would like to discuss physical meanings of the momentum eigenstates appearing in eqs.([eq : aux state as eigenstate ] ) and ( [ eq : tachyon bs as eigenstate ] ) . we henceforth concentrate on gluons . it is because analyses for the tachyon can be carried out in a parallel way to the gluon case and the formulae for gluons reduce to those of tachyons by setting the auxiliary parameters @xmath745 . let us regard the momentum eigenstate @xmath966 @xmath967 as a boundary state . the associated boundary action @xmath968 $ ] can be obtained by following the prescription in @xcite . it is given by @xmath969 } = \left.\left\langle p^{(b)}_{\mu}(\sigma ) = -\sum_{r=1}^{m}\left(k_{\mu}^{(r ) } -a_{r}a_{\mu}(k^{(r)})\partial_{\sigma_{r}}\right ) \delta(\sigma -\sigma_{r } ) \right| x_{n}\right\rangle~ , \label{eq : boundary action1}\ ] ] where @xmath970 is the coordinate eigenstate in ( [ coordinates 2 ] ) . overlaps @xmath971 can be computed for arbitrary eigenvalues by using the formulae ( [ eq : vevee ] ) in the oscillator representations of the eigenstates . the overlaps take the following forms : @xmath972~.\end{aligned}\ ] ] we apply the above formula to eq.([eq : boundary action1 ] ) with putting @xmath973 this enables us to find that the boundary action @xmath968 $ ] becomes , modulo constant terms , @xmath974 \nonumber \\ & & = -i \int^{2\pi}_{0 } d\sigma \left(\frac{1}{2 } b_{\mu\nu } x^{\mu}(\sigma)\partial_{\sigma}x^{\nu}(\sigma ) + x^{\mu}(\sigma ) \sum_{r=1}^{m}\left(k_{\mu}^{(r)}- a_{r}a_{\mu}(k^{(r)})\partial_{\sigma_{r}}\right ) \delta(\sigma-\sigma_{r } ) \right)~. \nonumber \\ \label{eq : boundary action2}\end{aligned}\ ] ] the above boundary action appears naturally in the path - integral formalism of world - sheet theory of string . let @xmath975 be the correlation function of @xmath260 gluons in the auxiliary forms with momenta @xmath976 and polarization vectors @xmath977 on the world - sheet disk . in the path - integral approach the correlation function can be expressed as @xmath978\left [ e^{i\left(k_{\mu}^{(1 ) } -a_{1}a_{\mu}(k^{(1)})\partial_{\sigma_{1}}\right ) x^{\mu}(\sigma_{1 } ) } \cdots e^{i\left ( k^{(m)}_{\mu } -a_{m}a_{\mu}(k^{(m)})\partial_{\sigma_{m}}\right ) x^{\mu}(\sigma_{m})}\right ] e^{-s[x]}~,\end{aligned}\ ] ] where @xmath979 $ ] is the world - sheet action ( [ eq : action2 ] ) . we can recast the rhs of this equation into @xmath980 e^{-s_{\mathrm{eff}}[x]}~,\ ] ] where @xmath981 $ ] is the sum of the bulk free action @xmath982 $ ] and the boundary action @xmath983 $ ] : @xmath984 = s_{0}[x ] + s_{b}[x]$ ] with @xmath985= \frac{1}{4\pi\alpha ' } \int_{\sigma } d\tau d\sigma \partial_{a}x^{m}(\sigma,\tau ) \partial^{a}x^{n}(\sigma,\tau ) g_{mn}~ , \\ & & s_{b}[x]= -i \int_{\partial \sigma } d\sigma \left [ \frac{1}{2 } b_{\mu\nu}x^{\mu}(\sigma)\partial_{\sigma}x^{\nu}(\sigma ) + x^{\mu}(\sigma ) \sum_{r=1}^{m } \left(k_{\mu}^{(r)}-a_{r}a_{\mu}(k^{(r)})\partial_{\sigma_{r } } \right ) \delta ( \sigma-\sigma_{r})\right]~.\nonumber\end{aligned}\ ] ] here we have used @xmath986 . we find that the above boundary action @xmath983 $ ] is identical with @xmath987 $ ] given in eq.([eq : boundary action2 ] ) . boundary conditions of @xmath71 can be read from variation of @xmath981 $ ] . it turns out to be @xmath988 , as expected . these observations on the boundary states and the momentum eigenstates seem to indicate a chance to interpret boundary states of open - string legs as momentum eigenstates of closed - string with eigenvalues being delta functions on the boundary circle . our expectation is , in fact , beyond this . let us recall that the momentum eigenstates are expressed in eqs.([eq : pnb - shift ] ) and ( [ eq : pn - shift ] ) by using operators of the form @xmath989 where @xmath508 are the closed - string coordinate operators and @xmath990 denotes the standard normal - ordering of closed - string . if we forget about @xmath508 being quantum operators , we could write down the exponential , without any hesitation , in the path - ordered form along the boundary circle , @xmath991 $ ] . since @xmath508 are quantum operators the path - ordered exponential becomes vague without a prescription . namely we need to regularize the path - ordered integral . relevant regularization in place of the above normal - ordering will be discretization of the boundary circle . in such a regularization scheme the eigenvalues @xmath26 are regarded as sums of delta functions on the original boundary circle . this leads us to the conjecture : _ momentum eigenstates of closed - string have expansions by means of boundary states with open - string legs _ . finally we would like to mention some related issues . as we carried out to a certain extent in this paper , our boundary states enable us to perform off - shell calculations . this suggests that these states can be used in a covariant formulation of a field theory of interacting open- and closed - strings . string field theory that has prediction power for the open - closed mixed systems has been required also in our understanding of unstable @xmath51-branes and tachyon condensations associated with them @xcite . it seems probable that boundary states with open - string legs constructed here are generalized to vertex functions between open- and closed - strings in such a field theory . we plan to discuss these issues elsewhere . t.n . would like to thank members of particle theory group of kek for their hospitality during his stay in summer , 2002 . the final part of this work was done there . k.m . would like to thank y. kitazawa , n. ishibashi , e. sezgin , t. suyama , t. masuda and p. sundell for discussions and comments . the work of k.m . is supported in part by nsf grant phy-0070964 . let @xmath198 and @xmath199 be a flat space - time metric and a constant two - form gauge field of closed - string . these are called closed - string tensors in the text . @xmath84 is given by their combination as @xmath992 . in @xcite , open - string metric @xmath993 and non - commutativity parameter @xmath754 are obtained respectively from symmetric and anti - symmetric parts of @xmath994 : @xmath995 where @xmath996 . tensors @xmath200 and @xmath201 are called open - string tensors in the text . the above relation implies : @xmath997 it is also possible to express the closed - string tensors by means of the open - string ones : @xmath998 the tensor @xmath999 is used frequently in the text . this tensor enjoys the following relations : @xmath1000 eq.([eq : n - ot ] ) implies @xmath1001 combination of eq.([eq : n - ot ] ) with eq.([eq : ot ] ) gives the following equalities : @xmath1002 we present formulae which become very useful for computations of string amplitudes in the text . as an illustration we derive @xmath1003 in eq.([result on fa ] ) ( and @xmath324 in eq.([result on f ] ) ) . let @xmath1004 , @xmath1005 , @xmath1006 and @xmath1007 be arbitrary complex @xmath668-vectors . let @xmath1008 and @xmath1009 be any complex @xmath1010-matrices . the following equality holds : @xmath1011 \exp \left[\frac{1}{n}\alpha_{-n}^{\mu}\omega^{(2)}_{\mu\nu } \tilde{\alpha}_{-n}^{\nu } + w_{\mu}\alpha^{\mu}_{-n } + y_{\mu}\tilde{\alpha}^{\mu}_{-n}\right ] |\mathbf{0}\rangle } \nonumber\\ & & = \frac{\det g_{\mu\nu } } { \det \left(g-\omega^{(2)t}g^{-1}\omega^{(1)}\right)_{\mu\nu } } \nonumber\\ & & \quad \times \exp \left [ n u_{\mu}\left(\frac{1}{g-\omega^{(2)}g^{-1}\omega^{(1)t } } \right)^{\mu\nu}w_{\nu } + n v_{\mu}\left(\frac{1}{g-\omega^{(2)t}g^{-1}\omega^{(1 ) } } \right)^{\mu\nu}y_{\nu } \right.\nonumber\\ & & \hspace{5em } + n u_{\mu}\left(g^{-1}\omega^{(2 ) } \frac{1}{g-\omega^{(1)t}g^{-1}\omega^{(2)}}\right)^{\mu\nu } v_{\nu } \nonumber\\ & & \hspace{5em } \left . + nw_{\mu}\left(g^{-1}\omega^{(1 ) } \frac{1}{g-\omega^{(2)t}g^{-1}\omega^{(1)}}\right)^{\mu\nu } y_{\nu } \right ] \nonumber\\ & & \quad \times \exp \left [ \frac{1}{n}\alpha^{\mu}_{-n } \left(\omega^{(2)}\frac{1}{g-\omega^{(1)t}g^{-1}\omega^{(2 ) } } g\right)_{\mu\nu } \tilde{\alpha}^{\nu}_{-n } \right . \nonumber\\ & & \hspace{5em } + \left\{v_{\lambda } { \left(g^{-1}\omega^{(2)t}\right)^{\lambda}}_{\mu } + w_{\mu } \right\ } { \left(\frac{1}{g-\omega^{(1)}g^{-1}\omega^{(2)t } } g \right)^{\mu}}_{\nu } \alpha^{\nu}_{-n } \nonumber\\ & & \hspace{5em } \left . + \left\ { u_{\lambda}{\left(g^{-1}\omega^{(2)}\right)^{\lambda}}_{\mu } + y_{\mu}\right\ } { \left(\frac{1}{g-\omega^{(1)t}g^{-1}\omega^{(2 ) } } g \right)^{\mu}}_{\nu } \tilde{\alpha}^{\nu}_{-n } \right ] |\mathbf{0}\rangle~.\label{eq : ee}\end{aligned}\ ] ] [ f-1 ] use of coherent states becomes convenient to see the above equality . let @xmath1012 be complex @xmath668-vectors @xmath1013 . coherent state @xmath1014 of the @xmath215-th level oscillators is defined as @xmath1015 |0\rangle~. \label{eq : coherent state}\ ] ] @xmath1012 becomes eigenvalues of the annihilation operators @xmath1016 and @xmath1017 respectively : @xmath1018 foe each @xmath215 , the coherent states @xmath1019 constitute a ( over)complete basis of the fock space built by @xmath1020 and @xmath1021 . the completeness relation reads @xmath1022 \left(\lambda^{+}_{n},\lambda^{-}_{n } \right|~. \label{eq : complete - cohe}\end{aligned}\ ] ] here @xmath1023 denotes the hermitian conjugate of the state @xmath1024 . it takes the form of @xmath1025~ , \label{eq : dual of coherent state}\ ] ] where @xmath1026 are complex conjugate to @xmath1027 and become eigenvalues of the creation operators @xmath1028 and @xmath1029 . formula [ f-1 ] can be shown by making use of the above partition of unity . we only describe an outline of the proof . we first insert the unity given in eq.([eq : complete - cohe ] ) between the two exponentials on the lhs of eq.([eq : ee ] ) . this makes the lhs into gaussian integrals with respect to @xmath1030 and @xmath1031 . these gaussian integrals are performed successively by using the relation @xmath1032 = \frac{\pi^{p+1}}{\det m^{\mu\nu}}~,\ ] ] for @xmath1033 and any @xmath1010 matrix @xmath1034 . then we obtain the rhs of eq.([eq : ee ] ) . the following formula is a corollary of formula [ f-1 ] : @xmath1035 \exp \left[\frac{1}{n}\alpha_{-n}^{\mu}\omega^{(2)}_{\mu\nu } \tilde{\alpha}_{-n}^{\nu } + w_{\mu}\alpha^{\mu}_{-n } + y_{\mu}\tilde{\alpha}^{\mu}_{-n}\right ] |\mathbf{0 } \rangle } \nonumber\\ & & = \frac{\det g_{\mu\nu } } { \det \left(g-\omega^{(2)t}g^{-1}\omega^{(1)}\right)_{\mu\nu } } \nonumber\\ & & \quad \times \exp \left [ n u_{\mu}\left(\frac{1}{g-\omega^{(2)}g^{-1}\omega^{(1)t } } \right)^{\mu\nu}w_{\nu } + n v_{\mu}\left(\frac{1}{g-\omega^{(2)t}g^{-1}\omega^{(1 ) } } \right)^{\mu\nu}y_{\nu } \right.\nonumber\\ & & \hspace{4em } + n u_{\mu}\left(g^{-1}\omega^{(2 ) } \frac{1}{g-\omega^{(1)t}g^{-1}\omega^{(2)}}\right)^{\mu\nu } v_{\nu } \nonumber\\ & & \hspace{4em } \left . + nw_{\mu}\left(g^{-1}\omega^{(1 ) } \frac{1}{g-\omega^{(2)t}g^{-1}\omega^{(1)}}\right)^{\mu\nu } y_{\nu } \right]~. \label{eq : vevee } \end{aligned}\ ] ] [ f-2 ] it is worth noting that a similar formula to formula [ f-1 ] is used in open string field theory ( see e.g. eq.(b.2 ) of @xcite ) . in particular , when we restrict @xmath1008 and @xmath1009 to symmetric matrices and change the variables from @xmath1036 and @xmath1037 to @xmath1038 and @xmath1039 ( @xmath1040 ) defined in eq.([eq:2harmonics ] ) , formula [ f-1 ] reduces to the formula ( b.2 ) of @xcite . formula [ f-1 ] can be regarded as a closed - string extension of it . in this appendix , we derive eqs.([result on f ] ) and ( [ result on fa ] ) . this also serves as an illustration of usage of the formulae . contributions of the massive states of closed - string propagating between boundary states of tachyons are denoted by @xmath1041 in eq.([pre tachyon amplitude by boundary state ] ) . those between boundary states of gluons are denoted by @xmath1042 in eq.([pre gluon amplitude by boundary state ] ) . their oscillator representations ( [ def of f ] ) and ( [ def of fa ] ) imply that @xmath774 reduces to @xmath324 by setting @xmath1043 . we therefore focus on eq.([result on fa ] ) . the representation ( [ def of fa ] ) allows us to evaluate @xmath1003 by applying formula [ f-2 ] with the following substitution : @xmath1044 in what follows , we will calculate each term in eq.([eq : vevee ] ) with the above substitution . in this course the following relations will be used implicitly : @xmath1045 first , the determinants in the rhs of eq.([eq : vevee ] ) become as follows : @xmath1046 as regards the exponential in the rhs of eq.([eq : vevee ] ) , the first two terms of the exponent are translated to @xmath1047 we can recast the third term as follows : @xmath1048 the last term of the exponent turns out to be the same as ( [ eq : uv ] ) with shifting the indices @xmath1049 and @xmath812 to @xmath1050 . contribution of the @xmath215-th level oscillators is obtained by gathering all the above results . contributions of the massive states are given by the infinite products taken over all the levels . these turn out to be written as follows : @xmath1051 \nonumber\\ & & \quad \times \prod_{m+1\leq r < s \leq m+n } \exp \bigg[-2\alpha'g^{\mu\nu } \left(k_{\mu}^{(r)}-a_{r } a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r}}\right ) \left(k_{\nu}^{(s)}-a_{s } a_{\nu}(k^{(s ) } ) \partial_{\sigma_{s}}\right ) \nonumber\\ & & \hspace{16em } \times \sum_{n=1}^{\infty } \frac { \left(q_{c}e^{i(\sigma_{r}-\sigma_{s})}\right)^{n } + \left(q_{c } e^{-i(\sigma_{r}-\sigma_{s})}\right)^{n } -2q_{c}^{n } } { n\left(1-q_{c}^{n}\right ) } \bigg ] \nonumber\\ & & \quad \times \prod_{r=1}^{m } \prod_{s = m+1}^{m+n } \exp \bigg [ -2\alpha'g^{\mu\nu } \left(k_{\mu}^{(r)}-a_{r } a_{\mu}(k^{(r ) } ) \partial_{\sigma_{r}}\right ) \left(k_{\nu}^{(s)}-a_{s } a_{\nu}(k^{(s ) } ) \partial_{\sigma_{s}}\right ) \nonumber\\ & & \hspace{13em } \times \sum_{n=1}^{\infty } \frac { \left(q_{c}^{\frac{1}{2}}e^{i(\sigma_{r}-\sigma_{s } ) } \right)^{n } + \left ( q_{c}^{\frac{1}{2 } } e^{-i(\sigma_{r}-\sigma_{s } ) } \right)^{n } -2q_{c}^{n } } { n\left(1-q_{c}^{n}\right)}\bigg ] \nonumber\\ & & \quad \times \exp \left [ -2\alpha ' g^{\mu\nu } \left\ { \left(\sum_{r=1}^{m+n}k_{\mu}^{(r)}\right ) \left(\sum_{s=1}^{m+n}k_{\nu}^{(s)}\right ) + \sum_{r=1}^{m+n}\left(a_{r}\right)^{2 } a_{\mu}(k^{(r ) } ) a_{\nu}(k^{(r ) } ) \right\}\right . \nonumber\\ & & \hspace{5em } \left . \times \sum_{n=1}^{\infty } \frac{q_{c}^{n}}{n\left(1-q_{c}^{n}\right ) } \right]~ , \label{eq : prefaresult}\end{aligned}\ ] ] where we have used the following rearrangement : @xmath1052 the infinite sums in eq.([eq : prefaresult ] ) can be translated into infinite products by using the following relation : @xmath1053 after these translations we obtain eq.([result on fa ] ) . we provide oscillator realizations of eigenstates of the closed - string operators @xmath1054 , @xmath609 and @xmath598 . these are used in the text . to start with , it is useful to recall coordinate and momentum eigenstates of a harmonic oscillator in quantum mechanics . description of this system is made by an annihilation and a creation operators @xmath1055 and @xmath1056 satisfying @xmath1057=1 $ ] . let @xmath1058 and @xmath589 be the coordinate and the momentum operators satisfying @xmath1059=1 $ ] . the operators @xmath1060 are related with the canonical pair @xmath1061 as @xmath1062 where @xmath1063 is chosen so that @xmath1064 becomes the mass of harmonic oscillator . here @xmath1065 is the frequency . let @xmath1066 and @xmath1067 be the eigenstates of @xmath1058 and @xmath589 . they are normalized by @xmath1068 and @xmath1069 . it is possible to realize these states on the fock vacuum @xmath1070 by using @xmath1056 . they are given by : @xmath1071 |0\rangle~,\nonumber\\ |p\rangle & = & \left(\frac{|\gamma|^{2}}{\pi}\right)^{\frac{1}{4 } } \exp \left [ \frac{\bar{\gamma}}{2\gamma } \hat{a}^{\dagger } \hat{a}^{\dagger } + i\sqrt{2}\bar{\gamma}p\hat{a}^{\dagger } -\frac{|\gamma|^{2}}{2}p^{2 } \right]|0\rangle~. \label{eq : ho - eigenstates}\end{aligned}\ ] ] the above realizations of coordinate and momentum eigenstates are generalized to the case of string . we first expand the coordinate and the momentum operators of closed - string by suitable canonical pairs , typically denoted by @xmath1072 ( @xmath1073 ; @xmath1074 ) . for the each pair , we introduce creation and annihilation operators , typically @xmath1075 . we then realize eigenstates of the canonical operators @xmath1076 and @xmath1077 by using the creation operators @xmath1078 . eigenstates of the coordinate and the momentum operators of closed - string are given by their infinite products . in order to obtain canonical pairs , we expand @xmath1054 , @xmath609 and @xmath598 by a real basis of the periodic functions on a circle : @xmath1079 , @xmath1080~ , \nonumber\\ \hat{p}^{(b)}_{\mu } ( \sigma ) & = & \frac{1}{2\pi } \left [ \hat{p}_{0\mu}+\sqrt{2 } \sum_{n=1}^{\infty } \left ( \hat{\varpi}^{(\mathrm{i})}_{n\mu } \cos n\sigma + \hat{\varpi}^{(\mathrm{ii})}_{n\mu } \sin n\sigma \right)\right]~. \label{eq : mode - expansion}\end{aligned}\ ] ] the canonical commutation relations between @xmath1054 and @xmath609 ( @xmath598 ) are converted into the following relations among the hermitian operators @xmath1081 , @xmath1082 and @xmath1083 ( @xmath1084 ; @xmath1085 ) : @xmath1086 = i\delta^{\mu}_{\nu}~ , \quad [ \hat{\phi}^{(i)\mu}_{m},\hat{\pi}^{(j)}_{n\nu } ] = i\delta^{\mu\nu}\delta^{i , j}\delta_{m , n}~ , \quad [ \hat{\phi}^{(i)\mu}_{m},\hat{\varpi}^{(j)}_{n\nu } ] = i\delta^{\mu}_{\nu}~,\ ] ] and the others are vanishing . these can be derived from the following relations as well : @xmath1087 \displaystyle \hat{\phi}^{(\mathrm{ii})\mu}_{n } = \frac{\sqrt{\alpha'}}{2n } \left ( \alpha^{\mu}_{n}-\tilde{\alpha}^{\mu}_{n } + \alpha^{\mu}_{-n}-\tilde{\alpha}^{\mu}_{-n } \right ) \end{array } \right . , ~ \left\ { \begin{array}{l } \displaystyle \hat{\pi}^{(\mathrm{i})}_{n\mu}=\frac{1}{2\sqrt{\alpha'}}g_{\mu\nu } \left(\alpha^{\nu}_{n}+\tilde{\alpha}^{\nu}_{n } + \alpha^{\nu}_{-n}+\tilde{\alpha}^{\nu}_{-n}\right)\\[1.5ex ] \displaystyle \hat{\pi}^{(\mathrm{ii})}_{n\mu}=\frac{-i}{2\sqrt{\alpha'}}g_{\mu\nu } \left(\alpha^{\nu}_{n}-\tilde{\alpha}^{\nu}_{n } -\alpha^{\nu}_{-n}+\tilde{\alpha}^{\nu}_{-n } \right)\\ \end{array } \right . , \nonumber\\ & & \left\ { \begin{array}{l } \displaystyle \hat{\varpi}^{(\mathrm{i})}_{n\mu } = \frac{1}{2\sqrt{\alpha'}}\left ( e_{\mu\nu}\alpha^{\nu}_{n } + e^{t}_{\mu\nu}\tilde{\alpha}^{\nu}_{n } + e_{\mu\nu}\alpha^{\nu}_{-n } + e^{t}_{\mu\nu}\tilde{\alpha}^{\nu}_{-n}\right)\\[1.5ex ] \displaystyle \hat{\varpi}^{(\mathrm{ii})}_{n\mu } = \frac{-i}{2\sqrt{\alpha ' } } \left ( e_{\mu\nu}\alpha^{\nu}_{n } -e^{t}_{\mu\nu}\tilde{\alpha}^{\nu}_{n } -e_{\mu\nu}\alpha^{\nu}_{-n } + e^{t}_{\mu\nu } \tilde{\alpha}^{\nu}_{-n } \right ) \end{array } \right .. \label{eq : among - modes}\end{aligned}\ ] ] let us begin with the canonical pairs @xmath1088 . for these pairs , we introduce annihilation and creation modes @xmath1089 by @xmath1090 they satisfy @xmath1091 = g^{\mu\nu } \delta^{i , j}\delta_{m , n}~. \label{eq : commutator of a}\ ] ] it follows from eqs.([eq : among - modes ] ) that these modes are expressed as @xmath1092 eqs.([eq : def of a2 ] ) take the same forms as eqs.([eq : def of a ] ) with @xmath1093 . therefore , eqs.([eq : ho - eigenstates ] ) enable us to write eigenstates @xmath1094 of @xmath1095 and @xmath1096 with eigenvalues @xmath1097 as follows : @xmath1098 |\mathbf{0}\rangle~. \label{eq : phiphi - a}\end{aligned}\ ] ] in the same way , eigenstates @xmath1099 of @xmath1100 and @xmath1101 with eigenvalues @xmath1102 become : @xmath1103 |\mathbf{0}\rangle~. \label{eq : pipi - a}\end{aligned}\ ] ] by the construction , these states are normalized as follows : @xmath1104 eqs.([eq:2harmonics ] ) make it possible to write these eigenstates in the forms of eqs.([eq : chi - chi n ] ) and ( [ eq : psi - psi ] ) . in the text we use complex variables @xmath1105 and write the coordinate eigenstates as @xmath1106 . as for the momentum eigenstates , we use complex variables @xmath1107 and write them as @xmath1108 . next we consider eigenstates of @xmath1109 . taking account of eqs.([eq : among - modes ] ) , let us write the expansion modes @xmath1110 ( @xmath1084 ; @xmath1085 ) as @xmath1111 where @xmath1112 and @xmath1113 denote @xmath1114 they turn out to satisfy @xmath1115 = g^{\mu\nu}\delta^{i , j}\delta_{m , n}~. \label{eq : commutator of eta}\ ] ] comparing eqs.([eq : def of eta ] ) and ( [ eq : commutator of eta ] ) with eqs.([eq : def of a2 ] ) and ( [ eq : commutator of a ] ) , we can find that eigenstates @xmath1116 of @xmath1117 and @xmath1118 are obtained from @xmath1119 by the following replacements : @xmath1120 thus we have : @xmath1121 |\mathbf{0}\rangle~. \end{aligned}\ ] ] substitutions of eqs.([eq : def of eta2 ] ) into the above realizations make the states in the forms ( [ eq : varrho - varrho ] ) . we use complex variables @xmath1122 in the text and write @xmath1123 . c .- s . chu and p .- m . ho , `` noncommutative open string and d - brane _ '' , nucl . phys . * b550 * ( 1999 ) 151 , [ hep - th/9812219 ] ; + `` _ constrained quantization of open string in background b field and noncommutative d - brane _ '' , nucl . * b568 * ( 2000 ) 447 , [ hep - th/9906192 ] .
|
system of a d - brane in bosonic string theory on a constant @xmath0 field background is studied in order to obtain further insight into the bulk - boundary duality .
boundary states which describe arbitrary numbers of open - string tachyons and gluons are given .
uv behaviors of field theories on the non - commutative world - volume are investigated by using these states .
we take zero - slope limits of generating functions of one - loop amplitudes of gluons ( and open - string tachyons ) in which the region of the small open - string proper time is magnified .
existence of @xmath0 field allows the limits to be slightly different from the standard field theory limits of closed - string .
they enable us to capture world - volume theories at a trans - string scale . in this limit
the generating functions are shown to be factorized by two curved open wilson lines ( and their analogues ) and become integrals on the space of paths with a gaussian distribution around straight lines .
these indicate a possibility that field theories on the non - commutative world - volume are topological at such a trans - string scale .
we also give a proof of the dhar - kitazawa conjecture by making an explicit correspondence between the closed - string states and the paths .
momentum eigenstates of closed - string or momentum loops also play an important role in these analyses .
mifp-02 - 09 + ou - het 418 + hep - th/0211232 + november 2002 * open wilson lines as states of closed string * + koichi murakami + toshio nakatsu + [ section ] [ section ] [ section ] [ section ] [ section ] [ section ]
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in this paper we describe a new dynamical instability in superfluids . this `` two - stream '' instability is analogous to the kelvin - helmholtz instability @xcite . its key distinguishing feature is that the two fluids are interpenetrating rather than in contact across an interface as in the standard scenario . the two - stream instability is well known in plasma physics [ where it is sometimes referred to as the `` farley - buneman '' instability @xcite ] , and it has also been discussed in various astrophysical contexts like merging galaxies @xcite and pulsar magnetopheres @xcite , but as far as we are aware it has not been previously considered for superfluids . in fact , the `` standard '' kelvin - helmholtz instability was only recently discussed in the context of superfluids @xcite . the similarity of the equations used in plasma physics [ a nice pedagogical description of the plasma two - stream instability can be found in @xcite ] to the ones for two - fluid superfluid models inspired us to ask whether an analogous instability could be relevant for superfluids . that this ought to be the case seemed inevitable . to prove the veracity of this expectation , we have adapted the arguments from the plasma problem to the superfluid case , and discuss various aspects of the two - stream instability in this paper . of particular interest to us is the possibility that the two - stream instability may operate in rotating superfluid neutron stars . mature neutron stars are expected to be sufficiently cold ( eg . below @xmath0 k ) that their interiors contain several superfluid / superconducting components . such loosely coupled components are usually invoked to explain the enigmatic pulsar glitches , sudden spin - up events where the observed spin rate jumps by as much as one part in @xmath1 @xcite . theoretical models for glitches @xcite have been discussed ever since the first vela pulsar glitch was observed in 1969 @xcite , but these event are still not well understood . after three decades of theoretical effort it is generally accepted that the glitches arise because a superfluid component can rotate at a rate different from that of the bulk of the star , and that a transfer of angular momentum from the superfluid to the crust of the star could lead to the observed phenomenon . the relaxation following the glitch is well explained in terms of vortex creep [ see for example @xcite ] , but the mechanism that triggers the glitch event remains elusive . in this context , it seems plausible that the superfluid two - stream instability may turn out to be relevant . we take as our starting point the two - fluid equations that have been used to discuss the dynamics of superfluid neutron stars @xcite . hence , we consider superfluid neutrons ( index @xmath2 ) coexisting with a conglomerate of charged components ( index @xmath3 ) . the constituents of the latter ( mainly ions and electrons in the neutron star crust and protons and electrons in the core ) are expected to be coupled by viscosity and the magnetic field on a very short timescale . hence , we assume that these charged components will move together and that it is appropriate to treat them as a single fluid . the corresponding equations are @xcite @xmath4 where @xmath5 represent the respective number densities and @xmath6 are the two velocities . here , and in the following , we use the constituent index @xmath7 which can be either @xmath2 or @xmath3 . the respective mass densities are obviously given by and we further introduce the relative velocity @xmath8 between the two fluids as @xmath9 the first law of thermodynamics is defined by the differential of the energy density or `` equation of state '' , , namely @xmath10 which defines the chemical potentials @xmath11 and the `` entrainment '' function @xmath12 as the thermodynamic conjugates to the densities and the relative velocity . with these definitions we can write the two euler - type equations : @xmath13 where we have introduced the dimensionless entrainment parameters @xmath14 the equation for the gravitational potential @xmath15 is @xmath16 where @xmath17 . when @xmath18 these equations make manifest the so - called entrainment effect . the entrainment arises because the bare neutrons ( or protons ) are `` dressed '' by a polarization cloud of nucleons comprised of both neutrons and protons . since both types of nucleon contribute to the cloud the momentum of the neutrons , say , is modified so that it is a linear combination of the neutron and proton particle number density currents ( the same is true for the proton momentum ) . this means that when one of the fluids begins to flow it will , through entrainment , induce a momentum in the other . because of entrainment a portion of the protons ( and electrons ) will be pulled along with the superfluid neutrons that surround the vortices by means of which the superfluid mimics bulk rotation . this motion leads to magnetic fields being attached to the vortices , and dissipative scattering of electrons off of these magnetic fields . this `` mutual friction '' is expected to provide one of the main dissipative mechanisms in superfluid neutron star cores @xcite . in order to establish the existence of the superfluid two - stream instability we consider the following model problem : let the unperturbed configuration be such that the `` protons '' remain at rest , while the neutrons flow with a constant velocity @xmath19 . for simplicity , we neglect the coupling through entrainment , i.e. we take @xmath20 , and we also neglect perturbations in the gravitational potential . under these assumptions , the two fluids are only coupled `` chemically '' through the equation of state . writing the two velocities as @xmath21\hat{x}$ ] and @xmath22 where @xmath23 and @xmath24 are taken to be suitably small , we get the perturbation equations @xmath25 and @xmath26 next , we assume harmonic dependence on both @xmath27 and @xmath28 , i.e. we use the fourier decomposition @xmath29 $ ] etcetera . this leads to the four equations @xmath30 we thus have four equations relating the six unknown variables @xmath31 , @xmath32 and @xmath33 . to close the system we need to provide an equation of state . given an energy functional @xmath34 we have @xmath35 and similarly @xmath36 finally , we define the two sound speeds by , cf . @xcite , @xmath37 and introduce the `` coupling parameter '' @xmath38 which also has the dimension of a velocity squared . for later convenience we have given the relation to the coefficients of the `` structure matrix '' @xmath39 used by @xcite . with these definitions we get @xmath40 and we can rewrite our set of equations as @xmath41 \ , \\ n_\p \delta v_\p & = & { \omega \over k } \delta n_\p = { k \over \omega } \left [ { n_\p \over n_\n } { \cal c } \delta n_\n + c_{\p}^2 \delta n_\p\right ] \ . \end{aligned}\ ] ] reshuffling we get @xmath42 \delta n_\n & = & { \cal c } \delta n_\p \ , \\ \left [ \left ( { \omega \over k}\right)^2 - c_\p^2 \right]\delta n_\p & = & { n_\p \over n_\n } { \cal c } \delta n_\n \ , \end{aligned}\ ] ] and a dispersion relation @xmath43\left [ \left ( { \omega \over k}\right)^2 - c_\p^2 \right ] = { n_\p \over n_\n } { \cal c}^2 \ .\ ] ] introducing the `` pattern speed '' ( the phase velocity ) @xmath44 we have @xmath45 \left [ \pat^2 - c_\p^2 \right ] = { n_\p \over n_\n } { \cal c}^2 \ . \label{disp0}\ ] ] not surprisingly , this local dispersion relation is qualitatively similar to the one for the plasma problem @xcite . we will now use it to investigate under what circumstances we can have complex roots , i.e. a dynamical instability . first of all , it is easy to see that ( [ disp0 ] ) leads to the simple roots @xmath46 in the uncoupled case , when @xmath47 . this establishes the interpretation of @xmath48 as the sound speeds . to investigate the coupled case , we introduce new variables @xmath49 and @xmath50 . then we get @xmath51[x^2-b^2 ] = 1 \label{fxy}\ ] ] where we have defined @xmath52 the onset of dynamical instability typically corresponds to the merger of two real - frequency modes . if this is the case , a marginally stable configuration will be such that ( [ fxy ] ) has a double root . this happens when an inflexion point of @xmath53 coincides with @xmath54 . this is a useful criterion for searching for the marginally stable modes of our system . as a `` proof of principle '' we consider the particular case of @xmath55 and @xmath56 ( we will motivate this particular choice in section iie ) . the real and imaginary parts of the mode - frequencies for these parameter values are shown as functions of @xmath57 in figure [ modes ] . we have complex roots ( an instability ) in the range @xmath58 . the corresponding mode frequencies lie in the range @xmath59 . the fastest growing instability occurs for @xmath60 for which we find that @xmath61 . in other words , in this particular case we encounter the two - stream instability once the rate of the background flow is increased beyond @xmath62 the corresponding frequency is given by @xmath63 from this we see that the instability is present well before the neutron flow becomes `` supersonic '' . this is crucial since one would expect the superfluidity to be destroyed for supersonic flows . ) for model parameters @xmath55 and @xmath56 . for these parameters the quartic dispersion relation has four real roots for both @xmath64 and @xmath65 , while it has two real roots and a complex conjugate pair for @xmath57 in the range @xmath66 . in this range , the two - stream instability is operating . the grey area corresponds to @xmath67 which is contained in the range of the instability criteria discussed in section iic . , height=226 ] we have thus established that the two - stream instability may , in principle , operate in superfluids . our example indicates the existence of a lower limit of the background flow for the instability . this turns out to be a generic feature . in contrast , the plasma two - stream instability can operate at arbitrarily slow flows . an ideal plasma is unstable to sufficiently long wavelengths for any given @xmath19 . in reality , however , one must also account for dissipative mechanisms . in the case of real plasmas one finds that the so - called landau damping stabilizes the longest wavelengths @xcite . thus the two - stream instability sets in below a critical wavelength in more realistic plasma models , and there is typically ( just like in the present case ) a range of flows for which the instability is present . we will discuss the effects of dissipation on the superfluid two - stream instability briefly in section iv . it is useful to consider whether we can derive a necessary condition for the two - stream instability.to approach this problem in full generality would likely be quite complicated , but we can make good progress for the simple one - dimensional toy problem discussed above . we begin by multiplying the euler equation ( [ eul_n ] ) for the neutrons by the complex conjugate @xmath68 . this leads to ( after also using the continuity equations to replace the perturbed number densities ) @xmath69 similarly , we obtain from the second euler equation ( [ eul_p ] ) @xmath70 next we combine these two equations to get @xmath71 where we have introduced the pattern speed @xmath72 . from this expression we see that the imaginary part of the left - hand side must vanish , so we should have i m @xmath73 . allowing the pattern speed to be complex , i.e. using @xmath74 we find that the following condition must be satisfied @xmath75 | \delta v_\n|^2 + \left [ 2 - { v_0 \over \sigma_r } \left ( 1 + { c_\p^2 \over |\pat|^2 } \right)\right ] mode , @xmath76 , the frequency clearly must be such that the factors multiplying the absolute values of the two velocities have different signs . let us first consider the case when the factor multiplying @xmath77 is negative . then we find that an instability is only possible if @xmath78 , and the following condition is satisfied : @xmath79 this shows that we must have @xmath80 which constrains the permissible frequencies to the range @xmath81 . thus we see that the flow must be subsonic , i.e. @xmath82 . in the case when the factor multiplying @xmath83 is negative we can only have an instability if @xmath84 . we also require @xmath85 or @xmath86 , the condition that must be satisfied is ( [ cond1 ] ) . it is useful to notice two things about this criterion . first of all , any unstable mode for which @xmath87 must lie in the range @xmath88 . secondly , when @xmath89 the permissible range will be well approximated by @xmath90 , cf . figure [ crit ] . as is clear from figure [ modes ] the unstable modes satisfy this last , and most severe , criterion . ) which is relevant for the example considered in section iib . this example is constructed by introducing @xmath91 , and then showing the four curves : @xmath92 , @xmath93 , @xmath94 and @xmath95 . here we have taken @xmath96 and @xmath97 . criterion ( [ cond1 ] ) is satisfied when @xmath98 and @xmath99 ( in the grey area ) . the corresponding range is well approximated by @xmath100 . , height=226 ] it is worth noting that the instability can be discussed in terms of a simple energy argument [ see @xcite and @xcite for similar discussions in other contexts ] . after averaging over several wavelengths , the kinetic energy of the protons is @xmath101 meanwhile we get for the neutrons @xmath102 which leads to @xmath103\ ] ] from which we see that the energy in the perturbed flow is smaller than the energy in the unperturbed case , which means that we can associate the wave with a `` negative energy '' , when @xmath104 a wave that satisfies @xmath105 moves forwards with respect to the protons but backwards according to an observer riding along with the unperturbed neutron flow . as we have seen above , the unstable modes in our problem satisfy this criterion and hence it is easy to explain the physical conditions required for the two - stream instability to be present . having established that the two - stream instability may be present in superfluids , we want to assess to what extent one should expect this mechanism to play a role for astrophysical neutron stars . to do this we will consider two particular equations of state . the results we obtain illustrate different facets of what we expect to be a rich problem . we begin by making contact with our recent analysis of rotating superfluid models @xcite as well as the study of oscillating non - rotating stars by @xcite . from the definitions above we have @xmath106 we combine these results with the explicit structure matrix given in eq . ( 144 ) of @xcite , which is based on a simple `` analytic '' equation of state . this leads to @xmath107 where @xmath108 is the proton fraction , and @xmath109 is defined by @xmath110 as discussed by @xcite , @xmath109 is related to the `` symmetry energy '' of the equation of state , cf . @xcite . ) . for slower flows , these critical curves approach the absolute instability regions . the region where the two - stream instability may operate in physical flows therefore lies between each dashed curve and the nearest grey area . for comparison we also indicate the curve in the @xmath111 plane traced out by the pal equation of state ( discussed in section iie ) as the density is varied from that near the crust / core interface ( @xmath112 ) to five times that of nuclear saturation @xmath113.,height=226 ] the instability regions for this model equation of state are illustrated in figure [ domain ] . a key feature of this figure is the presence of regions of `` absolute instability '' . this happens when @xmath114 . then there exist unstable solutions already for vanishing background flow , @xmath64 . that this is the case is easy to see . consider @xmath115 in the limit @xmath64 . in the limit we can solve directly for @xmath116 : @xmath117 from which it is easy to see that one of the roots for @xmath116 will be negative if @xmath114 . hence , one of the roots to the quartic ( [ fxy ] ) will be purely imaginary . the physics of this instability is quite different from the two - stream instability that is the main focus of this paper . yet it is an interesting phenomenon . from the above relations we find that @xmath118 corresponds to @xmath119 in the discussion by @xcite it was assumed that `` reasonable '' equations of state ought to satisfy this condition . we expected this to be the case since the structure matrix would not be invertible if its determinant were to vanish at some point . we now see that this constraint has a strong physical motivation : the condition is violated when @xmath114 , i.e. when we have an absolute instability . the regions where this instability is active are indicated by the grey areas in figure [ domain ] . in order to strengthen the argument that the two - stream instability may operate in astrophysical neutron stars , we have considered a `` realistic '' equation of state due to prakash , ainsworth and lattimer ( pal ) ( 1988 ) . the advantage of this model is that it is relatively simple . in particular , it leads to analytical expressions for the various quantities needed in our analysis . the energy density of the baryons for the pal equation of state can be written @xmath120 \ , \ ] ] where @xmath121 corresponds to the energy per nucleon , @xmath122 corresponds to the `` symmetry energy '' ( and is closely related to @xmath109 in the `` analytic '' equation of state discussed above ) , and @xmath123 with @xmath124 the nuclear saturation density . @xmath121 takes the following form : @xmath125 \ , \ ] ] with @xmath126 mev , @xmath127 mev , @xmath128 mev , @xmath129 , @xmath130 mev , @xmath131 mev , @xmath132 , and @xmath133 . the symmetry term is @xmath134 + s_0 f(u ) \ , \ ] ] with @xmath135 mev and @xmath136 mev . here @xmath137 is a function satisfying @xmath138 which is supposed to simulate the behaviour of the potentials used in theoretical calculations . in our study we have only considered @xmath139 , which is one of four possibilities discussed by @xcite . we further need to account for the energy contribution of the electrons , which is important since the electrons are highly relativistic inside neutron stars . hence , they can obtain high ( local ) fermi energies which may be comparable with the proton ( local ) fermi energies . considering only the electrons , the leptonic contribution to the energy density is given by ( in units where the speed of light is unity ) @xcite @xmath140 where @xmath141 is the electron mass ( in terms of the baryon mass @xmath142 ) , @xmath143 is the electron compton wavelength , and @xmath144^{1/2 } \left[1 + 2 x^2\right ] - { \rm ln}\left[x + \left(1 + x^2 \right)^{1/2}\right]\right ) \ , \ ] ] @xmath145 ^ 3 \right)^{1/3 } n_\p^{1/3 } \ .\ ] ] in doing this calculation we have assumed local charge neutrality , i.e. @xmath146 . the above energy term is added linearly in the equation of state . having obtained an expression for the total energy as a function of the density , we can derive explicit expressions for all quantities needed to discuss the two - stream instability . first we need to determine the proton fraction @xmath147 . we do this by assuming that the star is in chemical equilibrium , i.e. @xmath148 solving ( [ chemeq ] ) for @xmath147 provides us with the proton fraction as a function of @xmath149 . given this , and the relevant partial derivatives of @xmath150 we can readily evaluate the symmetry energy and well as the sound speeds @xmath151 , @xmath152 and the chemical coupling parameter @xmath153 . with this data we can determine the two parameters @xmath154 and @xmath155 which are needed if we want to solve the local dispersion relation ( [ fxy ] ) . the results we obtain for the proton fraction and the symmetry energy are indicated in figure [ domain ] . we consider the range from @xmath112 , presumed to correspond to the core - crust boundary , to @xmath156 which represents the deep core of a realistic neutron star . the corresponding results for the two - stream instability are shown in figure [ local ] . from this figure we can see that the two - stream instability may operate ( albeit at comparatively large relative flows ) in the region immediately below the crust . finally , we find that the conditions at the core - crust interface are such that @xmath55 and @xmath56 . these are the values we chose for the example in section iib and hence the results shown in figure [ modes ] correspond to a physically realistic model . . we indicate the location of the core - crust boundary ( @xmath157 ) by a vertical dashed line . our model is only relevant for the core fluid , i.e. to the right of the vertical line . finally , the horizontal dashed line indicates when the relative flow is equal to the ( neutron ) sound speed . we expect the superfluid degeneracy to be broken beyond this level , so an instability located above this line is unlikely to have physical relevance . the results indicate that there may be a region of instability immediately below the crust . right panel : the corresponding oscillation frequencies . particularly notable is the point near @xmath158 where the two critical curves cross . at this point the symmetry energy @xmath109 changes sign , cf . [ domain ] , and there exist a particular density such that the two fluids are uncoupled , cf . ( [ adef ] ) . , height=226 ] the fact that a superfluid neutron star may , in principle , exhibit the two - stream instability does not necessarily prove that this mechanism will be astrophysically relevant . yet , it is an intriguing possibility given that the mechanism underlying the enigmatic glitches observed in dozens of pulsars remains poorly understood . one plausible astrophysical role for the superfluid two - stream instability would be in this context : perhaps this instability serves as trigger mechanism for large pulsar glitches ? our aim in this section is to construct a toy problem that allows us to investigate a possible connection between the two - stream instability and pulsar glitches . a suitably simple problem corresponds to two fluids , allowed to rotate at different rates , confined within an infinitesimally thin spherical shell . by assuming that the shell is infinitesimal we ignore radial motion , i.e. we restrict the permissible perturbations of this system is such a way that the perturbed velocities must take the form @xmath159 where @xmath160 are the spherical harmonics and @xmath161 is the radius @xmath161 of the shell . this means that the system permits only toroidal mode - solutions . in other words , all oscillation modes of this shell model are closely related to the inertial r - modes of rotating single fluid objects @xcite . the perturbation equations for the configuration we consider have been derived in a different context and the complete calculation will be presented elsewhere @xcite . our primary interest here concerns whether the modes of this system may suffer the two - stream instability . the presence of the instability in this toy problem would be a strong indication that it will also be relevant when the shells are `` thick '' and radial motion is possible . that is , when we consider a rotating star that contains a partially decoupled superfluid either in the inner crust or the core . as discussed in section iic , the two - stream instability can be understood in terms of negative energy waves . in the current problem , the criterion for waves to carry negative energy according to one fluid but positive energy according to the other fluid is that the mode pattern speed [ we are assuming a decomposition @xmath162 @xmath163 lies between the two ( uniform ) rotation rates . in other words , a necessary condition for instability is @xmath164 where we have assumed that the superfluid neutrons lag behind the charged component , as is expected in a spinning down pulsar . after a somewhat laborious calculation , see @xcite for details , one finds that the dispersion relation for the toroidal two - fluid modes of the shell problem is @xmath165 + 2 m \omega_\n + m \eps_\n \left[2 - l \left(l + 1\right)\right ] \left[\omega_\p - \omega_\n\right]\right\}\cr & & \times \left\ { - l \left(l + 1\right ) ( 1 - \eps_\p ) \left[\omega + m \omega_\p\right ] + 2 m \omega_\p + m \eps_\p \left[2 - l \left(l + 1\right)\right]\left[\omega_\n - \omega_\p\right]\right\ } \cr & & - \left\ { l \left(l + 1\right)\right)^2 \eps_\n \eps_\p \left(\omega + m \omega_\n\right ) \left(\omega + m \omega_\p\right\ } = 0 \ .\end{aligned}\ ] ] this equation should be valid under the conditions in the outer core of a mature neutron star , where superfluid neutrons are permeated by superconducting protons . we rewrite this dispersion relation in terms of the entrainment parameter used by @xcite , namely @xmath166 the frequency as measured with respect to the rotation of the protons , @xmath167 and a dimensionless measure of the relative rotation , @xmath168 with these definitions we get @xmath169(\kappa+\y ) -2(1-x_\p)\y + x_\p \eps(l-1)(l+2 ) ( 1-\y ) \right\ } \nonumber\\ & & \times \left\ { l(l+1 ) [ 1-\eps](\kappa+1 ) -2 - \eps(l-1)(l+2 ) ( 1-\y ) \right\ } \nonumber \\ & & - [ l(l+1)]^2 x_\p \eps^2 ( \kappa+\y)(\kappa+1 ) = 0 \ . \label{disper}\end{aligned}\ ] ] in terms of these new variables a mode would satisfy the necessary instability criterion ( [ nec_crit ] ) if @xmath170 is such that @xmath171 as we will now establish , there exist modes that satisfy this criterion for reasonable parameter values . let us first consider the case of quadrupole oscillations , i.e. take @xmath172 . typical results for this case are shown in figures [ shell ] and [ rotfreq ] . the first figure illustrates the regions of the @xmath173 parameter space for which an instability is present in the case when ( i ) the neutrons rotate at a rate that is 90% faster than that of the charged component , and ( ii ) the neutrons lag behind by the same fraction . the second figure shows the mode - frequencies corresponding to the second case [ simply obtained by solving the quadratic ( [ disper ] ) ] for the particular value @xmath174 . this figure shows the presence of unstable modes within the range of values for @xmath175 that we take to be physically realistic @xcite : @xmath176 . from this figure we immediately deduce two things . first , we see that the unstable modes indeed satisfy the criterion ( [ yrange ] ) . secondly , the unstable modes may have imaginary parts as large as i m @xmath177 . it is useful to ask what this implies for the growth time of the instability . since our results only depend on the azimuthal index @xmath178 through the scaling @xmath179 we see that the fastest growth time corresponds to the @xmath180 modes . the e - folding time for the instability follows from @xmath181 where @xmath182 represents the observed rotation period of the pulsar ( presumably corresponding to the rotation of the charged component , i.e. @xmath183 ) . and relative rotation such that the neutrons rotate 90% slower ( left frame ) or faster ( right frame ) than the protons . the two - stream instability operates in regions 1 - 2 . the grey box corresponds to the `` physically reasonable '' part of parameter space ( for the core of a neutron star).,height=226 ] , @xmath184 and @xmath174 . the corresponding instability region can be deduced from the right panel of fig . [ shell ] . the grey area in the left frame indicates the region where an instability is permissible according to ( [ yrange ] ) . , height=226 ] the above example shows that the two - stream instability does indeed operate in this shell problem . in fact , the analysis goes beyond the local analysis of the plane parallel problem in section iib since we have now solved for the actual unstable modes ( satisfying the relevant boundary conditions ) . we see that , as expected for a dynamical instability , the growth time of an unstable mode may be very short . however , the relative rotation rates required to make the quadrupole modes unstable in the range @xmath185 and @xmath186 are likely far too large to be physically attainable . in this sense the results shown in figs . [ shell][rotfreq ] are , despite being instructive , somewhat extreme . a quantity of key importance for this discussion is the rotational lag between the two components . in order to be able to argue that the two - stream instability is relevant for pulsar glitches we need to consider lags that may actually occur in astrophysical neutron stars . to estimate the size of the rotational lag required to `` explain '' the observed glitches we assume that a glitch corresponds to a transfer of angular momentum from a partially decoupled superfluid component ( index @xmath2 ) to the bulk of the star ( index @xmath3 ) . then we have @xmath187 where @xmath188 are the two moments of inertia . now assume that the decoupled component corresponds 1% of the total moment of inertia , eg . the superfluid neutrons in the inner crust or a corresponding amount of fluid in the core . this would mean that @xmath189 , and we have @xmath190 combine this with the observations of large vela glitches to get @xmath191 in other words , we must have @xmath192 if we assume that the glitch brings the two fluids back into co - rotation , then we have @xmath193 and we see that the two rotation rates will maximally differ by one part in @xmath194 or so . rotational lags of this order of magnitude have often been discussed in the context of glitches . even though the key quantity in models invoking catastrophic vortex unpinning in the inner crust the pinning strength is very uncertain , and there have been suggestions that the pinning force is too weak to allow a build up of the required rotational lag @xcite , typical values considered are consistent with our rough estimate . in addition , frictional heating due to a difference in the rotation rates of the bulk of a neutron star and a superfluid component has been discussed as a possible explanation for the fact that old isolated pulsars seem to be somewhat hotter than expected from standard cooling models @xcite . @xcite argue that a lag of @xmath195 could explain the observational data . finally , the presence of rotational lags of the proposed magnitude is supported by a statistical analysis of 48 glitches in 18 pulsars @xcite . this study suggests that the critical rotational lag at which a glitch occurs is @xmath196 in order to make our shell model problem more realistic we consider the case when the superfluid neutrons lag behind the superconducting protons , and take @xmath197 , or @xmath198 , as a representative value . with this rotational lag , we see from ( [ yrange ] ) that the unstable modes must be such that @xmath199 . a series of results for this choice of parameters are shown in figures [ rotdomain]-[imag25 ] . figure [ rotdomain ] illustrates the fact that , if we decrease the rotational lag then the two - stream instability will not be active ( in the interesting region of parameter space ) for low values of @xmath200 . for a smaller rotational lag the instability acts on a shorter length scale . for @xmath198 we find that we must have @xmath201 in order for there to be a region of instability in the part of the @xmath173 plane shown in figure [ rotdomain ] . this means that the instability only operates on length scales shorter than @xmath202 m ( if we take the shell radius to be @xmath203 km ) . or @xmath198 . we show results for four different values of @xmath200 . there are no unstable modes in this part of parameter space unless @xmath201 . recall that @xmath204 and @xmath176 would be reasonable parameter values for a neutron star core . the grey diagonal line represents the singularity discussed in section iiid and the dashed curves indicate the instability regions in the large @xmath200 limit.,height=302 ] figure [ freq25 ] shows the real part of the mode frequencies for @xmath205 and various values of @xmath147 . from this figure we can see that the instability always occur in the region suggested by the instability condition ( [ yrange ] ) , i.e. two real frequency modes never merge to give rise to a complex conjugate pair of solutions outside the grey areas indicated in the various panels of figure [ freq25 ] . the imaginary parts for @xmath205 and the various values of @xmath147 considered in figure [ freq25 ] are shown in figure [ imag25 ] . from this figure we see that the imaginary part of @xmath170 typically reaches values of order @xmath206 . in fact , by comparing similar results for different values of @xmath200 we have found that the largest attainable imaginary part of @xmath170 varies by less than one order of magnitude as @xmath200 increases from 100 to 1000 . thus we estimate that the typical instability growth time for @xmath207 will be @xmath208 for a star rotating at the rate of the vela pulsar , @xmath209 ms , we would have @xmath210 s for @xmath211 . interestingly , this predicted growth time is significantly shorter than the resolved rise time of a large vela glitch @xmath212 s @xcite . ) as function of the entrainment parameter @xmath175 for @xmath205 , @xmath207 and various values of @xmath147 . the range in which an instability is permissible [ according to ( [ yrange ] ) ] is indicated by the grey areas.,height=302 ] ( determining the growth rate ) of the unstable modes for @xmath207 and @xmath205 is shown as a function of @xmath175 for several different values of @xmath147 ( as indicated in the figure ) . essentially , these results represent horizontal cuts through the frames in figure [ rotdomain ] . , height=226 ] the results obtained above indicate that the two - stream instability is likely to act on modes with rather short wavelengths . given this it makes sense to consider the large @xmath200 limit in more detail . keeping only the leading order term ( proportional to @xmath213 ) in ( [ disper ] ) we have the dispersion relation @xmath214\kappa \nonumber \\ & + & ( 1 - 2\eps - x_\p)\y + \eps x_\p + \eps(1-x_\p)\y^2 - 2x_\p\eps^2(1-\y)^2 = 0 \ . \end{aligned}\ ] ] we can readily write down the solutions to this quadratic : @xmath215^{1/2 } \right\ } \ . \label{kappas}\end{aligned}\ ] ] it turns out that we can learn a lot about the problem from this expression . the most obvious feature is the fact that @xmath216 have a singularity when @xmath217 . it is straightforward to show that one of the roots will become infinite at this point , while the second root becomes : @xmath218 \ .\ ] ] for the examples shown in figure [ rotdomain ] this special case corresponds to the intersection between the instability domain and the diagonal line @xmath217 , as indicated in figure [ rotdomain ] . it is also straightforward to deduce the regions of instability from the sign of the argument of the square - root . we see that we will have an unstable mode when @xmath219 or @xmath220 these regions are also indicated in figure [ rotdomain ] . out of these two possible instability regions , the first is most likely to be relevant for neutron stars since it allows the instability to be present already for small proton fractions . finally , we can use ( [ kappas ] ) to show the existence of an extremum at @xmath221 the corresponding imaginary part of @xmath170 would represent the fastest possible growth time for an unstable mode located in region 1 of figure [ rotdomain ] . this leads to the estimate @xmath222 or an estimate of the shortest growth time : @xmath223 which agrees well with the result we previously obtained for @xmath211 . hence , this simple formula can be used to estimate the fastest growth rate of the two - stream instability in our shell model for different parameter values . in this paper we have introduced the superfluid two - stream instability : a dynamical instability analogous to that known to operate in plasmas @xcite , which sets in once the relative flow between the two components of the system reaches a critical level . we have studied this instability for two different model problems . first we analysed a local dispersion relation derived for the case of a background such that one fluid was at rest while the other had a constant flow rate . this provided a proof of principle of the existence of the two - stream instability for superfluids . our analysis was based on the two - fluid equations that have been used to model the dynamics of the outer core of a neutron star , where superfluid neutrons are expected to coexist with superconducting protons and relativistic electrons . these equations are analogous to the landau model for superfluid helium he . ] , and should also ( after some modifications to incorporate elasticity and possible vortex pinning ) be relevant for the conditions in the inner crust of a mature neutron star . thus we expect the two - stream instability to be generic in dynamical superfluids , possibly limiting the relative flow rates of any multi - fluid system . our second model problem concerned two fluids confined within an infinitesimally thin spherical shell . the aim of this model was to assess whether the two - stream instability may be relevant ( perhaps as trigger mechanism ) for pulsar glitches . the results for this problem demonstrated that the entrainment effect could provide a sufficiently strong coupling for the instability to set in at a relative flow small enough to be astrophysically plausible . incidentally , the modes that become dynamically unstable in this problem are the superfluid analogues of the inertial r - modes of a rotating single fluid star . this is interesting since the r - modes are known to be secularly unstable due to the emission of gravitational radiation @xcite . in fact , the connection between the two instabilities goes even deeper than this since the radiation - driven secular instability is also a variation of the kelvin - helmholtz instability . in that case , the two fluids are the stellar fluid and the radiation . in order for an instability to be relevant the unstable mode must grow faster than all dissipation timescales . in the case of a superfluid neutron star core the main dissipation mechanisms are likely to be mutual friction and `` standard '' shear viscosity due to electron - electron scattering . since we have tried to build a plausible case for the two - stream instability to be relevant for pulsar glitches we would like to obtain some rough estimates of the associated dissipation timescales . to do this we first use an estimate of when mutual friction is likely to dominate the shear viscosity [ due to @xcite ] : @xmath224 where we assume that the wavelength of the mode is @xmath225 . we can write this as @xmath226 which suggests that the shear viscosity will be the dominant dissipation mechanism for large values of @xmath200 . for example , for a neutron star rotating with the period of the vela pulsar mutual friction would dominate for @xmath227 or so ( assuming @xmath228 ) . to estimate the shear viscosity damping we can use results obtained for the secular r - mode instability . in particular , @xcite have shown that for a uniform density star one has @xmath229 if we use the shear viscosity coefficient for electron - electron scattering @xmath230 we get @xmath231 we want to compare this damping timescale to the growth rate of the unstable modes in our shell toy - model . combining ( [ growth ] ) with ( [ tsv ] ) we estimate that in order to have an instability we must have @xmath232 let us now consider the case of the vela pulsar . estimating the core temperature as @xmath233 k [ roughly two orders of magnitude higher than the observed surface temperature @xcite ] we deduce that only modes with @xmath234 or so are likely to be stabilized by shear viscosity . given that our results indicate that the two - stream instability is active for much smaller values of @xmath200 , cf . the results shown in figure [ rotdomain ] , we conclude the dissipation is unlikely to suppress the instability in sufficiently young neutron stars . incidentally , the length scale corresponding to a mode with @xmath235 would be about ten meters . this is an interesting result since one can show that a large glitch could be explained by a small fraction ( @xmath236 ) of the neutron vortices moving a few tens of metres @xcite . obviously , the situation changes as the star cools further . based on the above estimates one can show that shear viscosity will suppress all modes with @xmath201 ( i.e. all unstable modes for the case considered in fig . [ rotdomain ] ) if the core temperature is below @xmath237 k. this means that the two - stream instability may not be able to overcome the viscous damping in a sufficiently cold neutron star , which is consistent with the absence of glitches in mature pulsars . we believe that the results of this paper suggest that the superfluid two - stream instability may be relevant in the context of pulsar glitches . if this is , indeed , the case then what is its exact role ? the answer to this question obviously requires much further work , but it is nevertheless interesting to speculate about some possibilities . most standard models for glitches are based on the idea of catastrophic vortex unpinning in the inner crust @xcite . this is an attractive idea since the glitch relaxation ( on a timescale of days to months ) would seem to be well described by vortex creep models @xcite . an interesting scenario is provided by the thermally induced glitch model discussed by @xcite . they have shown that the deposit of @xmath238 erg of heat would be sufficient to induce a vela type glitch . the mechanism that leads to the unpinning of vortices , eg . by the deposit of heat in the crust , is however not identified . we believe that the two - stream instability may fill this gap in the current theory . it should , of course , be pointed out that glitches need not originate in the inner crust . in particular , @xcite has argued that the vortex pinning is too weak to explain the recurrent vela glitches . if this argument is correct then the glitches must be due to some mechanism operating in the core fluid . since the model problems we have considered would be relevant for the conditions expected to prevail in the outer core of a mature neutron star , our results show that the two - stream instability may serve as a trigger for glitches originating there . the key requirement for the instability to operate is the presence of a rotational lag . it is worth pointing out that such a lag will build up both when there is a strong coupling between the two fluids ( i.e. when the vortices are pinned ) and when this coupling is weak . one would generally expect the strength of this coupling to vary considerably at various depths in the star @xcite , and it is not yet clear to what extent a rotational lag can build up in various regions . this is , of course , a key issue for future theoretical work on pulsar glitches . one final relevant point concerns the recent observation of a vela size glitch in the anomalous x - ray pulsar 1rxs j170849.0 - 400910 @xcite . this object has a spin period of 11 s , which means that any feasible glitch model must not rely on the star being rapidly rotating . what does this mean for our proposal that the two - stream instability may induce a glitch ? let us assume that the rotational lag builds up at the same rate as the electromagnetic spindown of the main part of the star ( i.e. that the superfluid component does not change its spin rate at all under normal circumstances ) . then the lag would be @xmath239 after time @xmath27 . if there is a critical value at which a glitch will happen ( corresponding to @xmath240 ) then the interglitch time @xmath241 could be approximated by @xmath242 where @xmath243 is the standard `` pulsar age '' . this argument implies the following : i ) for @xmath244 we would get @xmath245 . this ( roughly ) means that only pulsars younger than @xmath194 yr would be seen to glitch during 30 years of observation , which accords well with the fact that only young pulsars are active in this sense . ii ) there is no restriction on the rotation rate in this scenario ; a star spinning slowly may well exhibit a glitch as long as its spindown rate is fast enough . this means that one should not be surprised to find glitches in stars with extreme magnetic fields ( magnetars ) . this paper is only a first probe into what promises to be a rich problem area . future studies must address issues concerning the effects of different dissipation mechanisms , the nonlinear evolution of the instability , possible experimental verification for superfluid helium etcetera . these are all very interesting problems which we hope to investigate in the near future . na and rp acknowledge support from the eu programme improving the human research potential and the socio - economic knowledge base ( research training network contract hprn - ct-2000 - 00137 ) . na acknowledges support from the leverhulme trust in the form of a prize fellowship , as well as generous hospitality offered by the center for gravitational - wave phenomenology at penn state university . gc acknowledges partial support from nsf grant phy-0140138 .
|
this paper provides the first study of a new dynamical instability in superfluids .
this instability is similar to the two - stream instability known to operate in plasmas .
it is analogous to the kelvin - helmholtz instability , but has the distinguishing feature that the two fluids are interpenetrating .
the instability sets in once the relative flow between the two components of the system reaches a critical level .
our analysis is based on the two - fluid equations that have been used to model the dynamics of the outer core of a neutron star , where superfluid neutrons are expected to coexist with superconducting protons and relativistic electrons .
these equations are analogous to the standard landau model for superfluid helium .
we study this instability for two different model problems .
first we analyze a local dispersion relation for waves in a system where one fluid is at rest while the other flows at a constant rate .
this provides a proof of principle of the existence of the two - stream instability for superfluids .
our second model problem concerns two rotating fluids confined within an infinitesimally thin spherical shell .
the aim of this model is to assess whether the two - stream instability may be relevant ( perhaps as a trigger mechanism ) for pulsar glitches .
our results for this problem show that the entrainment effect could provide a sufficiently strong coupling for the instability to set in at a relative flow small enough to be astrophysically plausible .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it was calabi @xcite who first recognised the rich geometry that can be found on a hypersurface of @xmath6 when the latter is equipped with its natural cross product and @xmath3-structure . the realization , much later , of metrics with holonomy _ equal _ to @xmath3 allowed this theory to be extended , whilst retaining the key features of the `` euclidean '' theory . the second fundamental form or weingarten map @xmath7 of a hypersurface @xmath8 in a manifold @xmath9 with holonomy @xmath3 can be identified with the intrinsic torsion of the associated @xmath0-structure . the latter is defined by a 2-form @xmath10 and a @xmath11-form @xmath12 induced on @xmath8 , and @xmath7 is determined by their exterior derivatives . the symmetry of @xmath7 translates into a constraint on the intrinsic torsion ( equivalently , on @xmath13 and @xmath14 ) that renders the @xmath0-structure what is called _ half flat_. conversely , a @xmath15-manifold @xmath8 with an @xmath0-structure that is half flat can ( at least if it is real analytic ) be embedded in a manifold with holonomy @xmath3 @xcite . the metric @xmath16 on @xmath9 is found by solving a system of evolution equations that hitchin @xcite interpreted as hamilton s equations relative to a symplectic structure defined ( roughly speaking ) on the space parametrising the pairs @xmath17 . the simplest instance of this construction occurs when @xmath8 is a so - called _ nearly - khler _ space , in which case @xmath18 is a conical metric over @xmath8 , in accordance with a more general scheme described by br @xcite . the first explicit metrics known to have holonomy equal to @xmath3 were realized in this way . in this paper , we are concerned with the classification of left - invariant half - flat @xmath0-structures on @xmath5 , regarded as a lie group @xmath19 , up to an obvious notation of equivalence . one of these structures is the nearly - khler one that can be found on @xmath20 , for any compact simple lie group @xmath21 , by realizing the product as the 3-symmetric space @xmath22 . indeed , we verify that this nearly - khler structure is unique amongst invariant @xmath0-structures on @xmath23 ( see proposition [ prop : nkunique ] , that has a dynamic counterpart in proposition [ prop : nkunique ] ) . examples of the resulting evolution equations for @xmath3-metrics have been much studied in the literature @xcite , but one of our aims is to highlight those @xmath3-metrics that arise from half - flat metrics with specific intrinsic torsion , motivated in part by the approach in @xcite . nearly - khler corresponds to gray - hervella class @xmath24 , and it turns out that a useful generalization in our half - flat context consists of those metrics of class @xmath25 ; see section [ sec : symgrp ] . by careful choices of the coefficients in @xmath26 and @xmath27 , we obtain metrics on @xmath5 of the same class with zero scalar curvature . another aim is to develop rigorously the algebraic structure of the space of invariant half - flat structures on @xmath5 , and in section [ sec : para ] we show that the moduli space they define is essentially a finite - dimensional symplectic quotient . this is a description expected from @xcite , and in our treatment relies on elementary matrix theory . for example , the @xmath28-form @xmath26 can be represented by a @xmath29 matrix @xmath30 , and mapping @xmath10 to the 4-form @xmath31 corresponds to mapping @xmath30 to the transpose of its adjugate . we shall however choose to use a pair of symmetric @xmath32 matrices @xmath33 to parametrise the pair @xmath34 . the matrix algebra is put to use in section [ sec : flow ] to simplify and interpret the flow equations for the associated ricci - flat metrics with holonomy @xmath3 . the significance of the class @xmath35 becomes clearer in the evolutionary setting , as it generates known @xmath3-metrics . in our formulation , the equations ( for example in corollary [ cor : flow ] ) have features in common with two quite different systems considered in @xcite and @xcite , but both in connection with painlev equations . a more thorough analysis of classes of solutions giving rise to @xmath3-metrics is carried out in section [ sec : further ] . some of these exhibit the now familiar phenomenon of metrics that are asymptotically circle bundles over a cone ( `` abc metrics '' ) . all our @xmath3-metrics are of course of cohomogeneity one , and this allows us to briefly relate our approach to that of @xcite . in the final part of the paper , we present the tip of the iceberg that represents a numerical study of hitchin s evolution equations for @xmath23 . we recover metrics that behave asymptotically locally conically when @xmath36 belongs to a fixed @xmath28-dimensional subspace . more precisely , we show empirically that the planar solutions are divided into two classes , only one of which is of type abc . this can be understood in terms of the normalization condition that asserts that @xmath10 and @xmath27 generate the same volume form , and is a worthwhile topic for further theoretical study . for the generic case , the flow solutions do not have tractable asymptotic behaviour , but again the geometry of the solution curves ( illustrated in figure [ fig : g2sol3d ] ) is constrained by the normalization condition that defines a cubic surface in space . this paper grew out of an attempt to reconcile various contributions appearing in the literature . of particular importance concerning @xmath37-structures are schulte - hengesbach s classifications of half - flat structures ( * ? ? ? * theorem 1.4 , chapter 5 ) , and hitchin s notion of stable forms @xcite . in addition , the explicit constructions of @xmath38-metrics appearing in this paper are based on the work of brandhuber et al , cveti et al @xcite , as well as the contributions of dancer and wang @xcite . throughout the paper @xmath39 will denote the @xmath15-manifold @xmath40 . as this is a lie group , we can trivialise the tangent bundle . we describe left - invariant tensors via the identification @xmath41 relative to left multiplication . we keep in mind that there are lie algebra isomorphisms @xmath42 which at the group level can be phrased in terms of the diagram @xmath43 the cotangent space of @xmath39 , at the identity , consists of two copies of @xmath44 . we shall write @xmath45 and choose bases @xmath46 of @xmath47 and @xmath48 of @xmath49 such that @xmath50 here @xmath51 denotes the exterior differential on @xmath47 and @xmath52 induced by the lie bracket . we wish to endow @xmath39 with an @xmath0-structure . to this end it suffices to specify a suitable pair of real forms : a @xmath11-form @xmath27 , whose stabiliser ( up to a @xmath53-covering ) is isomorphic to @xmath54 , and a non - degenerate real @xmath55-form @xmath56 . these two forms must be compatible in certain ways . above all , @xmath12 must be a _ primitive _ form relative to @xmath10 , meaning @xmath57 . so as to obtain a genuine almost hermitian structure we also ask for volume matching and positive definiteness : @xmath58 these forms @xmath27 and @xmath59 are _ stable _ in the sense their orbits under @xmath60 are open in @xmath61 . the following well known properties ( cf . @xcite , and @xcite for the study of @xmath11-forms ) of stable forms will be used in the sequel : 1 . there are two types of stable @xmath11-forms on @xmath62 . these are distinguished by the sign of a suitable quartic invariant , @xmath63 , which is negative precisely when the stabiliser is @xmath64 ( up to @xmath65 ) ; each form of this latter type determines an almost complex structure @xmath66 . the stable forms @xmath59 and @xmath27 determine `` dual '' stable forms : @xmath59 determines the stable @xmath67-form @xmath68 , and @xmath27 determines the @xmath69-form @xmath70 characterised by the condition that @xmath71 be of type @xmath72 . as @xmath0-modules @xmath73 decomposes in the following manner : @xmath74\!]\cong{\lambda}^5t^*,\\ { \lambda}^2t^*\cong[\![{\lambda}^{2,0}]\!]{\oplus}[{\lambda}^{1,1}_0]{\oplus}{\mathbb r}\cong{\lambda}^4t^*,\\ { \lambda}^3t^*\cong[\![{\lambda}^{3,0}]\!]{\oplus}[\![{\lambda}^{2,1}_0]\!]{\oplus}[\![{\lambda}^{1,0}]\ ! ] , \end{gathered}\ ] ] using the bracket notation of @xcite . in terms of this decomposition ( see @xcite ) , the exterior derivatives of @xmath75 may now be expressed as @xmath76 where we have used a suggestive notation to indicate the relation between forms and the intrinsic torsion @xmath77 , i.e. , the failure of @xmath78 to reduce to @xmath0 . obviously , this expression depends on our specific choice of normalisation ( cf . ) . generally , @xmath77 takes values in the @xmath79-dimensional space @xmath80 our main focus , however , is to study the subclass of _ half - flat @xmath0-structures _ : these are characterised by the vanishing of @xmath81 , and @xmath82 , i.e. , @xmath83 to appreciate the terminology `` half flat '' , it helps to count dimensions : @xmath84 , @xmath85 , @xmath86 , @xmath87 . in particular , observe that for half - flat structures @xmath77 is restricted to take its values in @xmath88 dimensions out of @xmath89 possible . in this context , `` flat '' would mean _ holonomy_. for emphasis , we formulate : [ prop : intr_space ] for any invariant half - flat @xmath0-structure @xmath91 on @xmath39 the following holds : if @xmath92 then @xmath93 . if @xmath94 then @xmath95 . in particular , any structure with vanishing @xmath96 component has @xmath97=0\in h^3(m ) $ ] . in the case when @xmath92 we shall say the half - flat structure is _ coupled_. the second case above , @xmath98 , is referred to as _ co - coupled_. when the half - flat structure is both coupled and co - coupled , so @xmath99 , it is said to be _ nearly - khler_. [ [ examples - of - type- . ] ] examples of type . + + + + + + + + + + + + + + + + + + as the next two examples illustrate , it is not difficult to construct half - flat structures of type @xmath25 . [ ex : w1w3 ] in this example we fix a non - zero real number @xmath100 and consider the pair of forms @xmath17 given by : @xmath101 where @xmath102 is defined via the relation @xmath103 clearly , @xmath104 and @xmath105 . a calculation shows @xmath106 so that @xmath107 the @xmath11-form @xmath108 is given by @xmath109 note that the following normalisation condition is satisfied : @xmath110 in order to verify that the intrinsic torsion is of type @xmath111 , we calculate the exterior derivatives of @xmath112 , @xmath113 , and @xmath114 : @xmath115 finally , note that the associated metric is given by @xmath116 and one finds that the scalar curvature is positive : @xmath117 . [ ex : w1w3s0 ] consider the following pair of stable forms : @xmath118 we find that @xmath119 , and the @xmath11-form @xmath108 is given by @xmath120 the normalisation condition then reads @xmath121 the associated metric takes the form @xmath122 in this case one finds that the scalar curvature is zero . the author of @xcite uses lie algebra degenerations to study invariant hypo @xmath123-structures on @xmath124-dimensional nilmanifolds . in a similar way , one could study half - flat structures on the various group contractions of @xmath5 like @xmath125 , where @xmath126 is a compact quotient of the heisenberg group . ( see @xcite for partial studies of such contractions ) . the invariant half - flat structures on @xmath39 can be described in terms of symmetric matrices . in order to do this , we recall the local identifications and set @xmath127 , the space of real @xmath29 matrices , and @xmath128 , the space of real symmetric trace - free @xmath32 matrices . there is a well known correspondence between @xmath129 and @xmath130 ; a fact which is for example used in the description of the trace - free ricci - tensor @xmath131 on a riemannian @xmath132-manifold . [ lem : equiv - iso ] there is an equivariant isomorphism @xmath133 which maps a @xmath134 matrix @xmath135 to the matrix @xmath136 by fixing an oriented orthonormal basis @xmath137 of @xmath138 , we make the identifications @xmath139 , @xmath140 via @xmath141 the asserted isomorphism is then given by contraction on the middle two indices , as in the following example : @xmath142 table [ tab : comp - inv - cov ] summarises how invariants and covariants are related under the above isomorphism @xmath143 . cc ku & sv + k & s + 4(kk^t ) & ( s^2 ) + -2(k^t ) & ( s^2)_0 + -24(k ) & ( s^3 ) + 4(kk^t)k & ( s^2)s + 2kk^tk & 34(s^2)s-(s^3)_0 + 4((kk^t)^2 ) & 3(s)+14 ( s^4 ) + 2(kk^t)^2 & ( s)+14(s^4 ) + -24(k)k & ( s^3)s + 4(kk^t)(k ) & 13(s^3)s-(s^4)_0 + now , let us fix a cohomology class @xmath144 . we have : [ thm : half - flat - param ] the set @xmath145 of invariant half - flat structures on @xmath146 with @xmath97=c $ ] can be regarded as a subset of the _ commuting variety _ : @xmath147=0\right\}.\ ] ] recall @xmath148 , where @xmath149 so that we have @xmath150 the equation @xmath104 implies that @xmath151 which defines @xmath152 . also note @xmath153 lies in a space isomorphic to @xmath154 . we may assume that @xmath155 the condition @xmath156 implies @xmath157 lies in the kernel of some @xmath2-equivariant map @xmath158 which must correspond to @xmath159=qp - pq $ ] . consider the open subset set @xmath160 , @xmath161 , of the commuting variety given by pairs @xmath33 satisfying @xmath162 then @xmath145 is the hypersurface in @xmath160 characterised by the normalisation condition @xmath163 the space @xmath164 has a natural symplectic structure , and @xmath2 acts hamiltonian with moment map @xmath165 given by @xmath166.\ ] ] via ( singular ) symplectic reduction @xcite , we can the simplify the parameter space significantly : [ cor : param ] the set @xmath145 of half - flat structures modulo equivalence relations is a subset of the singular symplectic quotient @xmath167 for later use , we observe that in terms of the matrix framework , the dual @xmath11-form @xmath114 has exterior derivative given as follows : [ lem : hatgamma ] fix a cohomology class @xmath168 . for any element @xmath169 corresponding to an invariant half - flat structure , the associated @xmath55-form @xmath170 corresponds to the matrix @xmath171 , where @xmath172 in particular , if @xmath173 and we set @xmath174 then @xmath175 let @xmath176 : if @xmath33 corresponds to a coupled structure then @xmath177 and @xmath178 for a non - zero constant @xmath179 . if @xmath33 corresponds to a co - coupled structure then @xmath180 for a non - zero constant @xmath181 . obviously , the half - flat pair @xmath33 is of type @xmath111 if and only if the matrices @xmath182 and @xmath183 are proportional , i.e. , we have @xmath184 ; the type does not reduce further provided @xmath185 and @xmath186 . using these conditions it is easy to show that the structures of example [ ex : w1w3 ] and example [ ex : w1w3s0 ] have the type of intrinsic torsion claimed . indeed , in the first example , using lemma [ lem : hatgamma ] , we find that @xmath187 whilst the matrices of the second example satisfy @xmath188 [ ex : nk ] in this case , the following conditions should be satisfied : @xmath189 for some @xmath190 . this is equivalent to solving the equations @xmath191 where @xmath192 . we find this system of equations can be formulated as @xmath193 keeping in mind that we must have @xmath194 , we obtain only the following solutions @xmath195 : @xmath196 note that these solutions are identical after using a permutation ; the corresponding matrices @xmath36 are of the form @xmath197 respectively . the above example captures a well known fact about uniqueness of the invariant nearly - khler structure on @xmath5 . in our framework , this can be summarised as follows ( compare with ( * ? ? ? * proposition 2.5 ) and ( * ? ? ? * proposition 1.11 , chapter 5 ) ) . [ prop : nkunique ] modulo equivalence and up to a choice of scaling @xmath198 , there is a unique invariant nearly - khler structure on @xmath39 . it is given by the class @xmath199 $ ] where @xmath200 as observed in ( * ? ? ? * proposition 1.8 ) there are no invariant ( integrable ) complex structures on @xmath39 admitting a left - invariant holomorphic @xmath72-form . indeed , in terms of @xmath201 matrices this assertion is captured by in the notation of lemma [ lem : hatgamma ] , if @xmath202 then @xmath203 . although we have chosen to focus on the vector space @xmath130 and @xmath204 matrices , we conclude this section with a neat consequence of stability . consider @xmath205 . the cayley - hamilton theorem states that @xmath206 where @xmath207 , @xmath208 , and @xmath209 . consider now the adjugate @xmath210 so that @xmath211 . table [ tab : comp - inv - cov ] implies that the mapping @xmath212 corresponds to a multiple of @xmath213 . the following result describes a viable alternative to the square root of a @xmath214 matrix ; it can be proved directly using the singular value decomposition . any @xmath29 matrix with positive determinant equals @xmath215 for some unique @xmath216 . let @xmath217 be an interval . a @xmath3-structure and metric on the @xmath4-manifold @xmath218 can be constructed from a one - parameter family of half - flat structures on @xmath39 by setting @xmath219 where @xmath220 and @xmath221 . it is well known @xcite the holonomy lies in @xmath3 if and only if @xmath222 . for structures defined via a one - parameter family of half - flat structures , this can be phrased equivalently as : the riemannian metric associated with the @xmath3-structure has holonomy in @xmath3 if and only if the family of forms satisfies the equations : @xmath223 differentiation of @xmath224 and @xmath225 gives us : @xmath226 since the one - parameter family consists of half - flat @xmath227-structures , we have @xmath228 ( for each fixed @xmath229 ) , so the conditions @xmath230 reduce to the system . as explained in ( * theorem 8) , the evolution equations can be viewed as the flow of a hamiltonian vector field on @xmath231 . it is a remarkable fact that this flow does not only preserve the closure of @xmath59 and @xmath27 , but also the compatibility conditions . in order to show that a given @xmath3-metric on @xmath218 has holonomy equal to @xmath3 , one must show there are no non - zero parallel @xmath232-forms on the @xmath4-manifold ( see the treatment by bryant and the second author ( * ? ? ? * theorem 2 ) ) . for many of the metrics constructed in this paper , the argument is the same , or a variation of , the one applied in ( * ? ? ? * section 3 ) . in terms of matrices @xmath176 , we can rephrase the flow equations by [ prop : g2flow - matr ] as a flow , @xmath233 , in @xmath145 , the evolution equations take the form @xmath234 these equations are particularly simple when the cohomology class @xmath235 of @xmath27 satisfies the criterion @xmath173 . in this case , by lemma [ lem : hatgamma ] , we have : [ cor : flow ] for a flow , @xmath233 , in @xmath236 with @xmath173 , the equations take the form : @xmath237 when phrased as above , the preservation of the normalisation essentially amounts to jacobi s formula for the derivative of a determinant . proposition [ prop : g2flow - matr ] tells us that the @xmath3-metrics on @xmath218 that arise from the flow of invariant half - flat structures , can be interpreted as the lift of suitable paths @xmath238 to paths @xmath239 and moreover these paths lie on level sets of the ( essentially hamiltonian ) functional @xmath240 let @xmath33 be a ( normalised ) solution of the flow equations . then the trajectory @xmath241 lies on the level set @xmath242 inside the space @xmath243 . [ [ dynamic - examples - of - type- . ] ] dynamic examples of type . + + + + + + + + + + + + + + + + + + + + + + + + + + rephrasing results of @xcite , we now consider the one - parameter family of forms @xmath244 given by @xmath245 in this case , we find that @xmath246 and we shall assume @xmath247 and @xmath248 , so as to ensure @xmath249 . also note that @xmath250 in particular , the normalisation condition reads : @xmath251 in order to solve the flow equations , we also need the @xmath55-form @xmath252 based on the above expressions , the system becomes : @xmath253 these equations can be rewritten as a system of first order odes in @xmath254 and @xmath102 : @xmath255 as we require the normalisation to hold , we can not choose initial conditions @xmath256 freely . after suitable reparametrization , we find the explicit solution : @xmath257 where @xmath258 , and @xmath259 note that whilst @xmath260 is always non - zero , @xmath261 can be zero . indeed , this happens if @xmath262 is chosen such that the quadratic equation @xmath263 has a solution @xmath264 for some @xmath265 . this is the case for any non - zero @xmath266 : if @xmath267 the solution is obtained for @xmath268 and if @xmath269 the solution occurs when @xmath270 introducing @xmath271 , we can express the exterior derivatives of the defining forms via @xmath272 as @xmath273 and @xmath274 , this implies that the constructed one - parameter family of @xmath275-structures consists of members of type @xmath25 . the associated family of metrics takes the form @xmath276 and has scalar curvature given by @xmath277 zero scalar curvature is obtained for the solution which has @xmath278 . indeed , in this case the scalar curvature is zero when @xmath279 . finally , let us remark that the associated @xmath3-metric is of the form @xmath280 , or , phrased more explicitly , in terms of the parameter @xmath281 : @xmath282 & \frac{12}{1+as^{-3}}ds\otimes ds\\ & \hskip20pt+\sum_{i=1}^3\left(\frac{s^2(1+as^{-3})}{\sqrt{3}}(e^{2i-1}+e^{2i})\otimes ( e^{2i}+e^{2i-1 } ) + \sqrt{3}s^2(e^{2i-1}-e^{2i})\otimes ( e^{2i}-e^{2i-1})\right ) . \end{split}\ ] ] if @xmath283 this metric is conical whilst for @xmath284 , the metric is asymptotically conical : when @xmath285 it tends to a cone metric @xmath286 over @xmath39 . in terms of the classification @xcite , the metrics belong to the family ( i ) . in terms of the matrix framework , the one - parameter families of pairs @xmath33 take the form : @xmath287 in particular , we get another way of verifying the co - coupled condition : @xmath288 [ [ metrics - with - symmetry . ] ] metrics with symmetry . + + + + + + + + + + + + + + + + + + + + + + following mainly @xcite , we study examples that relate our framework to certain constructions of @xmath3-metrics appearing in the physics literature . our starting point in a one - parameter families half - flat pairs @xmath17 of the form : @xmath289 using the normalisation condition , we are able to express the associated one - parameter family of metrics on @xmath39 as follows : @xmath290 and the flow equations read : @xmath291 notice that the @xmath292 action which interchanges the two copies of @xmath293 preserves the metric provided the cohomology class @xmath97 $ ] is of the form @xmath294 , i.e. , @xmath97=(a ,- a ) $ ] . the action interchanges metrics of half - flat structures with @xmath97=(a,0 ) $ ] with those for which @xmath97=(0,-a ) $ ] . the latter observation is related to the notion of a _ flop _ @xcite . [ rem : volgrowth ] the quantity @xmath295 can be viewed as the ratio of the volume of @xmath296 relative to a fixed background metric on @xmath1 . as expected , we find that @xmath297 where we have used that @xmath298 , by the normalisation condition . a metric ansatz that has led to the discovery of new complete @xmath3-metrics ( see , for instance , @xcite ) can be expressed in terms of the condition @xmath173 . in this case , we find @xmath299 where @xmath300 or , alternatively , @xmath301 note that , up to a sign , we have @xmath302 . expressed in terms of the metric function @xmath303 , the flow equations become : @xmath304 the complete metrics constructed by brandhuber et al @xcite arise as a further specialisation of this system . indeed , if we take @xmath305 and @xmath306 and set @xmath307 , then the system reads @xmath308 which is the same as in ( * ? ? ? * equation ( 3.1 ) ) , where the authors find the following explicit holonomy @xmath3-metric : @xmath309 asymptotically this is the metric of a circle bundle over a cone , in short an _ abc metric_. in terms of the classification @xcite , it belongs to the family ( ii ) . [ [ cohomogeneity - one - ricci - flat - metrics . ] ] cohomogeneity one ricci flat metrics . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + any solution of gives us a cohomogeneity one ricci flat metric on @xmath218 . an important aspect of the cohomogeneity one terminology is to bridge a gap between our framework and the `` lagrangian approach '' appearing in the physics literature ( see , e.g. , ( * ? ? ? * section 4 ) ) . for example , consider the metric from the above example , assuming for simplicity that @xmath305 and @xmath310 . by @xcite , we know that the shape operator @xmath311 of the principal orbit @xmath312 satisfies the equation @xmath313 . for the given metric , we find that @xmath314 we also observe that @xmath315 in general , the ricci flat condition can now be expressed as : @xmath316 combined with another equation expressing the einstein condition for mixed directions . if we take the trace of the first equation in , and combine with the second one , we obtain the following conservation law : @xmath317 as explained in @xcite , the above system has a hamiltonian interpretation . it is this interpretation , in its lagrangian guise and phrased with the use of superpotentials , one frequently encounters in the physics literature . in this setting , the kinetic and potential energies are given by @xmath318 these definitions agree with those in @xcite up to a multiple of @xmath319 . in @xcite , the authors provide a relevant description of the superpotential ; in classical terms this is a solution of a time - independent hamilton - jacobi equation . in the concrete example , the superpotential @xmath320 can be viewed as a function of @xmath321 . concretely , we can take @xmath322 in terms of @xmath320 , the flow equations can then be expressed as follows : @xmath323 where @xmath324 ( assuming @xmath325 ) , @xmath326 and @xmath327 finally , we remark that the kinetic and potential terms can be expressed in the form @xmath328 as a further specialisation , let us consider the case when @xmath283 and @xmath329 , @xmath330 ; this is the nearly - khler case . then the shape operator is proportional to the identity : @xmath331 , and the kinetic and potential terms are @xmath332 respectively . so the total energy is zero @xmath333 for all @xmath334 . the superpotential is the fifth oder polynomial @xmath335 [ [ uniqueness - flowing - along - a - line . ] ] uniqueness : flowing along a line . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in the case when @xmath336 , the flow equations turn out to have a unique ( admissible ) solution satisfying for which @xmath36 belongs to a fixed one - dimensional subspace . [ prop : nkunique ] assume @xmath337 is a solution of . then @xmath36 belongs to a fixed @xmath338-dimensional subspace of @xmath339 if and only if the associated @xmath3-metric is the cone metric over @xmath5 endowed with its nearly - khler structure . it is easy to see that the solution of which corresponds to the cone metric over @xmath5 ( with its nearly - khler structure ) is represented by @xmath340 so , in this case , @xmath36 indeed belongs to a fixed @xmath338-dimensional subspace of @xmath339 . conversely , let us assume we are given a solution such that @xmath341 then the system reads : @xmath342 these equations show that there is a purely algebraic constraint to having a solution : @xmath343 where @xmath344 . uniqueness of the `` nearly - khler cone '' , as a flow solution , now follows by observing that these algebraic equations have the following set of solutions : @xmath345 the solutions with @xmath346 are not `` admissible '' whilst the remaining solutions all result in one - parameter families of pairs equivalent to . as indicated in the earlier parts of this paper , previous studies of @xmath3-metrics on @xmath218 have focused mainly on metrics with isometry group ( at least ) @xmath347 . in addition , most of the attention has been centred around solutions in @xmath145 for @xmath348 . a technique that seems effective if one is specifically looking for complete metrics is to choose the initial values of the flow equations to obtain a singular orbit at that point ( meaning , in our context , one whose stabilizer has positive dimension in @xmath349 ) . this approach was adopted in @xcite for @xmath350 holonomy . however , this final section shifts the focus of our investigation in order to illustrate some more generic behaviour of the flow on the space of invariant half - flat structures on @xmath40 . we first look for solutions in @xmath351 for which @xmath36 takes the form @xmath352 where @xmath353 are smooth functions on an interval @xmath354 . a solution of is then uniquely specified by the quadruple @xmath355 we have solved the system for a wide range of initial conditions . a selection of solutions are shown in figure [ fig : g2sol2d ] . apart from the nearly - khler straight line , these solutions are new . plotting the metric functions , we find that some of the new metrics have one stabilising direction when @xmath356 and no collapsing directions ( they are therefore abc metrics of the sort mentioned in connection with ) . the others have shrinking directions which cause the volume growth to slow down as shown in figure [ fig : g2sol2dvgrowth ] . more precisely , in the case @xmath357 , the normalisation forces @xmath358 , written as @xmath359 , to lie on the curve @xmath360 which has two branches separated by the line @xmath361 . one branch corresponds to positive - definite metrics , including the nearly - khler solution @xmath362 the abc metrics are those for which @xmath363 , and appear on the top left of the nearly - khler line in figure [ fig : g2sol2dqcurvea ] , in green in the coloured version . when @xmath364 , the nearly - khler solution is excluded . nevertheless , the overall picture remains valid , meaning one branch of the normalisation curve corresponds to positive - definite metrics , and this branch itself has two half pieces , one corresponding to abc curves and one to the other solutions . in the trace - free case , @xmath365 , all solutions degenerate at a point @xmath366 . the abc solutions are `` half complete '' , meaning that away from the degeneration they are complete in one direction of time . ( see @xcite for other examples of half - complete @xmath3-metrics ) . the other solutions reach another degeneracy point @xmath367 in finite time . the singularity at @xmath368 can not be resolved . in particular , it is not possible to find complete @xmath3-metrics . one way to circumvent this issue is to consider flow solutions for which @xmath97\neq0 $ ] ; solutions of this form include the metrics discovered by brandhuber et al @xcite . now , turning to `` less symmetric '' @xmath3-metrics , we consider for solutions in @xmath351 with @xmath36 of the ( generic ) form : @xmath369 where @xmath370 are smooth functions on an interval @xmath354 . a solution of is then uniquely specified by the sextuple @xmath371 as in the case of planar solutions , we have solved the flow equations for a large number of initial conditions . in contrast with the planar case , we have not been able to find metrics with one stabilising directions as @xmath372 . we shall confine our presentation to the class of solutions with the same initial point @xmath373 as the nearly - khler solution , but with varying velocity vector @xmath374 similar to the planar case , the flow lines are governed by the normalization condition , and is replaced by the cubic surface @xmath375 the asymptotic planes corresponding to the vanishing of @xmath376 separate the surface into four hyperboloid - shaped components , and only the one with all factors negative is relevant to our study of positive - definite metrics with holonomy @xmath3 . the nearly - khler solution @xmath377 ( cf . ) corresponds to its centre point . families of solutions are shown in figure [ fig : g2sol3d ] which , like those in figure [ fig : g2sol2d ] , were plotted using _ mathematica _ and the command _ ndsolve_. to obtain the curves , it was convenient to further reduce attention to the case in which @xmath378 are all negative . the corresponding subset of is now a curved triangle @xmath379 with truncated vertices . by issuing a plotting command for @xmath380 , we obtained an abundant sample of mesh points to feed into as initial values . one can then regard each curve as the continuing trajectory of a particle launched towards a point of @xmath381 , which fits in close to the apex of figure [ fig : g2sol3dqcurvea ] . all the solutions , apart from the central nearly - khler one , are new . they tend to have shrinking directions , causing the volume growth to slow down . the @xmath382 solution curves in figure [ fig : g2sol3dqcurvea ] are plotted for the range @xmath383 since many develop singularities close to @xmath384 ( and close to @xmath385 though positive @xmath386 is not shown ) . in the coloured `` cocktail umbrella '' picture , they are separated into groups distinguished by the value of the function @xmath387 of the initial condition , with the nearly - khler line @xmath388 and its close neighbours in red . solutions resulting from one of the coordinates being positive can be short - lived in comparison to the others , leading to less coherent plots , and this is why they are absent . the view looking down the nearly - khler line from a point @xmath389 with @xmath390 is shown in figure [ fig : g2sol3dqcurveb ] . the @xmath391 symmetry obtained by permuting the coordinates is evident . the splitting behaviour at the three `` ends '' is to some extent artificial , reflecting as it does the truncation that has resulted from our decision to restrict attention to the negative octant . the abc two - function solutions of figure [ fig : g2sol2dqcurvea ] in the previous subsection arise when two of @xmath378 coincide and assume a common value greater than @xmath392 . the projection of these planar curves orthogonal to the nearly - khler line can be seen in figure [ fig : g2sol3dqcurvec ] . computations confirm that , unlike the generic curves of figure [ fig : g2sol3dqcurveb ] emanating from @xmath393 , these can be extended for all @xmath394 . in addition to the solutions in @xmath395 , we have investigated solutions in @xmath396 . regarding the asymptotic behaviour of the associated @xmath3-metrics , the overall picture appears not dissimilar to the one we have described by deforming the nearly - khler velocity . taking account also of the numerical analysis in @xcite , we conjecture that the only solutions that can be extended for @xmath397 or @xmath398 lie in a plane . both authors thank mark haskins for discussions that helped initiate this research , and in particular for bringing @xcite to their attention . the first author gratefully acknowledge financial support from the danish council for independent research , natural sciences . 30 v. apostolov , s. salamon , khler reduction of metrics with holonomy @xmath3 . ( 2004 ) , no . 1 , 4361 . m. atiyah , j. maldacena , c. vafa , an m - theory flop as a large n duality . strings , branes , and m - theory . j. math . 42 ( 2001 ) , no . 7 , 32093220 . c. br , real killing spinors and holonomy . 154 ( 1993 ) , no . 3 , 509521 . l. bedulli , l. vezzoni , the ricci tensor of @xmath399-manifolds . j. geom 57 ( 2007 ) , no . 4 , 11251146 . a. brandhuber , @xmath400 holonomy spaces from invariant three - forms . nuclear phys . b 629 ( 2002 ) , no . 1 - 3 , 393416 . a. brandhuber , j. gomis , s. gubser , s. gukov , gauge theory at large n and new @xmath400 holonomy metrics . nuclear phys . b 611 ( 2001 ) , no . 1 - 3 , 179204 . r. bryant , non - embedding and non - extension results in special holonomy . the many facets of geometry , 346367 , oxford university press , oxford , 2010 . r. bryant , s. salamon , on the construction of some complete metrics with exceptional holonomy . duke math . j. 58 ( 1989 ) , no . 3 , 829850 . j .- butruille , espace de twisteurs dune varit presque hermitienne de dimension 6 . fourier ( grenoble ) 57 ( 2007 ) , no . 5 , 1451485 . j .- butruille , homogeneous nearly khler manifolds . handbook of pseudo - riemannian geometry and supersymmetry , 399423 , irma lect . phys . , 16 , eur . , zrich , 2010 . e. calabi , construction and properties of some 6-dimensional almost complex manifolds . 87 1958 407438 . s. chiossi , a. fino , conformally parallel @xmath401 structures on a class of solvmanifolds . z. 252 ( 2006 ) , no . 4 , 825848 . s. chiossi , s. salamon , the intrinsic torsion of su(3 ) and @xmath400 structures . differential geometry , valencia , 2001 , 115133 , world sci . publ . , river edge , nj , 2002 . z. chong , m. cveti , g. gibbons , h. l , c. pope , p. wagner , general metrics of @xmath3 holonomy and contraction limits . nuclear phys . b 638 ( 2002 ) , no . 3 , 459482 . d. conti , su(3)-holonomy metrics from nilpotent lie groups . arxiv:1108.2450 [ math.dg ] . m. cveti , g. gibbons , h. l , c. pope , supersymmetric m3-branes and @xmath3 manifolds . nuclear phys . b 620 ( 2002 ) , no . 1 - 2 , 328 . m. cveti , g. gibbons , h. l , c. pope , a @xmath400 unification of the deformed and resolved conifolds . b 534 ( 2002 ) , no . 1 - 4 , 172180 . m. cveti , g. gibbons , h. l , c. pope , cohomogeneity one manifolds of @xmath402 and @xmath3 holonomy . d ( 3 ) 65 ( 2002 ) , no . 10 , 106004 , 29 pp . m. cveti , g. gibbons , h. l , c. pope , orientifolds and slumps in @xmath3 and spin(7 ) metrics . physics 310 ( 2004 ) , no . 2 , 265301 . a. dancer , mckenzie wang , painlev expansions , cohomogeneity one metrics and exceptional holonomy . geom . 12 ( 2004 ) , no . 4 , 887926 . a. dancer , mckenzie wang , superpotentials and the cohomogeneity one einstein equations . ( 2005 ) , no . . j. eschenburg , mckenzie wang , the initial value problem for cohomogeneity one einstein metrics . j. geom 10 ( 2000 ) , no . 1 , 109137 . e. ferapontov , b. huard , a. zhang , on the central quadric ansatz : integrable models and painlev reductions . j. phys . a 45 ( 2012 ) , no . 19 , 195204 , 11 pp . m. fernndez , a. gray , riemannian manifolds with structure group @xmath3 . ann . pura appl . ( 4 ) 132 ( 1982 ) , 1945 ( 1983 ) . n. hitchin , stable forms and special metrics . global differential geometry : the mathematical legacy of alfred gray ( bilbao , 2000 ) , 7089 , contemp . , 288 , amer . soc . , providence , ri , 2001 . e. lerman , r. montgomery , r. sjamaar , examples of singular reduction . symplectic geometry , 127155 , london math . lecture note ser . , 192 , cambridge univ . press , cambridge , 1993 . w. reichel , ber die trilinearen alternierenden formen in 6 und 7 vernderlichen , dissertation , greifswald , ( 1907 ) . f. reidegeld , exceptional holonomy and einstein metrics constructed from aloff - wallach spaces . 102 , no . 6 , 11271160 ( 2011 ) . r. reyes carrin , a generalization of the notion of instanton . differential geom . ( 1998 ) , no . 1 , 120 . s. salamon , riemannian geometry and holonomy groups . pitman research notes in mathematics series , 201 . longman scientific & technical , harlow ; copublished in the united states with john wiley & sons , inc . , new york , 1989 . isbn : 0 - 582 - 01767-x . f. schulte - hengesbach , half - flat structures on lie groups . phd thesis ( 2010 ) , hamburg . r. westwick , real trivectors of rank seven . linear and multilinear algebra 10 ( 1981 ) , no . 3 , 183204 .
|
we describe left - invariant half - flat @xmath0-structures on @xmath1 using the representation theory of @xmath2 and matrix algebra .
this leads to a systematic study of the associated cohomogeneity one ricci - flat metrics with holonomy @xmath3 obtained on @xmath4-manifolds with equidistant @xmath5 hypersurfaces .
the generic case is analysed numerically .
* half - flat structures on * thomas bruun madsen and simon salamon _ keywords : _ @xmath3- and @xmath0-structures , einstein and ricci - flat manifolds , special and exceptional holonomy , stable forms , superpotential . _
2010 mathematics subject classification : _
primary 53c25 , 53c29 ; secondary 53c44 , 53d20 , 83e15 , 83e30 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the study of stellar systems based on the `` @xmath0to@xmath1 '' approach ( where @xmath0 is the material density and @xmath1 is the associated phase space distribution function , hereafter df ) , @xmath0 is given , and specific assumptions on the internal dynamics of the model are made ( e.g. see @xcite , @xcite ) . in some special cases inversion formulae exist and the df can be obtained in integral form or as series expansion ( see , e.g. , @xcite@xcite ) . once the df of the system is derived , a non negativity check should be performed , and in case of failure the model must be discarded as unphysical , even if it provides a satisfactory description of data . indeed , a minimal but essential requirement to be met by the df ( of each component ) of a stellar dynamical model is positivity over the accessible phase space . this requirement ( also known as phase space consistency ) is much weaker than the model stability , but it is stronger than the fact that the jeans equations have a physically acceptable solution . however , the difficulties inherent in the operation of recovering analytically the df prevent in general a simple consistency analysis . fortunately , in special circumstances phase space consistency can be investigated without an explicit recovery of the df . for example , analytical necessary and sufficient conditions for consistency of spherically symmetric multi component systems with osipkov merritt ( hereafter om ) anisotropy ( @xcite , @xcite ) were derived in @xcite ( see also @xcite ) and applied in several investigations ( e.g. , @xcite@xcite ) . moreover , in @xcite we derived analytical consistency criteria for the family of spherically symmetric , multi component generalized cuddeford @xcite systems , which contains as very special cases constant anisotropy and om systems . another necessary condition for consistency of spherical systems is given by the `` central cusp anisotropy theorem '' by an & evans @xcite , an inequality relating the values of the _ central _ logarithmic density slope @xmath2 and of the anisotropy parameter @xmath3 of _ any _ consistent spherical system : * theorem * in every consistent system with constant anisotropy @xmath4 necessarily @xmath5 moreover the same inequality holds asymptotically at the center of every consistent spherical system with generic anisotropy profile . in the following we call @xmath6 @xmath7 the _ global _ density slope anisotropy inequality : therefore the an & evans theorem states that constant anisotropy systems obey to the global density slope - anisotropy inequality . however , constant anisotropy systems are quite special , and so it was a surprise when we found ( @xcite ) that the necessary condition for model consistency derived in @xcite for om anisotropic systems can be rewritten as the global density slope anisotropy inequality . in other words , the global inequality holds not only for constant anisotropy systems , but also for each component of multi component om systems . prompted by this result , in @xcite we introduced the family of multi component generalized cuddeford systems , a class of models containing as very special cases both the multi component om models and the constant anisotropy systems . we studied their phase space consistency , obtaining analytical necessary and sufficient conditions for it , and we finally proved that the global density slope anisotropy inequality is again a necessary condition for model consistency ! the results of @xcite and @xcite , here summarized , revealed the unexpected generality of the global density slope anisotropy inequality . in absence of counter examples ( see in particular the discussions in @xcite ) it is natural to ask whether the global inequality is just a consequence of some special characteristics of the df of generalized cuddeford systems , or it is even more general , i.e. it is necessarily obeyed by all spherically symmetric two integrals systems with positive df . here we report on two new interesting analytical cases of models , not belonging to the generalized cuddeford family , supporting the latter point of view . we also present an alternative formulation of the global density slope anisotropy inequality . therefore , even if a proof of the general validity of the global density slope anisotropy inequality is still missing , some relevant advance has been made , and we now have the proof that entire new families of models do obey the global inequality ( see @xcite for a full discussion ) . the om prescription to obtain radially anisotropic spherical systems assumes that the associated df depends on the energy and on the angular momentum modulus of stellar orbits as @xmath8 where @xmath9 is the so called anisotropy radius ( e.g. see @xcite ) . in the formula above @xmath10 is the relative energy per unit mass , @xmath11 is the relative ( total ) potential , and @xmath12 for @xmath13 . a multi component om system is defined as the superposition of density components , each of them characterized by a df of the family ( [ fom ] ) , but in general with different @xmath9 . therefore , unless all the @xmath9 are identical , a multi component om model is not an om system . it is easy to prove that the radial dependence of the anisotropy parameter associated to such models is @xmath14 i.e. systems are isotropic at the center and increasingly radially anisotropic with radius . consistency criteria for multi component om models have been derived in @xcite , while in @xcite it was shown that a necessary condition for phase space consistency of each density component can be rewritten as the global density slope - anisotropy inequality @xmath15 i.e. not only constant anisotropy systems but also multi component om models follow the global inequality . an interesting generalization of om and constant anisotropy systems was proposed by cuddeford ( @xcite ; see also @xcite ) , and is obtained by assuming @xmath16 where @xmath17 is a real number and @xmath18 is defined as in equation ( [ fom ] ) . therefore , both the om models ( @xmath19 ) , and the constant anisotropy models ( @xmath20 ) , belong to the family ( [ f ] ) . in particular , it is easy to show that from equation ( 5 ) @xmath21 remarkably , also for these models a simple inversion formula links the df to the density profile ( @xcite ) . such inversion formula still holds for multi component , generalized cuddeford systems , that we have introduced in @xcite . _ each _ density component of a generalized cuddeford model has a df given by the sum of an arbitrary number of cuddeford dfs with arbitrary positive weights @xmath22 and possibly different anisotropy radii @xmath23 ( but same @xmath24 function and angular momentum exponent ) , i.e. @xmath25 of course , the orbital anisotropy distribution characteristic of df ( [ sumcud ] ) is _ not _ a cuddeford one , and quite general anisotropy profiles can be obtained by specific choices of the weights @xmath22 , the anisotropy radii @xmath23 , and the exponent @xmath26 . however , near the center @xmath27 , and @xmath28 for @xmath29 , independently of the specific values of @xmath22 and @xmath23 . in @xcite , we have found necessary and sufficient conditions for the consistency of multi component generalized cuddeford systems . at variance with the simpler case of om models , the new models admit a _ family _ of necessary conditions , that can be written as simple inequalities involving repeated differentiations of the augmented density expressed as a function of the total potential . _ surprisingly , we also showed that the first of the necessary conditions for phase space consistency can be reformulated as the global density slope anisotropy inequality ( 4 ) _ , which therefore holds at all radii for each density component of multi component generalized cuddeford models . the natural question posed by the analysis above is whether the global density slope anisotropy inequality is a peculiarity of multi component generalized cuddeford models : after all , only models in this ( very large ) family have been proved to obey the global inequality . we now continue our study by showing , by direct computation , that two well - known anisotropic models , whose analytical df is available and not belonging to the generalized cuddeford family , indeed obey to the global density slope anisotropy inequality . a full discussion of the following cases , and their place in a broader context , will be presented in @xcite . dejonghe @xcite , by using the augmented density approach , studied a family of ( one component ) anisotropic plummer models , with normalized density potential pair @xmath30 both the radial trend of orbital anisotropy and the model df were recovered analytically : @xmath31 where @xmath32 belongs to the family of hypergeometric functions . in @xcite it is shown that the consistency requirement @xmath33 imposes the limitation @xmath34 . well , a direct computation of the logarithmic density slope of the plummer model ( [ rhodej ] ) , together with equation ( [ betadej ] ) , proves that these models obey to the global density slope anisotropy inequality when @xmath34 . baes & dejonghe @xcite considered a family of one component anisotropic hernquist models whose normalized density potential pair is @xmath35 with @xmath36 . the corresponding anisotropy parameter and df are @xmath37 so that @xmath38 and @xmath39 are the anisotropy values at the center and at large radii of the system , respectively ; note that in this family of models the orbital anisotropy decreases moving away from the center . in equation ( [ betabadej ] ) @xmath40 are hypergeometric functions and , in accordance with the `` cusp slope central anisotropy theorem '' , the request of non negativity imposes @xmath41 ( see @xcite ) . note that , as in the previous case , the df is not of the generalized cuddeford family . again a comparison of the logarithmic density slope of hernquist profile ( [ rhobadej ] ) with equation ( [ betabadej ] ) shows that , when @xmath36 and @xmath41 also these models obey the global inequality ( 4 ) ! while we refer the reader to @xcite for a full discussion of the new results , and for how these find place in a more general context , here we show that the density slope anisotropy inequality can also be expressed as a condition on the radial velocity dispersion . in fact , the relevant jeans equation in spherical symmetry reads @xmath42 ( e.g. , @xcite ) . introducing the logarithmic density slope and rearranging the terms , one finds @xmath43 as an equivalent , alternative formulation of the density slope anisotropy inequality . of course , the proof that a given family of self consistent models obeys inequality ( [ jeansgamma ] ) is not easier than the proof that would be obtained by working on phase space . we have shown analytically that two more models , in addition to the whole family of multi - component generalized cuddeford systems , satisfy the global density slope anisotropy inequality as a necessary condition for phase space consistency . this reinforces the conjecture that the global slope anisotropy relation ( 4 ) could be a universal necessary condition for consistent spherical systems . we recall that additional evidences supporting such idea exist : for example michele trenti kindly provided us with a large set of numerically computed @xmath44 models @xcite , and all of them , without exception , satisfy the inequality @xmath45 at all radii . additional numerical findings are mentioned in @xcite . j. h. an , and w. evans , _ apj _ * 642 * , 752758 ( 2006 ) . g. bertin , _ dynamics of galaxies _ , cambridge univ . press , cambridge , 2000 . j. binney , and s. tremaine , _ galactic dynamics _ ( 2 ed . ) , princeton univ . press , princeton , 2008 . w. fricke , _ astron . nachr . _ * 280 * , 193216 ( 1952 ) . d. lynden bell , _ mnras _ * 123 * , 447458 ( 1962 ) . l. p. osipkov , _ pisma astron.zh._ * 5 * , 7780 ( 1979 ) . d. merritt , _ aj _ * 90 * , 10271037 ( 1985 ) . h. dejonghe , _ phys . rep . _ * 133 * , no . 3 - 4 , 217313 ( 1986 ) . h. dejonghe , _ mnras _ * 224 * , 1339 ( 1987 ) . p. cuddeford , _ mnras _ * 253 * , 414426 ( 1991 ) . c. hunter , and e. qian , _ mnras _ * 262 * , 401428 ( 1993 ) . l. ciotti , and g. bertin , _ a&a _ * 437 * , 419427 ( 2005 ) . l. ciotti , and s. pellegrini , _ mnras _ * 255 * , 561571 ( 1992 ) . s. d. tremaine , d. o. richstone , y. i. byun , a. dressler , s. m. faber , c. grillmair , j. kormendy , and t. r. lauer , _ aj _ * 107 * , 634644 ( 1994 ) . l. ciotti , _ apj _ * 471 * , 6881 ( 1996 ) . l. ciotti , _ apj _ * 520 * , 574591 ( 1999 ) . l. ciotti , and b. lanzoni , _ a&a _ * 321 * , 724732 ( 1997 ) . l. ciotti , l. , and l. morganti , _ mnras _ * 393 * , 179191 ( 2009 ) . l. ciotti , l. morganti , and p. t. de zeeuw , _ mnras _ * 393 * , 491500 ( 2009 ) . l. ciotti , and l. morganti , _ mnras _ * 401 * , 10911098 ( 2010 ) . l. ciotti , and l. morganti , l. , in preparation ( 2010 ) . l. ciotti , _ lecture notes on stellar dynamics _ , scuola normale superiore , pisa , 2000 . m. baes , and h. dejonghe , _ a&a _ * 393 * , 485497 ( 2002 ) . g. bertin , and m. trenti , _ apj _ * 584 * , 729734 ( 2003 ) .
|
starting from the central density slope
anisotropy theorem of an & evans @xcite , recent investigations have shown that the involved density slope
anisotropy inequality holds not only at the center , but at all radii ( i.e. globally ) in a very large class of spherical systems with positive phase space distribution function . here
we present some additional analytical cases that further extend the validity of the global density slope
anisotropy inequality .
these new results , several numerical evidences , and the absence of known counter examples , lead us to conjecture that the global density slope anisotropy inequality could actually be a universal property of spherical systems with positive distribution function .
address = dept . of astronomy ,
univ . of bologna , + via ranzani 1 , 40127 bologna , italy address = dept . of astronomy ,
univ . of bologna , + via ranzani 1 , 40127 bologna , italy , , altaddress = max - planck - institut fr ex .
physik , garching , germany
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
transition metal dichalcogenides ( tmds ) @xmath15 , where @xmath3 is a transition metal such as w and mo , and @xmath16 is a chalcogen such as s , se , and te , receive considerable attention due to their important mechanical and electronic properties@xcite . molybdenum disulfide mos@xmath0 , a prototypical example of tmds , is a layered system where mo atoms form hexagonal layers@xcite . each of the mo hexagonal layer is sandwiched between two similar lattices of s atoms , forming a trilayer@xcite . the atoms within each trilayer are held together by strong covalent bonds , while the trilayers of mos@xmath0 interact primarily through weak van der waals interactions . it is this sandwiched structure that endows mos@xmath0 with the important mechanical properties for solid lubricants@xcite . the electronic , optical , and lattice dynamical properties have been under intense investigations@xcite . the research on multilayers of mos@xmath0 , among many other multilayers of tmds , have been fueled by their novel properties intrinsic to 2d materials . for example , successes of mos@xmath0 multilayers have been demonstrated for the purposes of energy - efficient field - effect transistor@xcite , advanced electrocatalysts@xcite , thermoelectric devices@xcite with a large and tunable seebeck coefficient , phototransistors@xcite , superconductivity@xcite , etc . mos@xmath0 is joining the rank of other low - dimensional companions , demanding both efficient and accurate treatment of a first - principles approach@xcite . even though the mechanical , electronic , and lattice dynamical properties of the equilibrium structure of mos@xmath0 have been studied extensively@xcite , there are relatively few first - principles studies of the anharmonic effects@xcite that contribute to the thermal properties such as thermal conductivity and thermal expansion coefficients ( tecs ) . the linear tecs of 2h - mos@xmath0 have been measured @xcite where it was found that the tec along the @xmath2 direction is larger than that along the @xmath1 direction . on the theoretical side , tecs may be calculated by solving the vibrational self - consistent - field equations@xcite or the nonequilibrium green s function method@xcite . tecs may also be determined from a quasiharmonic approximation ( qha ) calculation where a set of calculations is to be carried out over a grid or mesh of lattice - parameter points , where the dimensionality of the grid depends on the number of independent lattice parameters@xcite . recently ding _ et al._@xcite chose six volumes to perform phonon calculations to first obtain the volumetric tec . another relation involving the linear tecs for @xmath1 and @xmath2 ( eq . 15 of ref . [ ] ) was set up and the values of tecs were solved . in this work , we develop a direct approach based on the grneisen formalism to calculate the tecs in the @xmath1 and @xmath2 directions . our tec results are then compared with experiment . the outline of this paper is as follows : section [ sec : method ] discusses the methodology used to efficiently calculate the thermal expansion coefficients of a general hexagonal system . section [ sec : results ] reports the results and discussions on the application of the method to mos@xmath0 . section [ sec : conclusions ] contains the conclusions . we shall first present the expressions for tecs for a general hexagonal system obtained from the grneisen formalism.@xcite results specific to the hexagonal mos@xmath0 will be presented later . the linear tecs of the crystal along the @xmath9 , @xmath10 and @xmath8 directions , denoted by @xmath17 , @xmath18 , and @xmath19 , at temperature @xmath3 can be described by a matrix equation @xmath20 where @xmath21 is the equilibrium volume of the primitive cell , @xmath22 is the elastic compliance matrix@xcite . the values @xmath23 are the matrix elements of the elastic constant matrix @xmath24 that corresponds to a hexagonal system@xcite where @xmath25 the integrated quantities in eq . [ eq : alpha ] are given by @xmath26 where the integral is over the first brillouin zone ( bz ) . the frequency @xmath27 of a phonon mode depends on the mode index @xmath28 and wavevector @xmath29 . the heat capacity contributed by a phonon mode with frequency @xmath6 at temperature @xmath3 is @xmath30 with @xmath31 , @xmath32 and @xmath33 are the planck and boltzmann constants , respectively . the grneisen parameter @xmath34 measures the relative change of a phonon frequency @xmath35 as a result of an @xmath7 type deformation with strain size @xmath36 applied to the crystal . for example , if a uniaxial strain is applied in the @xmath9 direction then the strain parameters are @xmath37 ( in the voigt notation@xcite ) , i.e. , @xmath38 , and @xmath39 , for @xmath40 . we apply uniaxial strains in the @xmath9 , @xmath10 , and @xmath8 directions to give @xmath41 , @xmath42 , and @xmath43 , respectively . grneisen parameters are evaluated using a central - difference scheme , where a change in the dynamical matrices before and after deformation is used in the perturbation theory to deduce the changes in eigenfrequencies@xcite . by a proper sampling in the @xmath44-space , we may calculate the phonon density of states as @xmath45 next we introduce a related quantity , @xmath46 , the phonon density of states weighted by the grneisen parameters as @xmath47 the usefulness of @xmath5 is that we may obtain @xmath4 in eq . [ eq : integ ] from another relation @xmath48 to calculate the linear tecs , it appears that a set of three uniaxial deformations in the @xmath9 , @xmath10 , and @xmath8 directions are needed . however , due to the symmetry of the hexagonal system , we should have @xmath49 on physical ground and hence @xmath50 , so that the tec eq . [ eq : alpha ] reduces to @xmath51 & c_{13 } \\ 2c_{13 } & c_{33 } \\ \end{pmatrix}^{-1 } \begin{pmatrix } i_{1}\\ i_{3 } \end{pmatrix } \label{eq:2d1}\ ] ] or @xmath52 \\ \end{pmatrix } \begin{pmatrix } i_{1}\\ i_{3 } \end{pmatrix}\ ] ] where @xmath53 . therefore , for a hexagonal system two uniaxial strains , the first one either in the @xmath9 or @xmath10 direction , and a second one in the @xmath8 direction are sufficient to determine the linear tecs . for mos@xmath0 , the symmetry of the hexagonal system is not altered ( the space group remains as @xmath54 @xmath55 ) when a uniaxial strain is applied in the @xmath8 direction . however , the symmetry is significantly lowered from hexagonal with a space group of @xmath54 @xmath55 to base - centered orthorhombic with a space group of @xmath56 @xmath57 after a uniaxial strain is applied in the @xmath9 or @xmath10 direction . this will result in an increase of the computational cost compared to that which preserves the hexagonal symmetry where a phonon calculation is to be performed . for example , after applying a uniaxial strain in the @xmath9 direction , a @xmath58 @xmath59-mesh required in a phonon calculation will result in @xmath60 irreducible @xmath59-points and @xmath61 irreducible representations or @xmath61 self - consistent - field calculations . this is to be compared with the symmetry - preserving deformations ( e.g. , a uniaxial strain in the @xmath8 direction ) where the number of irreducible @xmath59-points is @xmath62 and the number of irreducible representations is @xmath63 , which clearly shows a substantial computational saving . more savings are expected when complicated crystal structures are treated . we propose to use a computationally efficient , symmetry - preserving biaxial strain in the @xmath11 plane ( hereafter it shall be called an @xmath11 biaxial strain ) where @xmath64 to evaluate the grneisen parameters @xmath65 and use eq . [ eq : integ ] or eq . [ eq : integ2 ] to obtain @xmath66 . due to the underlying symmetry , @xmath67 . however it should be noted that grneisen parameters due to an @xmath11 biaxial strain may not the same as the average of the grneisen parameters due to @xmath9 and @xmath10 uniaxial strains . these points will be elaborated later . the phonon spectra of mos@xmath0 are calculated with density functional perturbation theory ( dfpt)@xcite . for the unstrained structure , a @xmath59-mesh of @xmath68 is used for the phonon calculations , which is equivalent to evaluating the force constants@xcite using a @xmath68 supercell . the phonon calculations proceed by evaluating dynamical matrices at a number of irreducible @xmath59 points . from the the dynamical matrices the inter - atomic force constants in the real space are obtained by an inverse fourier transform , and these force constants are used to construct dynamical matrices at any @xmath69 to calculate the phonon eigenfrequencies @xmath27 . for the strained structures , a @xmath59-mesh of @xmath58 is also used . for the unstrained structure , a larger @xmath59-mesh of @xmath70 is used to confirm that a @xmath59-mesh of @xmath68 is sufficient for the purpose of tec calculations . the bulk mos@xmath0 belongs to @xmath71 nonsymmorphic space group ( # @xmath72 ) , with two inequivalent atoms where a mo atom occupies a @xmath73 site and a s atom occupies @xmath74 site , @xmath75 . this gives a total of six atoms in the hexagonal primitive cell . density - functional theory ( dft ) calculations are carried out using a plane - wave basis code quantum espresso@xcite . the orientation of the crystal adopted in this work is dictated by the choice of the primitive lattice vectors where @xmath76 , @xmath77 , and @xmath78 , @xmath1 and @xmath2 are the hexagonal lattice parameters . we use @xmath79 ry as the cutoff energy for the plane - wave basis set . the local density approximation ( lda ) is used to describe the exchange and correlation . pseudopotentials for mo and s are generated from pslibrary.1.0.0 based on the rappe - rabe - kaxiras - joannopoulos@xcite scheme . a @xmath80 monkhorst - pack @xmath44-point mesh is used . the hexagonal lattice parameters and the atomic positions are fully relaxed . the force tolerance is taken to be @xmath81 ev / . we obtain @xmath82 and @xmath83 , which is in good agreement with the experimental result@xcite of @xmath84 , and @xmath85 . this is also consistent with the fact that lda tends to overbind in crystals . we perform elastic constants calculations@xcite to obtain the elastic constants @xmath86 gpa . these results are in very good agreement with other computational results@xcite where the values of @xmath87 gpa . the agreement of our results with the experimental results@xcite is rather good except @xmath88 and @xmath89 ( where the experimental values of @xmath90 gpa and @xmath91 gpa ) that may be due to the fact these two values are not directly determined in experiment , as discussed in ref . along the high - symmetry directions for hexagonal mos@xmath0 , for ( a ) @xmath9 uniaxial , ( b ) @xmath10 uniaxial , ( c ) @xmath8 uniaxial , and ( d ) @xmath11 biaxial strains . the corresponding densities of grneisen parameters , @xmath92 , are shown on the right . a mesh of @xmath93 for the @xmath44-point sampling is used to calculate @xmath92 . , width=302 ] we perform uniaxial deformations in @xmath9 , @xmath10 , and @xmath8 directions with strains set to @xmath94 . for the @xmath11 biaxial deformations , @xmath95 . the grneisen parameters @xmath96 along the high symmetry directions due to these deformations are shown in fig . [ fig : gp](a)@xmath97(d ) . to quantify more clearly the distribution of grneisen parameters in the brillouin zone , we calculate the density of grneisen parameters according to @xmath98 which is shown on the right panels of fig . [ fig : gp ] . the density of grneisen parameters due to @xmath9 uniaxial , @xmath10 uniaxial , and @xmath11 biaxial strains are also displayed in the inset of fig . [ fig : gamma](b ) for a direct comparison . from figs . [ fig : gp](a ) and ( d ) , some negative grneisen parameters are observed near the @xmath99 point , which correspond to the lowest transverse acoustic ( za ) modes . however , the plots for densities of grneisen parameters , @xmath92 , show most grneisen parameters are populated between a small range of @xmath100 to @xmath101 , and negative grneisen parameters are completely suppressed . the @xmath102 plots also show that large grneisen parameters , say , @xmath103 , are totally negligible when sampling is taken . from fig . [ fig : gp](c ) we note that most grneisen parameters are very small for @xmath8 uniaxial deformation , which is consistent with the fact weak van der waals interactions exist between mos@xmath0 trilayers . for the unstrained mos@xmath0 structure . ( b ) the phonon density of states weighted by the grneisen parameters , @xmath5 . the indices @xmath104 correspond to @xmath9 uniaxial , @xmath10 uniaxial , @xmath8 uniaxial , and @xmath11 biaxial strains , respectively . the insert shows the densities of grneisen parameters , @xmath105 , for @xmath106 . , width=302 ] the phonon density of states of the unstrained structure is shown in fig . [ fig : gamma](a ) where there is a frequency gap from @xmath107 to @xmath108 @xmath109 . the density of states weighted by the grneisen parameters , @xmath5 , are shown in fig . [ fig : gamma](b ) , where a gap is inherited from fig . [ fig : gamma](a ) . we note that , @xmath110 , which is due to a @xmath8 uniaxial strain , has a broad peak near @xmath111 @xmath109 due to fact these frequencies are associated with more significant grneisen parameters ( see eq . [ eq : gnu ] ) . it is interesting to see that , while the density of grneisen parameters due to the @xmath11 biaxial strain , @xmath112 , is quite different from that due to the @xmath9 or @xmath10 uniaxial strains as shown in the inset of fig . [ fig : gamma](b ) ( or even the average of grneisen parameters due to @xmath9 and @xmath10 uniaxial strains ) , @xmath113 , @xmath114 , and @xmath115 are essentially the same numerically , as shown in fig . [ fig : gamma](b ) . this justifies the proposal to use an @xmath11 biaxial strain to replace an @xmath9 uniaxial or @xmath10 uniaxial strain to calculate the integrated quantity @xmath116 . along @xmath117 path . ( b ) shows two doubly degenerate phonon modes for the unstrained structure ( at @xmath99 , the frequencies are @xmath118 and @xmath119 @xmath109 for @xmath120 and @xmath121 , respectively ) . ( c ) , ( d ) , and ( e ) show the detailed variations of the phonon frequencies for the @xmath9 uniaxial , @xmath10 uniaxial , and @xmath11 biaxial strained structures , respectively . the strain values for uniaxial and biaxial strains are @xmath122 and @xmath123 , respectively . , width=302 ] we now provide two evidences that explain the difference between grneisen parameters obtained with an @xmath11 biaxial strain and the average of grneisen parameters obtained with @xmath9 and @xmath10 uniaxial strains . [ fig : ph - ga - brgo - zoom.pdf.pdf ] shows the phonon dispersions of the equilibrium structure and the strained systems along that @xmath124 path . [ fig : ph - ga - brgo - zoom.pdf.pdf](b)-(e ) we have focused on the change of frequencies around @xmath125 @xmath109 . it can be seen from figs . [ fig : ph - ga - brgo - zoom.pdf.pdf](b ) and ( e ) that a @xmath11 biaxial strain can not destroy the degeneracies of two doubly degenerate @xmath120 and @xmath121 phonon modes . however , as seen in fig . [ fig : ph - ga - brgo - zoom.pdf.pdf](c ) , under an @xmath9 uniaxial deformation , the two doubly degenerate @xmath120 and @xmath121 phonon modes around @xmath125 @xmath109 split into four nondegenerate phonon modes . fig . [ fig : ph - ga - brgo - zoom.pdf.pdf](d ) shows the same splittings for a @xmath10 uniaxial strain . this explains the average of grneisen parameters due to @xmath9 and @xmath10 strains is not the same as that due to an @xmath11 biaxial strain . however , the integrations according to eq . [ eq : gnu ] associated with @xmath9 uniaxial , @xmath10 uniaxial , and @xmath11 biaxial strains give rise to the same phonon density of states weighted by the gruneisen parameters , a fact which is expected on physical ground . uniaxial strains . the chosen paths are ( a ) @xmath126 and @xmath127 , and ( b ) @xmath126 and @xmath128 . the paths are shown in the insets . , width=302 ] fig . [ fig : kgm ] furnishes another evidence that under an @xmath9 uniaxial strain , the planar brillouin zone now has a 4-fold rotation symmetry , in contrast to the 6-fold rotation symmetry for the case of biaxial strain ( where the hexagonal symmetry is preserved ) . the agreement of grneisen parameters along @xmath129 and @xmath130 paths in fig . [ fig : kgm](a ) shows that there is a reflection symmetry around @xmath10 axis . similarly , the agreement of grneisen parameters between that along @xmath131 and @xmath132 paths shows that there is a reflection symmetry around @xmath9 axis . under the 6-fold rotation symmetry of the planar brillouin zone , we expect grneisen parameters to be the same along @xmath129 and @xmath133 , or along @xmath131 and @xmath134 paths . however , fig . [ fig : kgm](b ) shows that there is no agreement of grneisen parameters along @xmath129 and @xmath133 paths , or along @xmath131 and @xmath134 paths . as a function of temperature . ( b ) the linear tecs of mos@xmath0 along the @xmath1 and @xmath2 directions denoted by @xmath135 and @xmath136 , respectively , as a function of temperature . the volumetric tec is denoted by @xmath137 . the insert shows the comparison of @xmath135 with the in - plane tec of a single - trilayer mos@xmath0 , obtained using a qha approach@xcite . , width=302 ] the integrated quantities @xmath41 , @xmath42 , and @xmath116 shown in fig . [ fig : ltec](a ) are essentially identical , and they are much larger than @xmath138 , which is due to a @xmath8 uniaxial strain . this may be traced to the fact that interactions between mos@xmath0 trilayers are weak compared to in - plane interactions that result in smaller grneisen parameters ( i.e. , smaller frequency changes ) for @xmath8 strains . the linear tecs in @xmath1 and @xmath2 directions are shown in fig . [ fig : ltec](b ) . even though @xmath139 is about three times smaller than @xmath66 for , say , @xmath140 k , @xmath141 is larger than @xmath142 . at @xmath143 k , @xmath144 k@xmath145 , @xmath146 k@xmath145 , and the volumetric tec @xmath147 k@xmath145 . the main reason for this is that the value of @xmath148 ( @xmath149 gpa ) is much smaller than the value of @xmath150 ( @xmath151 gpa ) , therefore , according to eq . [ eq:2d1 ] it is possible that @xmath136 is larger than @xmath135 . the result that @xmath136 is indeed larger than @xmath135 is consistent with the physical fact that it is easier to perform a deformation in the @xmath8 direction than in the in - plane direction , which is again , attributed by the weak interactions in the out - of - plane direction . this is also confirmed by the results shown in the inset of fig . [ fig : ltec](b ) where there is a striking similarity between the temperature dependence of in - plane tecs for both the bulk mos@xmath0 and a single - trilayer mos@xmath0 , which is obtained from a qha - lda treatment@xcite . el - mahalawy and evans@xcite measured the linear tecs of 2h - mos@xmath0 between @xmath152 and @xmath153 k and reported @xmath154 k@xmath145 and @xmath155 k@xmath145 , which are consistently lower than our values . the same group@xcite again measured the tec of 2h - mos@xmath0 between @xmath156 and @xmath157 k and found larger @xmath158 k@xmath145 , which brings a better agreement with our result , and @xmath159 k@xmath145 , which is somewhat larger . we find the behavior of the rate of change of @xmath136 with @xmath3 to be different from that of @xmath135 where the former increases rapidly at low @xmath3 while the latter increases more gradually with @xmath3 . et al._@xcite pointed out that the lattice constant @xmath1 ( @xmath2 ) increases linearly ( nonlinearly ) with temperature , which is consistent with our tec results at low @xmath3 . in summary , we have proposed a direct way to calculate the linear thermal expansion coefficients ( tecs ) of a hexagonal system based on the grneisen formalism . we have also proposed a way to replace the inefficient symmetry - lowering uniaxial strains by the efficient symmetry - preserving biaxial strains . we successfully implemented the computational schemes and applied them to a technologically important material mos@xmath0 . we found that mos@xmath0 has a large tec anisotropy where the thermal expansion coefficient in the @xmath2 direction is @xmath13 times larger than that in the @xmath1 direction at high temperatures . we highlighted that even though the integrated quantities @xmath4 required by the tec calculations can be obtained via a symmetry - preserving biaxial strain , the grneisen parameters from a biaxial strain may not be a simple average of the grneisen parameters from uniaxial @xmath9 and @xmath10 strains . we demonstrated that we only need a minimum of two symmetry - preserving deformations to directly calculate the tecs of a general hexagonal system . in contrast , the quasiharmonic approximation , when dealing with a two - parameter system , may require an expensive search in the two - dimensional search space . therefore , we expect the strategies adopted in this paper to treat a general two lattice - parameter hexagonal system can be similarly applied to treat other two lattice - parameter systems such as trigonal or tetragonal systems , thus opening the door for a truly predictive tec calculations for many important materials . we also expect the tec calculations based on the gruneisen formalism via symmetry - preserving deformations may be readily incorporated in any phonon related codes such as phonopy@xcite . we acknowledge stimulating discussions with ching hua lee . we thank the national supercomputing computer , singapore ( https://www.nscc.sg ) for computing resources . y.l . acknowledges support from the singapore national science scholarship .
|
using density - functional perturbation theory and the grneisen formalism , we directly calculate the linear thermal expansion coefficients ( tecs ) of a hexagonal bulk system mos@xmath0 in the crystallographic @xmath1 and @xmath2 directions .
the tec calculation depends critically on the evaluation of a temperature ( @xmath3 ) dependent quantity @xmath4 , which is the integral of the product of heat capacity and @xmath5 , of frequency @xmath6 and strain type @xmath7 , where @xmath5 is the phonon density of states weighted by the grneisen parameters .
we show that to determine the linear tecs we may use minimally two uniaxial strains in the @xmath8 , and either @xmath9 or @xmath10 direction .
however , a uniaxial strain in either @xmath9 and @xmath10 direction drastically reduces the symmetry of the crystal from a hexagonal one to a base - centered orthorhombic one .
we propose to use an efficient and accurate symmetry - preserving biaxial strain in the @xmath11 plane to derive the same result for @xmath12 .
we highlight that the grneisen parameter associated with a biaxial strain may not be the same as the average of grneisen parameters associated with two separate uniaxial strains in the @xmath9 and @xmath10 directions due to possible preservation of degeneracies of the phonon modes under a biaxial deformation .
large anisotropy of tecs is observed where the linear tec in the @xmath2 direction is about @xmath13 times larger than that in the @xmath1 or @xmath14 direction at high temperatures .
our theoretical tec results are compared with experiment . the symmetry - preserving approach adopted here
may be applied to a broad class of two lattice - parameter systems such as hexagonal , trigonal , and tetragonal systems , which allows many complicated systems to be treated on a first - principles level .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the light - matter interaction has been subject of central interest since the early years of quantum mechanics . in @xmath0 , one of the first attempts to explain the results coming from experiments was the rabi model , that describes the simplest dipole semiclassical interaction between light and matter @xcite , and is reduced to a pseudospin-@xmath1 system driven by a monochromatic classical radiation field . however , the advent of quantum technologies such as cavity qed @xcite , have allowed us to access the quantum regime of the radiation field , where the dynamical description is given by the celebrated jaynes - cummings model ( jcm ) @xcite . this model predicts collapses and revivals of the population inversion , the appearance of jaynes - cummings doublets as a consequence of the excitation number conservation , and it has found a testbed in several hybrid setups such as trapped ions @xcite , quantum dots @xcite , and circuit quantum electrodynamics ( circuit qed ) @xcite . circuit qed has been growing both theoretically and experimentally , and their complex proposals such as the generation of multipartite entanglement @xcite rely on the fundamentals of the jcm and the tavis - cummings model @xcite . this can be justified because ratios between the coupling strength @xmath2 and the resonator frequency @xmath3 may grow from typical quantum optical values of @xmath4 to circuit qed values of @xmath5 . note that , in the latter , the rotating - wave approximation ( rwa ) can still be applied . nowadays , two key experiments with superconducting circuits @xcite have made a significant improvement in the coupling strength , reaching values @xmath6 in the so - called ultrastrong coupling ( usc ) regime @xcite . in this case , the rwa is not longer valid and all dynamical and statical properties have to be explained through the quantum rabi model ( qrm ) h_r=_z + a + g_x(a+ ) , [ hamr ] where @xmath7 and @xmath8 are pauli matrices , @xmath9 is the annihilation ( creation ) operator , @xmath10 is half of the qubit energy , @xmath3 is the resonator frequency , and @xmath2 stands for the coupling strength . in addition , a recent proposal considers the case where the coupling strength @xmath2 becomes comparable or larger than the mode frequency @xmath3 , @xmath11 , which is called deep strong coupling ( dsc ) regime @xcite . in this case , the dynamics can be intuitively explained as photon number wavepackets propagating along a defined parity chain . although the state - of - the - art in circuit qed does not provide this coupling strength yet , the main features of the dsc regime have been observed in an analog quantum simulation @xcite . the qrm described by the hamiltonian ( [ hamr ] ) , has substantial differences as compared to the jcm , in fact only recently the properties of the qrm have been completely understood @xcite . in the qrm , a discrete @xmath12-symmetry replaces the continuous @xmath13-symmetry of the jcm . therefore the excitation number is no longer a conserved quantity and the hilbert space splits into two infinite - dimensional invariant subspaces , the parity chains @xcite . each eigenstate can be labeled with a @xmath12-quantum number , the parity @xmath14 , taking values @xmath15 . furthermore , it is possible to give analytical expressions for these eigenstates as elements of the bargmann space @xcite , that allows the well - defined computation of their norms and overlaps with eigenstates of the harmonic oscillator , without truncation of the hilbert space @xcite . the aim of this work is to present a new insight of the quantum rabi model by stressing the use of dynamical correlation functions coming from the analytical solution obtained in ref . @xcite . in this manner , we are able to explain relevant features such as the validity of the rwa in two well - defined regions , ranging from the jc regime to higher - coupling regimes of the quantum rabi model . in addition , as the coupling strength @xmath2 enters into the dsc regime , we find that the true eigenstates of the system can be well approximated by the shifted oscillator basis in each invariant parity chain without the need for a more complicated basis . this fact is supported by the calculation of the wigner function of the eigenstates , whose unexpected fidelity can be understood in terms of the analytical form of the eigenfunctions @xcite , resembling closely to fock states . in this context , we also find stationary schrdinger cat - like states in the dsc regime . finally , we present our concluding remarks . in this section , we shall use the exact dynamics of the quantum rabi model to characterize two coupling regions that we may call _ lower coupling region _ and _ higher coupling region _ , respectively . the first region comprises the jc regime , where the rwa holds , together with the perturbative usc regime , where small deviations from the jc occur ( @xmath16 ) . the second identified region comprises a jump towards the higher - coupling regime , @xmath17 , and forms a precursor of the dsc regime ( @xmath18 ) . the intermediate region , that is for @xmath19 , determines a kind of _ dark zone _ where no intuitive physics has been identified up to now . we begin by studying a suitable dynamical observable , the time average photon number |n_0 = _ t_0^tt n_0(t ) , [ time - av ] which measures the break of the @xmath13-symmetry in the jcm . as we will show , this quantity exhibits significant features to characterize the regions mentioned before . since @xmath20 commutes with the parity operator @xmath21 , it is useful to investigate the dynamics within a fixed parity chain . let @xmath22 with @xmath23 correspond to the states @xmath24 and @xmath25 of the two - level system , respectively . then the subspace @xmath26 with parity @xmath27 is spanned by states @xmath28 where @xmath29 denotes the ( anti-)symmetric part of @xmath30 which is an element of the hilbert space @xmath31 of the radiation mode . @xmath31 is spanned by fock states @xmath32 , which are ( anti-)symmetric if @xmath33 is ( odd ) even . there exists a transformation @xmath34 that maps the element @xmath35 of @xmath31 onto a parity eigenstate @xmath36 that belongs to a parity chain f_|= |,= _ s|+_a| . [ isop ] the dynamical quantities in each chain depend only on the initial distribution of photons , which fixes @xmath37 at time @xmath38 . the state at later times follows as solution of the schrdinger equation i_t|(t)=h_|(t ) , [ schp ] where @xmath39 acts on functions in @xmath31 @xcite . the natural observable within the invariant subspaces is thus the photon number . we consider first the time - dependent expectation value , @xmath40 for some initial @xmath41 and positive parity . for @xmath42 and an initial fock state @xmath43 , the average photon number at time @xmath44 reads ( @xmath45 ) n_m(t)=m+2 ^ 2(1-(t ) ) , [ ndelta0 ] which entails that the difference @xmath46 is independent of the initial photon number , always greater than zero and oscillates with the mode frequency @xmath47 . because fock states have definite reflection symmetry , the initial state in the full hilbert space will be the product @xmath48 . the jc result of eq . and the exact result . panel ( b ) shows the difference @xmath49 . the black line shows where the jcm matches exactly the analytical solution . , scaledwidth=44.0% ] starting now from the photon vacuum @xmath50 we consider values @xmath51 . then , the correlation function @xmath52 depends on the initial state and shows a complicated oscillatory behavior . still the time evolution can be globally characterized by the time average quantity defined in eq . ( [ time - av ] ) . its dependence on @xmath53 for fixed @xmath54 shows features that allow to discern a regime which corresponds to the validity of the rwa ( the jc - regime ) from a region where the non - conservation of the excitation number in the qrm becomes relevant . as shown in fig . [ fig1](a ) , the jaynes - cummings model exhibits a sharp peak of @xmath55 at @xmath56 with vanishing width as @xmath57 , and @xmath58 . the maximum value 1/2 is reached at resonance and corresponds to rabi oscillations within the first jc - doublet , @xmath59 . one photon is periodically exchanged between qubit and radiation field which contains 1/2 photon on average . for general values of @xmath2 and @xmath60 , the value of @xmath61 in the jcm reads @xmath62 on the other hand , the exact result exhibits an unbounded @xmath55 due to the broken @xmath13-symmetry , together with a pronounced broadening of the resonance . figure [ fig1](b ) shows the difference of the time - averaged photon number @xmath55 calculated from the exact analytical solution and from the jcm . notice that for @xmath63 , there is good agreement between both quantities until a value @xmath64 , establishing what we called before the lower - coupling region . this means that with regard to photon production , the rwa result is valid for couplings associated with the usc , at least exactly at @xmath56 . for @xmath65 the deviations set in much earlier . interestingly , for @xmath66 , _ less _ photons are generated than expected from the rwa calculation . this corresponds to a shift of the resonant @xmath60 to values below @xmath67 . finally , for couplings @xmath68 , what we called before the higher - coupling region , the resonance is completely gone and the photon production exceeds the rwa prediction for all @xmath60 , growing rapidly with the coupling . this marks the crossover to the deep strong coupling regime . within the dsc , @xmath69 , the qualitative behavior of the system changes again , accompanied by the stabilization of schrdinger cat - like states ( see section [ dsc - regime ] ) we study the time evolution of the autocorrelation function @xmath70 where we start with an initial fock state with @xmath71 . from this the revival probability is obtained as @xmath72 . in fig . [ fig2 ] , we observe the characteristic collapse and revivals of @xmath73 in the deep strong coupling limit as in @xcite . this feature is not present in the usc but appears only for sufficiently strong g. starting in the vacuum as in fig . [ fig2](a ) , the revivals are explained by the fact that the initial state @xmath74 is a coherent state expressed in terms of the ground state of the quantum rabi model @xmath75 for @xmath76 . for large @xmath77 the physical behavior of the @xmath76 case is already present . that the physics of this limit is present already in a considerable distance from it is discussed in sec . [ secapproxbasis ] . collapse and revivals are also observed for higher initial fock states as seen in fig . [ fig2](b ) . then the revivals become sharper but have additional contributions from higher harmonics . of state @xmath78 and in ( b ) the revival probability @xmath79 of state @xmath80 for @xmath81 and three values of @xmath54 . the result depicted is obtained by solving eq . for positive parity.,scaledwidth=44.0% ] in fig . [ fig3 ] , we show the time evolution of the distribution obtained by projection on the complete fock state basis , i.e. the photon number distribution . again the collapse and revival oscillations are visible in form of the red nodes of the interference patterns particularly pronounced in fig . [ fig3](d ) . the second feature that is observed is the bouncing of photon number wave packets " @xcite . again the phenomenon can be understood in terms of the limit @xmath76 where the eigen basis of the quantum rabi model becomes the basis of a shifted oscillator . this is discussed in detail in the next section , sec . [ secapproxbasis ] . for @xmath82 ( a , b ) and @xmath83 ( c , d ) and different initial fock states @xmath78 ( a , c ) and @xmath84 ( b , d ) . the boundaries for the `` bouncing of photon number wave packets '' can be obtained in a simple way using fig . [ fig4](b ) . , scaledwidth=46.0% ] the natural starting point for an approximation of the full quantum rabi model at large @xmath2 is the well known adiabatic approximation @xcite . the hamiltonian to zeroth order for fixed parity reads h_0=a^a + g(a^+a ) . [ hnull ] the characteristic rabi physics in the dsc can already be extracted by use of the eigenbasis @xmath85 of this simple hamiltonian . any more sophisticated approximation to the eigenbasis improves the quality only marginally and does not reveal new physical behavior . for convenience we restrict ourselves to the positive parity subspace . the spectrum in @xmath86 is given by the set of zeros @xmath87 of the function @xcite , g_+(x)=_m=0^k_m(x ) ^m , [ sol ] as @xmath88 . the functional form of @xmath89 reads g_+(x ) = g_+^0(x ) + _ m=1^ [ gg0 ] with @xmath90 and the @xmath91 vanish for @xmath42 . @xmath92 is of slow variation on the scale given by @xmath47 , therefore the location of the zeros of @xmath89 is determined by the poles at @xmath93 . this leads to an almost equidistant distribution of the @xmath87 for small @xmath94 : the @xmath33-th root @xmath87 lies on the left or on the right of @xmath95 according to the sign of @xmath96 . because the @xmath96 grow with @xmath97 , we conclude that @xmath98 for small @xmath94 . the dynamics of the model _ for fixed parity _ is encoded in this smooth distribution of eigenvalues together with the fidelity of the adiabatic basis . there is a natural association of the shifted oscillator basis @xmath99 to the true eigenbasis @xmath100 that can be inferred from the representation @xcite , [ eqrepbarg ] _ m=0^k_m(x_n ) m;g . as the quantum rabi model approaches the adiabatic limit , the spectrum converges to the spectrum of the shifted oscillator , i.e. x_n n g / this limit is encoded in the representation of the rabi eigenstates by the divergence of ( x_n - n)^-1 g/. this factor selects the @xmath33th shifted oscillator in the sum in eq . for sufficiently large @xmath101 . we conclude that the rabi eigenbasis in a fixed parity subspace is close to diagonal in the basis of shifted harmonic oscillators . this behavior is illustrated in fig . [ fig4](a ) where we plot the contributions from the projections of the eigenstates on the shifted oscillators @xmath102 on a logarithmic scale . the resulting matrix shows a rapid decrease of the off - diagonal elements . this reduction is already present for small values of @xmath33 and @xmath103 and becomes more pronounced for higher values . ( a ) and fock states @xmath104 ( b ) for @xmath82 and @xmath81 . this is for positive parity , i.e. @xmath105 is an eigenstate of @xmath106 . the arrows in panel ( b ) show how to determine the boundaries for the `` bouncing photon number wave packets '' that are visible in fig . , scaledwidth=46.0% ] in fig . [ fig4](b ) , by contrast , we plot the representation of the eigenstates in terms of fock states . the figure displays a clear parabolic shape . this shape can be completely understood within the adiabatic limit . considering the representation of the shifted oscillator states in terms of fock states , a perfectly symmetric parabola is seen . this is due to the fact that the fock basis is transformed into the shifted oscillator basis by the operator @xmath107 and the group property of @xmath108 , @xmath109 . because the matrix element @xmath110 reads @xmath111 , where @xmath112 denotes a laguerre polynomial , it follows @xmath113 , furthermore , @xmath114 and therefore @xmath115 . the finite value of @xmath81 then leads to the slight asymmetry visible in the parabola of fig . [ fig4](b ) . surprisingly , also fig . [ fig4](a ) shows an almost symmetric shape which can not be explained by the preceding argument . for @xmath82 ( a ) and @xmath83 ( b ) . @xmath52 is evaluated as sum over the photon distribution of fig . [ fig3](a ) and ( c ) : @xmath116 where @xmath117 . ( i ) solid lines depict numerically exact results calculated with this expression , ( ii ) dashed lines are calculated with @xmath118 replaced by @xmath119 and ( iii ) dotted lines are calculated with the additional replacement of the spectrum by its first order corrected perturbation theory value used in eq . . for @xmath82 , the adiabatic approximation allows a good description of the time evolution only if the exact spectrum is employed as seen in panel ( a ) . for @xmath83 , the adiabatic approximation is highly precise also if the approximated spectrum is used . , scaledwidth=44.0% ] the effect of bouncing photon number wave packets mentioned earlier and shown in fig . [ fig3 ] can be described solely by the presence of the parabola in fig . [ fig4](b ) . to illustrate the determination of the boundaries for the bouncing phenomenon in fig . [ fig3](b ) , we have added the arrows in fig . [ fig4](b ) . the upwards pointing arrow indicates the initial state @xmath120 . the horizontal arrows then lead to the minimum and maximum photon numbers contained in the eigenstates . in fig . [ fig5 ] , we present results for the time evolution of the quantum rabi model calculated using the adiabatic approximation . fig . [ fig5](b ) shows that for @xmath121 the adiabatic approximation gives a detailed picture of the full time dependence . [ fig5](a ) by contrast shows that for moderate values of the coupling , here @xmath122 , the full adiabatic approximation , i.e. approximated spectrum and approximated basis , completely misses a physical description of the dynamics . this insufficiency of the full adiabatic approximation is mainly due to deviations in the spectrum which yield wrong phases . by contrast , the approximated basis combined with the exact spectrum leads to a qualitatively correct description of the dynamics . to further illustrate the point that the adiabatic basis contains already all of the physics even far away from the limit @xmath123 , while the adiabatic spectrum leads to unphysical results , we define a distance of the state @xmath119 of the approximate basis from the true eigenstate @xmath118 by [ distance ] d_n(g,)= 1- ^2 . a distance of the first order corrected eigen energy from the true eigen energy is given by the difference [ distance2 ] d^e_n(g,)= n - g+ ( -1)^n n , g n ,- g - e_n . these functions are shown in fig . [ fig6 ] for @xmath124 . while their magnitude can not be compared directly , their dependence on @xmath2 and @xmath60 allows to conclude the following . the slope of the contour lines is much steeper in case of the basis ( fig . [ fig6](a ) ) compared to the spectrum ( fig . [ fig6](b ) ) . this means that the quality of the approximate basis increases much faster with @xmath101 than that of the first order spectrum . this is surprising , as the basis is only a zeroth order approximation while the spectrum is approximated to first order . it is possible to improve the adiabatic basis in a systematic fashion @xcite or use the rwa on top of the adiabatic approximation @xcite . there are other approaches which reduce the approximative diagonalization of @xmath20 in @xmath26 to diagonalization of finite matrices @xcite . all these techniques are equivalent to a break - up of @xmath26 into a set of invariant subspaces with finite dimension . this means that a continuous symmetry is superimposed on the quantum rabi hamiltonian , leading to an effective model of jaynes - cummings type @xcite . in the quantum rabi model , such an additional ( hidden ) symmetry is not even present in an approximate sense , because otherwise fig . [ fig4](a ) would display a block - diagonal pattern . instead , the off - diagonal components of the true eigenstates in terms of shifted oscillator states are distributed rather smoothly on both sides of the central diagonal being at the same time smaller by at least two orders of magnitude . it can be concluded that no self - consistent reduction to finite - dimensional invariant subspaces @xcite is possible which improves the adiabatic basis without creating a strong ( and unphysical ) symmetry . most of the physics in the dsc regime ( in fact , already for @xmath125 ) is captured by the simple adiabatic basis . of the shifted oscillator ground state from the ground state of the quantum rabi model in the positive parity subspace is shown in panel ( a ) . the corresponding difference of the eigen energies @xmath126 in panel ( b ) . the contour lines in the plots are drawn at values of @xmath127 and @xmath128 . comparing the slope of the contour lines between panel ( a ) and ( b ) shows that the quality of the basis increases faster than the quality of the spectrum . , scaledwidth=46.0% ] besides these physical considerations , we point out the technical fact that the spectrum is much easier to compute than the basis . combining the exact spectrum with the simple basis allows us to predict correctly dynamical quantities of the qrm in the blue sketched parameter region of fig . [ fig6](a ) , as demonstrated in fig . [ fig5 ] . the above discussion in the dsc regime led us to find the shifted basis as a good approximation to the real one . the feature can be further confirmed by analyzing the wigner functions associated to each eigenstate in both parity chains , @xmath129 . [ fig7 ] shows the wigner functions corresponding to the first eigenstates in the parity chain @xmath130 . we see that these functions resemble the fock states @xmath50 , @xmath131 , @xmath132 , and @xmath133 , though shifted from the center , as expected from the shifted oscillator basis . notice the appearance of some additional interference pattern coming from other fock states which contribute to the whole solution , see fig . [ fig4](a ) . it is noteworthy to mention that exactly the same behavior can be observed in the other parity chain @xmath134 . both chains behave very similar regarding the form of the eigenvectors , the ground state in @xmath135 looks almost the same as the first exited state in @xmath86 . , and considering the dsc regime . ( a ) grouns state , ( b ) first excited , ( c ) second excited , and ( d ) third excited states . we can see that they resemble the wigner functions of fock states @xmath50 , @xmath131 , @xmath132 , @xmath133 , respectively , but displaced from the center as expected from the shifted oscillator basis.,scaledwidth=52.0% ] fig . [ fig6](a ) shows that the ground state of @xmath106 at strong coupling is well approximated by @xmath136 . if the system is prepared in this state , it will only weakly depend on time . using the transformation @xmath34 from eq . , the state @xmath137 is mapped to the following states in the original basis with fixed parity @xmath27 : cc |c_+= & e^-((a^)0 |e-(a^)0 |g ) + [ cat - states ] these schrdinger cat - like states contain qubit - field entanglement @xcite , where the terms @xmath138 and @xmath139 are ( anti-)symmetric superpositions of the semiclassical states @xmath140 and @xmath141 @xcite . indeed , as the average photon number @xmath142 in @xmath143 is @xmath144 , an experimentally realizable value of @xmath145 corresponds to a dsc value @xmath146 . these states behave almost like eigenstates regarding expectation values of observables , e.g. the revival probability @xmath147 . as seen in fig . [ fig8 ] , @xmath148 remains almost constant for @xmath149 and exhibits only small high - frequency variations , but not the collapse - and - revival behavior with frequency @xmath47 as the fock states in fig . [ fig1 ] . [ fig8](b ) shows that , for @xmath149 , the oscillations with @xmath47 are completely gone and all fluctuations occur on time scales much shorter than the oscillator period @xmath150 . averaged over such a short time , @xmath147 remains constant and behaves effectively a conserved quantity although the corresponding state is not an eigenstate . the phase variable of the protected coherent states depends directly on the coupling @xmath54 and confines the effect to the dsc regime if the coherent states are required to contain more than two photons on average . of coherent states @xmath151 for different values of @xmath152 and @xmath81 . if @xmath153 , the time evolution of the coherent state remains almost constant , @xmath154 for all times . this is to be compared with the time evolution depicted in fig . [ fig1](b ) of the initial fock state @xmath155 that yields the same photon number @xmath156 as @xmath157 . , scaledwidth=46.0% ] we have studied the dynamical properties of the quantum rabi model . by using the recently developed analytical solutions @xcite , we obtained qualitative features of the spectrum and the eigenstates in two parts of the qrm : a lower and a higher coupling region , respectively . it turns out that it is mandatory to pay attention to the sole and important @xmath12-symmetry of the qrm , which leads to the separation of the hilbert space into two invariant subspaces , the parity chains @xcite . within each chain , the dynamics appears to be rather simple . especially in the deep strong coupling regime , @xmath158 , we find a quite regular periodic behavior of the photon number distribution which can be traced back to the almost equidistant separation of energy levels . this is no longer true if the initial state has no definite parity . under such circumstances , e.g. if the initial state is a product of a coherent state and a qubit eigenstate , the time development may be more complicated , as both chains interfere @xcite . we have identified a simple observable , the average photon generation from the vacuum at positive parity , to discern the convenient lower - coupling and higher - coupling regions . this quantity shows a remarkable sharp peak just for @xmath159 , which demonstrates in a direct way the resonant enhancement of coupling between qubit and cavity mode for small @xmath2 , which was the original motivation to introduce the rwa in the first place @xcite . this determines the lower - coupling region . for stronger couplings , the resonance gets broader until the peak vanishes around @xmath160 , establishing the higher - coupling region . here , the average generated photon number becomes larger than the jc limit of 1/2 due to the counter - rotating terms in ( [ hamr ] ) , which break the conservation law of the jcm . as early as for @xmath161 , the characteristic features of the deep strong coupling regime begin to manifest . the dynamics for fixed parity are dominated by the adiabatic basis and the almost equidistant spectrum . interestingly , the nontrivial effects , which separate the qrm in this regime from the simple adiabatic limit , can be incorporated by using the exact spectrum together with the adiabatic basis . in fact , the exact eigenstates are very close to their adiabatic approximants whereas deviations in the eigenvalues lead to phase differences which become apparent after longer times ( fig . [ fig5 ] ) . the fidelity of the adiabatic basis is even more visible in the wigner representation , which demonstrates the similarity of parity eigenstates with fock states in the deep strong coupling regime ( fig . [ fig7 ] ) . finally , the dsc allows for a special class of states which are not exact eigenstates but lead to expectation values fluctuating with small amplitudes on very short time scales . in this sense , they form stationary schrdinger cat - like states , unaffected by the system interaction . we acknowledge funding from the deutsche forschungsgemeinschaft through trr 80 , spanish micinn juan de la cierva , fis2009 - 12773-c02 - 01 , basque government it472 - 10 , upv / ehu ufi 11/55 , solid , ccqed , and promisce european projects . h. paik , d. i. schuster , l. s. bishop , g. kirchmair , g. catelani , a. p. sears , b. r. johnson , m. j. reagor , l. frunzio , l. i. glazman , s. m. girvin , m. h. devoret , and r. j. schoelkopf , phys . lett . * 107 * , 240501 ( 2011 ) . m. mariantoni , h. wang , r. c. bialczak , m. lenander , e. lucero , m. neely , a. d. oconnell , d. sank , m. weides , j. wenner , t. yamamoto , y. yin , j. zhao , j. m. martinis , and a. n. cleland , nature phys . * 7 * , 287 ( 2011 ) .
|
we study the quantum rabi model within the framework of the analytical solution developed in phys .
rev . lett . * 107 * , 100401 ( 2011 ) .
in particular , through time - dependent correlation functions , we give a quantitative criterion for classifying two regions of the quantum rabi model , involving the jaynes - cummings , the ultrastrong , and deep strong coupling regimes .
in addition , we find a stationary qubit - field entangled basis that governs the whole dynamics as the coupling strength overcomes the mode frequency .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
late time accelerated expansion of the universe was indicated by measurements of distant type ia supernovae ( sne ia ) @xcite . this was confirmed by observations of cosmic microwave background ( cmb ) anisotropies by the wilkinson microwave anisotropy probe ( wmap ) @xcite , and the large - scale structure in the distribution of galaxies observed in the sloan digital sky survey ( sdss ) @xcite . it is not possible to account for this phenomenon within the framework of general relativity containing only matter . therefore , a number of models containing `` dark energy '' have been proposed as the mechanism for the acceleration . there are currently many dark energy models , including cosmological constant , scalar field , quintessence , and phantom models @xcite . however , dark energy , the nature of which remains unknown , has not been detected yet . the cosmological constant , which is the standard candidate for dark energy , can not be explained by current particle physics due to its very small value , and it is plagued with fine - tuning problems and the coincidence problem . an alternative method for explaining the current accelerated expansion of the universe is to extend general relativity to more general theories on cosmological scales . instead of adding an exotic component such as a cosmological constant to the right - hand side ( i.e. , the energy - momentum tensor ) of einstein s field equation , the left - hand side ( i.e. , the einstein tensor , which is represented by pure geometry ) can be modified . typical models based on this modified gravity approach are @xmath1 models @xcite and the dvali gabadadze porrati ( dgp ) model @xcite ( for reviews , see @xcite ) . in @xmath1 models , the scalar curvature @xmath2 in the standard einstein hilbert gravitational lagrangian is replaced by a general function @xmath1 . by adopting appropriate function phenomenologically , @xmath1 models can account for late - time acceleration without postulating dark energy . the dgp model is an extra dimension scenario . in this model , the universe is considered to be a brane ; i.e. , a four - dimensional ( 4d ) hypersurface , embedded in a five - dimensional ( 5d ) minkowski bulk . on large scales , the late - time acceleration is driven by leakage of gravity from the 4d brane into 5d spacetime . naturally , there is no need to introduce dark energy . on small scales , gravity is bound to the 4d brane and general relativity is recovered to a good approximation . according to various recent observational data including that of type ia supernovae @xcite , it is possible that the effective equation of state parameter @xmath3 , which is the ratio of the effective pressure @xmath4 to the effective energy density @xmath5 , evolves from being larger than @xmath6 ( non - phantom phase ) to being less than @xmath6 ( phantom phase @xcite ) ; namely , it has currently crossed @xmath6 ( the phantom divide ) . @xmath1 models that realize the crossing of the phantom divide have been studied @xcite . on the other hand , in the original dgp model @xcite and a phenomenological extension of the dgp model described by the modified friedmann equation proposed by dvali and turner @xcite , the effective equation of state parameter never crosses the @xmath3 = @xmath6 line . in this paper , we develop the `` phantom crossing dgp model '' by further extending the modified friedmann equation by dvali and turner @xcite . in our model , the effective equation of state parameter of dgp gravity crosses the phantom divide line , as indicated by recent observations . this paper is organized as follows . in the next section , we summarize the original dgp model , and check the behavior of the effective equation of state . in section [ sec:3 ] , we describe the modified friedmann equation by dvali and turner @xcite , and we also demonstrate that the effective equation of state does not cross the @xmath3 = @xmath6 line in this framework . in section [ sec:4 ] , we construct `` the phantom crossing dgp model '' by extending the modified friedmann equation proposed by dvali and turner . we show that the effective equation of state parameter of our model crosses the phantom divide line , and investigate the properties of our model . finally , a summary is given in section [ sec:5 ] . the dgp model @xcite assumes that we live on a 4d brane embedded in a 5d minkowski bulk . matter is trapped on the 4d brane and only gravity experiences the 5d bulk . the action is @xmath7 where the subscripts ( 4 ) and ( 5 ) denote quantities on the brane and in the bulk , respectively . @xmath8 ( @xmath9 ) is the 5d ( 4d ) planck mass , and @xmath10 represents the matter lagrangian confined on the brane . the transition from 4d gravity to 5d gravity is governed by a crossover scale @xmath11 . @xmath12 on scales larger than @xmath11 , gravity appears 5d . on scales smaller than @xmath11 , gravity is effectively bound to the brane and 4d newtonian dynamics is recovered to a good approximation . @xmath11 is the single parameter in this model . assuming spatial homogeneity and isotropy , a friedmann - like equation on the brane is obtained as @xcite @xmath13 where @xmath14 is the total cosmic fluid energy density on the brane . @xmath15 represents the two branches of the dgp model . the solution with @xmath16 is known as the self - accelerating branch . in this branch , the expansion of the universe accelerates even without dark energy because the hubble parameter approaches a constant , @xmath17 , at late times . on the other hand , @xmath18 corresponds to the normal branch . this branch can not undergo acceleration without an additional dark energy component . hence in what follows we consider the self - accelerating branch ( @xmath16 ) only . for the second term on the right - hand side of eq . ( [ dgp_fri ] ) , which represents the effect of dgp gravity , the effective energy density is @xmath19 and the effective pressure is @xmath20 where @xmath21 , the differential of the hubble parameter with respect to the cosmological time @xmath22 . using eqs . ( [ rho_eff ] ) and ( [ p_eff ] ) , the effective equation of state parameter of dgp gravity is given by @xmath23 vs. redshift @xmath24 . the red ( solid ) , green ( dashed ) , blue ( dotted ) lines represent the cases for @xmath25 , and @xmath26 , respectively ( corresponding to @xmath27 , and @xmath28 , respectively ) . [ fig : dgp_wrc ] ] fig . [ fig : dgp_wrc ] shows the behavior of the effective equation of state of dgp gravity @xmath29 versus the redshift @xmath24 for @xmath25 , and @xmath26 . assuming that the total cosmic fluid energy density @xmath14 of eq . ( [ dgp_fri ] ) contains matter and radiation , from eq . ( [ rch0omegam ] ) , these values of @xmath30 correspond to @xmath27 , and @xmath28 , respectively . @xmath31 where @xmath32 is the normalized energy density of matter and @xmath33 is the radiation on the brane ; i.e. , @xmath34 and @xmath35 . ( @xmath36 , @xmath37 ) . the subscripts @xmath38 designate the present value . the effective equation of state of dgp @xmath29 can also be exactly expressed in terms of the energy densities of matter and radiation , @xmath32 and @xmath33 @xcite . @xmath39 in realistic ranges of the energy density , @xmath40 and @xmath41 , the value of the effective equation of state can not be less than or equal to @xmath6 . that is , the effective equation of state never crosses the phantom divide line in the original dgp model . dvali and turner @xcite phenomenologically extended the friedmann - like equation ( eq . ( [ dgp_fri ] ) ) of the dgp model . this model interpolates between the original dgp model and the pure @xmath42cdm model with an additional parameter @xmath43 . the modified friedmann - like equation is @xcite @xmath44 for @xmath45 , this agrees with the original dgp friedmann - like equation , while @xmath46 leads to an expansion history identical to that of @xmath42cdm cosmology . differentiating both sides of eq . ( [ dt_fri ] ) with respect to the cosmological time @xmath22 , we obtain the following differential equation . @xmath47 where a dot indicates the derivative respect to the cosmological time . the quantity @xmath48 is the total cosmic fluid pressure on the brane . for the second term on the right - hand side of eq . ( [ dt_fri ] ) , which represents the effect of dgp gravity , the effective energy density is @xmath49 and from eq . ( [ hdot2 ] ) the effective pressure is @xmath50 . \label{p_dt}\ ] ] from eqs . ( [ rho_dt ] ) and ( [ p_dt ] ) , the effective equation of state parameter of the dgp model extended by dvali and turner is given by @xmath51 fig . [ fig : dt_wa ] shows a plot of the behavior of the effective equation of state of the dgp model by dvali and turner @xmath52 versus the redshift @xmath24 for @xmath53 , and @xmath54 ( assuming @xmath55 ) . , vs. redshift @xmath24 of the dgp model extended by dvali and turner for @xmath53 , and @xmath54 ( top to bottom ) assuming @xmath55 . [ fig : dt_wa ] ] in general , for equation of state @xmath56 , the energy density @xmath14 varies as @xmath57 . this leads to the following proportional relation . @xmath58 at the same time , from eq . ( [ rho_dt ] ) , we find @xmath59 . in the radiation - dominated epoch , from the proportional relation on hubble parameter @xmath60 , we obtain the following relation . @xmath61 as compared the right - hand side of eq . ( [ rad1 ] ) to that of eq . ( [ rad2 ] ) , during the earlier radiation - dominated epoch ( @xmath62 ) , the effective equation of state can also be represented with @xmath43 @xcite . @xmath63 in the same way , during the matter - dominated epoch ( @xmath64@xmath24@xmath65@xmath66 ) , from the proportional relation on hubble parameter @xmath67 , @xmath68 at the present time , @xmath52 is a stationary value close to @xmath6 . from these results , in the case of @xmath69 , the effective equation of state @xmath70 in all era , even in the radiation - dominated epoch . that is to say , the component of the dgp gravity works as the driving force of the accelerated expansion of the universe in all epochs . on the other hand , for @xmath71 , there is era when the effective equation of state becomes @xmath72 . that is , the dgp gravity does not drive the accelerated expansion in all epochs . the case of @xmath73 corresponds to the original dgp model described in the previous section . thus , in the original dgp model , the effective equation of state @xmath74 in the radiation - dominated epoch . and after the radiation - dominated epoch , becomes @xmath75 . in other words , the dgp gravity acts as the driving force of the accelerated expansion just after the radiation - dominated epoch . however , when @xmath43 is positive , the effective equation of state @xmath52 will exceed @xmath6 at all times . for negative @xmath43 , @xmath52 is always less than @xmath6 . in the case of @xmath46 , @xmath52 is @xmath6 constantly . based on this analysis , crossing of the phantom divide does not occur in the dgp model extended by dvali and turner . we propose the `` phantom crossing dgp model '' that extends the modified friedmann equation ( eq . ( [ dt_fri ] ) ) proposed by dvali and turner . our model can realize crossing of the phantom divide line for the effective equation of state of the dgp gravity . as mentioned in the previous section , the effective equation of state parameter of the dgp model by dvali and turner @xmath52 , takes the value of over @xmath6 for positive @xmath43 , and it is less than @xmath6 for negative @xmath43 . when @xmath46 , @xmath52 becomes @xmath6 . on the basis of these results , we consider a model in which @xmath43 varies being positive to being negative . to keep the model as simple as possible , we make the following assumption , @xmath76 where @xmath77 is the scale factor ( normalized such that the present day value is unity ) . the quantity @xmath0 is a constant parameter . in the period when the scale factor @xmath77 is less than the parameter @xmath0 ( @xmath78 ) , the effective equation of state exceeds @xmath6 . at the point when the scale factor @xmath77 equals @xmath0 , ( @xmath46 ) , the equation of state s value will be @xmath6 . in the period when the scale factor @xmath77 exceeds the parameter @xmath0 ( @xmath79 ) , the equation of state will be less than @xmath6 . in this way , crossing of the phantom divide is realized in our model . replacing @xmath43 by @xmath80 in eq . ( [ dt_fri ] ) , the friedmann - like equation in our model is given by @xmath81 differentiating both sides of eq . ( [ hirano_fri ] ) with respect to the cosmological time @xmath22 , the following differential equation is obtained . @xmath82 for the second term on the right - hand side of eq . ( [ hirano_fri ] ) representing the effect of dgp gravity , the effective energy density is @xmath83 and from eq . ( [ hirano_hdot2 ] ) , the effective pressure is @xmath84 . \label{p_hirano}\ ] ] using eqs . ( [ rho_hirano ] ) and ( [ p_hirano ] ) , the effective equation of state of our model is given by @xmath85 , vs. redshift @xmath24 . the red ( solid ) , green ( dashed ) , blue ( dotted ) lines represent the cases of @xmath86 , and @xmath87 , respectively ( assuming @xmath55 ) . [ fig : dgp_hirano],width=445 ] depicted in fig . [ fig : dgp_hirano ] near recent epochs . [ fig : dgp_hirano_kakudai],width=445 ] fig . [ fig : dgp_hirano ] shows a plot of the effective equation of state of our model @xmath88 versus the redshift @xmath24 ( see also fig . [ fig : dgp_hirano_kakudai ] which shows an enlarged view of this diagram ) . our model is an extension of the dgp model and realizes crossing of the phantom divide . the effective equation of state @xmath88 of models for @xmath89 and @xmath87 ( assuming @xmath55 ) crosses the phantom divide line when the redshift @xmath90 and @xmath91 , respectively . we find that the smaller the parameter @xmath0 is , the older epoch crossing of the phantom divide occurs in . @xmath0 is not necessarily equal to the scale factor at the time of crossing the phantom divide , even though eq . ( [ beta ] ) is assumed . in the eq . ( [ hirano_fri ] ) , the value of @xmath80 that is the power index of @xmath92 varies with respect to time . as the power index of the differential equation changes with time , furthermore , in parallel , the differential equation is solved with respect to time . hence , the time lag occurs , the scale factor at the time of crossing the phantom divide is more than the value of @xmath0 . in a way similar to the derivation of eq . ( [ wa_rad ] ) , we represent the effective equation of state of phantom crossing dgp model with @xmath0 . in the radiation - dominated epoch , the scale factor @xmath77 is taken to be @xmath38 in comparison with the value of @xmath0 . therefore , as @xmath93 in eq . ( [ beta ] ) , during the radiation - dominated epoch ( @xmath62 ) , the effective equation of state is approximately @xmath94 that is , in the case of @xmath95 , the effective equation of state @xmath96 in all era , including the radiation - dominated epoch . on the other hand , for @xmath97 , there is era when the effective equation of state becomes @xmath98 . and absolute value of the effective pressure @xmath99 ( note that @xmath100 ) of our model vs. redshift @xmath24 , for ( @xmath101 ) = ( @xmath102 ) . [ fig : rhop_hirano ] ] fig . [ fig : rhop_hirano ] shows the effective energy density @xmath103 and absolute value of the effective pressure @xmath99 ( note that @xmath100 ) of our model for ( @xmath101 ) = ( @xmath102 ) versus the redshift @xmath24 , normalized such that the effective energy density is unity at the time of phantom crossing . it shows that the absolute value of the effective pressure @xmath99 exceeds the effective energy density @xmath103 at the time of crossing of the phantom divide . the recent observational data for type ia supernovae @xcite show that crossing of the phantom divide line occurs at a redshift @xmath104 @xcite . in our model , for @xmath105 ( when @xmath55 ) , crossing of the phantom divide occurs at @xmath104 . + in a proposed model in which the phantom divide is crossed at @xmath104 , ( @xmath101 ) = ( @xmath102 ) , we investigate and show the property of phantom crossing dgp model . relative to that of a constant expansion cosmology @xmath106 , vs. the redshift @xmath24 . models and parameters are ( from top to bottom ) : ( 1 ) @xmath42cdm model , @xmath107 = 0.30 ; ( 2 ) phantom crossing dgp model , @xmath0 = 0.50 , @xmath107 = 0.30 ; ( 3 ) dgp model by dvali and turner , @xmath43 = 0.50 , @xmath107 = 0.30 ; ( 4 ) original dgp model , @xmath107 = 0.30 . [ fig : snia ] ] , matter @xmath32 , and dgp gravity @xmath108 , vs. the redshift @xmath24 in the phantom crossing dgp model with the proposed parameter ( @xmath101 ) = ( @xmath102 ) . [ fig : omega ] ] fig . [ fig : snia ] shows the distance modulus @xmath109 relative to that of a constant expansion cosmology @xmath106 , versus the redshift @xmath24 . that is , when @xmath110 is positive , cosmic expansion is accelerating . the distance modulus is defined by @xmath111 where @xmath112 is the hubble free luminosity distance given by @xmath113 @xmath114 being the hubble constant @xmath115 in units of @xmath116 . we adopt @xmath117 @xcite . in fig . [ fig : snia ] , models and parameters are ( from top to bottom ) : ( 1 ) @xmath42cdm model , @xmath107 = 0.30 ; ( 2 ) phantom crossing dgp model , @xmath0 = 0.50 , @xmath107 = 0.30 ; ( 3 ) dgp model by dvali and turner , @xmath43 = 0.50 , @xmath107 = 0.30 ; ( 4 ) original dgp model , @xmath107 = 0.30 . phantom crossing dgp model can realize late - time acceleration of the universe very similar to that for @xmath42cdm model , without dark energy . [ fig : omega ] shows the normalized energy density of radiation @xmath33 , matter @xmath32 , and dgp gravity @xmath108 versus the redshift @xmath24 in the phantom crossing dgp model with the proposed parameter ( @xmath101 ) = ( @xmath102 ) . where , @xmath118 . @xmath119 is the effective energy density of dgp gravity defined by eq . ( [ rho_hirano ] ) . we find that the universe is dgp gravity - dominated near recent epochs . therefore , in the phantom crossing dgp model , the late - time acceleration is driven by the effect of dgp gravity . , vs. the redshift @xmath24 . models and parameters are same as fig . [ fig : snia ] . [ fig : weff ] ] fig . [ fig : weff ] shows the effective equation of state @xmath120 versus the redshift @xmath24 . models and parameters are same as fig . [ fig : snia ] . only our phantom crossing dgp model can realize crossing of the phantom divide line at @xmath104 as indicated by recent observations . * we confirmed that the effective equation of state does not cross the phantom divide line in the original dgp model . + * we also demonstrated that crossing of the phantom divide does not occur in the dgp model by dvali and turner . + * we constructed the phantom crossing dgp model . this model realizes crossing of the phantom divide . we found that the smaller the value of the new introduced parameter @xmath0 is , the older epoch crossing of the phantom divide occurs in . our model can realize late - time acceleration of the universe very similar to that of @xmath42cdm model , without dark energy , due to the effect of dgp gravity . in the proposed model ( ( @xmath101 ) = ( @xmath102 ) ) , crossing of the phantom divide occurs at @xmath104 as indicated by recent observations . riess , et al . j. * 116 * , 1009 ( 1998 ) s. perlmutter , et al . , astrophys . j. * 517 * , 565 ( 1999 ) r. knop , et al . , astrophys . j. * 598 * , 102 ( 2003 ) a.g . riess , et al . , astrophys . j. * 607 * , 665 ( 2004 ) a.g . riess , et al . , astrophys . j. * 659 * , 98 ( 2007 ) p. astier , et al . astrophys . * 447 * , 31 ( 2006 ) g. miknaitis , et al . , astrophys . j. * 666 * , 674 ( 2007 ) w.m . wood - vasey , astrophys . j. * 666 * , 694 ( 2007 ) j.a . frieman , et al . j. * 135 * 338 ( 2008 ) d.n . spergel , et al . , astrophys . * 148 * , 175 ( 2003 ) e. komatsu , et al . [ wmap collaboration ] , astrophys . * 180 * , 330 ( 2009 ) m. tegmark , et al . [ sdss collaboration ] , astrophys . j. * 606 * , 702 ( 2004 ) m. tegmark , et al . d * 69 * , 103501 ( 2004 ) b. ratra , p.j.e . peebles , phys . d * 37 * , 3406 ( 1988 ) p.j.e . peebles , b. ratra , astrophys . j. * 325 * , l17 ( 1988 ) r.r . caldwell , phys . b * 545 * , 23 ( 2002 ) r.r . caldwell , m. kamionkowski , n.n . weinberg , phys . lett * 91 * , 071301 ( 2003 ) k. hirano , k. kawabata , z. komiya , astrophys . space sci . * 315 * , 53 ( 2008 ) j.g . hartnett , k. hirano , astrophys . space sci . * 318 * , 13 ( 2008 ) k. hirano , et al . , proceedings of 59th yamada conference `` inflating horizons of particle astrophysics and cosmology '' ( universal academy press ) , p.219 ( 2005 ) z. komiya , k. kawabata , k. hirano , h. bunya , n. yamamoto , j. korean astron . soc . * 38 * , 157 ( 2005 ) z. komiya , k. kawabata , k. hirano , h. bunya , n. yamamoto , astron . astrophys . * 449 * , 903 ( 2006 ) s. nojiri , s.d . odintsov , int . phys . * 4 * , 115 ( 2007 ) s. nojiri , s.d . odintsov , arxiv:0801.4843 [ astro - ph ] . s. nojiri , s.d . odintsov , arxiv:0807.0685 [ hep - th ] . dvali , g. gabadadze , m. porrati , phys . b * 485 * , 208 ( 2000 ) c. deffayet , phys . b * 502 * , 199 ( 2001 ) c. deffayet , g.r . dvali , g. gabadadze , phys . d * 65 * , 044023 ( 2002 ) k. koyama , gen . . grav . * 40 * , 421 ( 2008 ) . u. alam , v. sahni , a.a . starobinsky , jcap * 0406 * , 008 ( 2004 ) s. nesseris , l. perivolaropoulos , jcap * 0701 * , 018 ( 2007 ) p.u . yu , phys . b * 643 * , 315 ( 2006 ) h.k . jassal , j.s . bagla , t. padmanabhan , astro - ph/0601389 . s. nojiri , s.d . odintsov , phys . b * 562 * , 147 ( 2003 ) k. bamba , c.q . geng , phys . b * 679 * , 282 ( 2009 ) k. bamba , c.q . geng , s. nojiri , s.d . odintsov , phys . d * 79 * , 083014 ( 2009 ) g. dvali , m. turner , astro - ph/0301510 . k. nozari , m. pourghassemi , jcap * 0810 * , 044 ( 2008 ) k. nozari , n. behrouz , t. azizi , b. fazlpour , prog . . phys . * 122 * , 735 ( 2009 ) a. lue , r. scoccimarro , g.d . starkman , phys . d * 69 * , 124015 ( 2004 ) a. lue , phys . rep . * 423 * , 1 ( 2006 ) w.l . freedman , et al . , astrophys . j. * 553 * , 47 ( 2001 )
|
we propose a phantom crossing dvali gabadadze porrati ( dgp ) model . in our model ,
the effective equation of state of the dgp gravity crosses the phantom divide line .
we demonstrate crossing of the phantom divide does not occur within the framework of the original dgp model or the dgp model developed by dvali and turner . by extending their model , we construct a model that realizes crossing of the phantom divide .
we find that the smaller the value of the new introduced parameter @xmath0 is , the older epoch crossing of the phantom divide occurs in .
our model can account for late - time acceleration of the universe without dark energy .
we investigate and show the property of phantom crossing dgp model .
example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the noise excess free ( phase - insensitive ) quantum record / upload of single - mode light into the atomic memory can be defined by the transformation @xmath1 of the light quadratures @xmath2 and @xmath3 to the atomic quadratures @xmath4 and @xmath5 in the heisenberg picture . here @xmath6 stands for the transmission coefficient of the record / upload . the quadrature noisy operators @xmath7 and @xmath8 describe the added noises in the state transfer . the variance of vacuum noise is considered to be unity . from the heisenberg uncertainty principle , the minimal variance of both @xmath7 and @xmath8 is @xmath9 ( @xmath10 ) . in this case , there is no noise excess in the record / upload process . it corresponds to transmission exhibiting only a pure loss , which can be modeled by a virtual beam splitter between light and atoms with vacuum in the free port . such the noise excess free record / upload is advantageous because it can preserve some important non - classical properties of quantum states . it will never completely break entanglement of gaussian state , vanish sub - poisson behavior of single - photon state @xcite or quantum superposition of coherent states @xcite . also only loss in the record / upload will not break security of the continuous - variable key distribution protocol with coherent states if the reverse reconciliation is used @xcite . basically , information encoded into the quadratures and gaussian entanglement or security of key distribution are not changed if the recorded / upoload state is only unitarily transformed by a known gaussian operation like the phase shift , displacement and squeezing . in this case , the record / upload is noise excess free up to that unitary operation . particularly , we focus on the noise excess free record / upload up to the squeezing operation : @xmath11 , @xmath12 . for an application , either the squeezing could be actively post - corrected in the memory , as will be proposed below , or if it is not necessary , it can be finally simply corrected on the measured data . to increase a quality of the upload , also a pre - squeezing operation @xmath13 , @xmath14 on the light mode before record / upload can be considered . it transforms the input state to its squeezed version before the record / upload into the memory . it will be shown , both the pre - squeezing of input state and the post - squeezing correction can remarkably help to reach the noise excess free record or even the lossless upload of quantum state into the memory . consider now the quantum memory experiment in ref . @xcite , see also ref . @xcite for more details . in that experimental setup , there are two simultaneously available quantum non - demolition ( qnd ) interactions between the light and atoms inside the cells . basically , they couple together the mode of light described by two complementary quadratures @xmath2 and @xmath3 and the effective collective atomic mode having complementary quadratures @xmath4 and @xmath5 . both the qnd transformations can be simply described in heisenberg picture : @xmath15 and @xmath16 where @xmath17 is an effective coupling constant @xcite . either coupling ( [ coupl1 ] ) or ( [ coupl2 ] ) can be separately activated in the same set - up @xcite . both are the particular qnd interactions , but the following analysis is generally valid for any kind of the qnd transformation between the quadratures of light and atomic memory . if the coupling ( [ coupl1 ] ) is considered , the quadrature @xmath3 can be directly written into the memory by the light - atom interaction , up to the added noise from the atomic quadrature @xmath4 . to write the complementary quadrature @xmath2 , the light pulse passing through the memory is measured by homodyne detection and the photocurrent controls the magnetic field applied with an adjustable gain to the atomic cells . by this feed - forward technique , the atomic quadrature @xmath5 can be displaced whereas the quadrature @xmath4 is not disturbed . this is standard record mechanism used in the ref . ( @xcite ) . now we assume the record of unknown quantum state up to the squeezing of the recorded state . after such the procedure , the transformation of the atomic quadratures takes the following form @xmath18 where @xmath19 is an overall gain of the feed - forward correction and @xmath20 is a scaling factor representing the squeezing of recorded state . the effective atomic mode is considered initially in vacuum state with the unit variance . to obtain the noise excess free record up to the squeezing , the parameters have to satisfy the following equations @xmath21 the solution of ( [ eq1 ] ) : @xmath22 and @xmath23 , gives a possibility to achieve the noise excess free record described by eqs . ( [ noiseless ] ) up to the squeezing . the record transmission coefficient @xmath24 of the transfer from the light mode to atomic mode shows that an unknown state written inside the memory is always squeezed by the factor @xmath23 . but even for very small @xmath25 , any state can be ( up to the squeezing ) written into the memory with no noise excess . for a feasible gain around @xmath26 , the upload can be up to the noise excess free attenuation @xmath27 up to the squeezing . such the noise excess free record even without the squeezing correction can be useful , for example , for the manipulations with gaussian entanglement . using computable measure of entanglement for gaussian state @xcite , it is straightforward to prove the following comparison . let us compare the cases after the record , without squeezing correction and with perfect squeezing correction . consider single mode from two - mode gaussian entangled state which is recorded into the memory . for any single - mode noisy gaussian operation after the noise excess free record ( up to the squeezing ) , even with an arbitrary small @xmath25 , no matter that the uncorrected squeezing reduces entanglement it will never help to subsequent gaussian channel to completely break entanglement . it means that for both the cases , the thresholds to pass entanglement through the record procedure plus consecutive gaussian channel are identical . it illustrates a practical advantage of the noise excess free record up to the squeezing . such the result is generally impossible to obtain if there is a noise excess in the record . to actively compensate the squeezing of the recorded state inside the memory , the squeezing post - correction is necessary . the squeezing post - correction should make transformation @xmath28 and @xmath29 to reduce effect of the squeezing in the atomic memory . since the total record process could exhibit pure loss , the squeezing post - correction can be implemented at a cost of an additional loss in the record . to build post - correction operation , the second kind of the coupling ( [ coupl2 ] ) with the coupling constant @xmath30 is now assumed in the same experimental set - up . now , the light mode described by @xmath31 and @xmath32 is considered in vacuum state . thus only bright local oscillator is injected into the cells . the homodyne measurement of @xmath33 quadrature of the pulse outgoing from the cells is followed by the feed - forward displacement of the atomic quadrature @xmath4 by the magnetic field . to achieve a desirable squeezing post - correction @xmath28 and @xmath29 ( @xmath34 is considered to compensate the squeezing in eqs . ( [ record ] ) ) up to a pure loss ( but without any noise excess ) , the transformations @xmath35 have to satisfy the following conditions @xmath36 it can be achieved if @xmath37 and then @xmath38 is given for any @xmath39 . the additional losses introduced by the post - correction are characterized by the transmission @xmath40 . since @xmath34 , the squeezing post - correction can be performed only in the @xmath5 quadrature . to precisely compensate the squeezing introduced in the record , it is necessary to adjust @xmath41 . from this follows that @xmath42 and total transmission coefficient of the record plus squeezing correction is given by @xmath43 the maximal transmission @xmath44 is achieved for @xmath26 ( @xmath45 ) which is feasible in the experiment in ref . @xcite . as the result , any quantum state can be in principle deterministically recorded without noise excess as is described by eq . ( [ noiseless ] ) up to the transmission @xmath46 . another open possibility how to compensate the squeezing in the recorded state is a pre - squeezing @xmath47 and @xmath13 of the state which should be recorded . from a simple calculation follows that this configuration does not lead in principle to the noise excess free record . but if both the pre - squeezing and squeezing post - correction are considered together , the local pre - squeezing helps to increase the coupling constant @xmath17 . taking both the pre - squeezing and squeezing post - correction into the consideration , the record mechanism is described by the transformations : @xmath48 where @xmath49 stands for the scaling factor corresponding to the pre - squeezing of the light mode . to approach the noise excess free record , it is necessary to adjust electronic gain @xmath50 and then the squeezing correction after the record is determined by @xmath51 . thus for a given @xmath17 , the squeezing post - correction has to be larger ( @xmath20 should be smaller ) as @xmath49 is bigger . interestingly , the transmission coefficient @xmath52 of the record up to the squeezing is effectively increased as the squeezing factor @xmath49 is larger . remarkably , by only local pre - squeezing of the state , which should be recorded into memory , the coupling constant @xmath17 is increased . since @xmath53 approaches unity for sufficiently large @xmath49 , a lossless record up to the squeezing can be approached . it is even possible for an arbitrary small @xmath25 if the sufficient pre - squeezing is available . on the other hand , if the uploaded state inside the memory should be not squeezed , the squeezing correction has to be applied . then , to reach the limit @xmath44 , @xmath54 has to be satisfied and it can be approached for any @xmath17 as the pre - squeezing is increasing . this deterministic record can be used to upload not only coherent states and gaussian entangled states , but also highly nonclassical single - photon state as an useful quantum resource . the single - photon state can be prepared by the single - photon subtraction from the squeezed light @xcite . actually , the state prepared by such the procedure approaches the squeezed single - photon state @xcite . this squeezing would be advantageous for the upload procedure as has been discussed . but , to have exactly single - photon state inside memory , it is necessary to actively apply the squeezing post - correction . this operation can be naturally included into the uploading scheme proposed above . in a result , to get single - photon state inside the memory , the single - photon subtraction can be simply applied on the squeezed state followed by the uploading procedure proposed above . but still the upload has some uncertainty , we do not know if the single photon state is actually inside the memory . therefore , the uploaded state is a mixture @xmath55 where @xmath56 which optimally approaches @xmath57 as @xmath54 . the wigner function of that state is never negative since @xmath58 @xcite , but still any mixture ( [ mix ] ) with @xmath59 is a non - classical state @xcite . for example , the statistics of ( [ mix ] ) is always sub - poissonian for arbitrary @xmath59 . in experiments with the single photons , this method can be used only if the post - selection on the fixed number of photons is finally applied to have complete control how many photons participated in the experiment . note , there is a similar requirement for the linear optical experiments with the single photons . they work correctly only if all the photons are finally detected . fortunately , there is a better way how to upload the single - photon state . instead of the deterministic electro - optical feed - forward correction , a proper post - selection of measured data from the homodyne detector is applied . it makes the upload only probabilistic , but since the uploaded state is resource , it is acceptable if the rate is not too small . for the upload of the gaussian states , there is no gain from this probabilistic method over the previous deterministic feed - forward correction . but for the non - classical non - gaussian states , for example , for the squeezed single - photon state produced by the single - photon subtraction @xcite the post - selection of the measured quadrature @xmath33 in a tiny interval @xmath60 around value @xmath61 will remarkably help . to simply understand why it will help , the unitary coupling ( [ coupl2 ] ) between light and atoms can expand for very small @xmath62 , where @xmath63 is an interaction time , as @xmath64 , where @xmath65 and @xmath66 are annihilation operators of the light and atomic modes . in this convention , the vacuum noise has variance @xmath0 . the ideal projection on the eigenstate @xmath67 of @xmath33 corresponding to eigenvalue @xmath61 can be rewritten in a form of @xmath68 . initially , light is in the fock state @xmath69 and atomic mode is in the vacuum state @xmath70 . straightforwardly , we can prove that @xmath71 and from it follows that in the limit of weak coupling and very tight post - selection @xmath72 , it is possible to upload single - photon state to the memory without any loss and noise . to calculate the upload for a finite @xmath73 , we use the formalism of wigner functions . the squeezed single - photon state is represented by the wigner function @xmath74 where @xmath20 is squeezing parameter . the atomic memory is considered initially in vacuum state described by the wigner function @xmath75 . the application of the coupling ( [ coupl2 ] ) on the product @xmath76 followed by the homodyne detection and above mentioned post - selection will generate the wigner function @xmath77+\mbox{erf}\left[\frac{\sqrt{2}}{a } \left(b+\kappa x\right)\right ] \right)-\nonumber\\ \frac{4\sqrt{2}}{\sqrt{\pi^3 } s d } \exp\left(-2\frac{b^2}{a^2}-2x^2\frac{d^2}{a^2}-2p^2\frac{a^2}{d^2 } \right)\times\nonumber\\ \left(b\mbox{cosh}\left[\frac{4b\kappa x}{a^2}\right]+\kappa x \mbox{sinh}\left[\frac{4b\kappa x}{a^2}\right]\right)\nonumber\\\end{aligned}\ ] ] of the state uploaded into the memory , where @xmath78 and @xmath79 $ ] is error function . the success rate of the upload is @xmath80-\sqrt{\frac{2}{\pi } } \exp\left(-\frac{2b^2}{d^2}\right)\frac{2a^2b}{d^3}.\ ] ] if the squeezing post - correction @xmath81 and @xmath82 is applied , then as @xmath73 goes to zero , the wigner function of atomic state approaches @xmath83 where @xmath84 is exactly the wigner function of the single photon . the squeezing post - correction does not depend on the post - selection threshold @xmath73 . to approach controllable upload , the squeezing post - correction proposed in the previous section can not be used , since it can work only up to additional losses which destroys the desired control of upload . but , fortunately , if both @xmath85 are small then the squeezing post - correction can be approximately omitted and the uploaded state is @xmath86 as the result , the single - photon state is probabilistically uploaded into the memory with full control and remarkably , is not attenuated comparing to the previous deterministic case . thus the post - selection compensates the loss which occurs in the deterministic upload . since the probabilistic upload is obtained for a small @xmath17 , the post - selection also compensates strength of the coupling between light and atoms . but as @xmath17 decreases , the post - selection threshold @xmath73 has to decrease and it reduces the success rate of the upload . to simply measure quality of the single - photon state uploaded into the memory versus the success rate of the upload , the fidelity and negativity of wigner function can be used . the fidelity between the uploaded state described by the wigner function @xmath87 and target single - photon state with the corresponding wigner function @xmath84 is defined as @xmath88 whereas the negativity in the origin of the phase space is given simply by @xmath89 . the expression for the fidelity is rather complex but for both the @xmath17 and @xmath73 small , the expansion @xmath90 shows how the result of a sufficiently tight post - selection practically does not depend on a small @xmath17 . without the squeezing post - correction , the maximal fidelity is exactly @xmath91 in the limit @xmath92 . for a small @xmath73 , the negativity of wigner function is @xmath93 the maximum of negativity @xmath94 can be always approached if @xmath73 is sufficiently small . the @xmath95 is independent on @xmath20 and @xmath17 because the negativity of wigner function is not changed by the pre - squeezing . both the fidelity and negativity can be achieved with the success rate approaching @xmath96 which grows as pre - squeezing parameter @xmath20 is smaller . here the pre - squeezing plays also positive role , it significantly enhances the probability of success . the quantitative results of the single photon upload are depicted in fig.1 . the pre - squeezing @xmath97 can evidently help to increase the success rate for both the fidelity as well as for the negativity , three curves are depicted for @xmath98 . thus squeezing of the single - photon state from the squeezed light by the single photon subtraction is actually an advantage for the upload into the memory . contrary to the deterministic case , the post - selection can effectively increase the coupling between light and atoms . to approach such the lossless upload of single photon state , the in - coupling and out - coupling losses have to be minimized . the in - coupling losses will mix the single - photon state with vacuum state and in a result , a less sub - poissonian state is actually uploaded . the efficiency of homodyne detector can be can be very close to unity and also electronic noise of homodyne detection can be negligible , if the power of local oscillator is high . the out - coupling losses can be joined with the efficiency of the homodyne detection to an effective total out - coupling efficiency @xmath99 . the fidelity ( [ fi ] ) is a decreasing function of @xmath99 , as can be seen from fig . 2 . to approach the losses upload , the coupling constant @xmath17 has to be reasonably high and then , the pre - squeezing in the opposite direction ( @xmath100 ) can help to reach unit fidelity of the single - photon state at a cost of the success rate . a superposition of the coherent states is considered as an another resource for the highly non - linear quantum operations and quantum computing @xcite . it is rather a superposition of many fock states , than just single fock state , therefore it can be interesting to investigate whether the upload can transfer such the state of light into the memory . let us consider the following superposition @xmath101 of coherent states which is pre - squeezed before the upload , where @xmath102 . then the state can be described by the wigner function @xmath103,\end{aligned}\ ] ] where @xmath104 . at the beginning , the vacuum state is inside the atomic memory . after the coupling ( [ coupl2 ] ) and successful post - selection of the measured @xmath33 quadrature in the interval @xmath105 around @xmath61 , the atomic mode is projected to the state with the following wigner function @xmath106\mbox{erf}(p)+\mbox{cerf}(p)\right),\nonumber\end{aligned}\ ] ] where @xmath107 + \mbox{erf}\left[\frac{\sqrt{2}}{a}\left(b+a\kappa p\right)\right],\nonumber\\\end{aligned}\ ] ] and @xmath108\right]+\nonumber\\ & & \mbox{re}\left[\mbox{erf}\left[\frac{\sqrt{2}}{a}\left(ix_0a+b+\kappa p\right)\right]\right].\end{aligned}\ ] ] the success rate of such the uploading process is @xmath109+\right.\nonumber\\ \left.\mbox{erf}[\frac{\sqrt{2}(ix_0a+b)}{d}]-\mbox{erf } [ \frac{\sqrt{2}(ix_0a - b)}{d}]\right).\nonumber\\\end{aligned}\ ] ] if the squeezing post - correction @xmath110 and @xmath111 is applied , the wigner function ( [ wiga ] ) approaches the eq . ( [ cat ] ) as @xmath73 goes close to zero , but with a reduced amplitude @xmath112 . if @xmath73 is enough small and @xmath17 is sufficiently smaller than @xmath20 ( @xmath113 ) then the squeezing post - correction can be approximately omitted . but then also amplitude @xmath114 of the states in the uploaded superposition decreases . fortunately , this amplitude can be successfully compensated by a sufficient pre - squeezing ( @xmath20 is small ) . thus sufficiently large superposition of coherent states can be also uploaded into the quantum memory without any loss . in a contrast to a simple phase - insensitive character of the single photon state , the superposition of coherent states exhibits more complex phase - space behavior . in one quadrature there two symmetrically located gaussian peaks like for the classical mixture of the coherent states . on the other hand , the complementary quadrature shows multiple interference fringes covered by a gaussian envelope . because the fidelity averages both the interference as well as distinguishable peaks , it is not good measure of a quality of the uploaded superposition . from this reason , the marginal probability distribution @xmath115 exhibiting the interference is rather evaluated separately , to directly observe the interference in the superposition . the explicit form of marginal distribution @xmath116 can be approximated by @xmath117\ ] ] for a small post - selection threshold @xmath73 . assuming @xmath118 , the final distribution is @xmath119.\ ] ] for @xmath120 ( @xmath121 ) , @xmath122 approaches the exact marginal distribution of the superposition of the coherent state only the amplitude @xmath123 is reduced . as a result , the attenuation in the deterministic scheme , which for the coherent - state superposition introduces an excess noise in the uploaded state , is substituted here by only a reduction of @xmath124 without any additional noise . the uploaded state closely approaches the ideal pure state superposition with the reduced amplitude . the pre - squeezing plays here also a remarkable role . in fig . 2 ( upper picture ) , there is a marginal distribution of the uploaded state without pre - squeezing . evidently , the interference is vanishing . but if the sufficient pre - squeezing ( -6db ) is applied ( lower picture ) then the fringes can be observed and also the success rate is higher . in conclusion , the deterministic noise excess free record of unknown state and the probabilistic lossless upload of the resource state ( single - photon state , superposition of coherent states ) into quantum memory are proposed , both based on the original quantum memory experiment in ref . first , the deterministic scheme is able to record unknown quantum state only with a reduced transmission ( maximally @xmath0 ) . the pre - squeezing of the state of light going to the memory can compensate small coupling strength between light and atomic memory . but this record is without any trigger controlling the uploading process what is crucial , for example , for the upload of single photons . therefore , the probabilistic scheme which is able to upload single - photon state is proposed . it approaches unit fidelity of the upload at the cost of success rate , even for a weak coupling between the light and atoms . in this case , the pre - squeezing enhances the success rate of the upload . the same method can be used to upload the pure coherent - state superposition , at the cost of the success rate and reduced amplitude of the superposition , but with preservation of the purity of the superposition . in this case , the pre - squeezing not only enhances the success rate , but also effectively enlarges the reduced amplitude of the coherent states in the superposition and thus increases non - classicality of the uploaded state . since the considered scheme has been used in the real experiment set - up , both the methods can be directly used to upload the non - classical resources into quantum memory . the proposed schemes are direct steps towards to storage and operations on highly non - classical quantum states in the continuous - variable quantum memory . * acknowledgments * this research has been supported by projects msm 6198959213 and lc06007 of the czech ministry of education and grant 202/08/0224 of ga cr . i also acknowledges a support by the alexander von humboldt foundation and eu grant fp7 212008 - compas .
|
in recent continuous - variable quantum memory experiment , one quadrature of light pulse is directly uploaded by the light - atom coupling whereas complementary quadrature is obtained by homodyne measurement of the out - coupled light .
subsequently , information from homodyne measurement is written into the memory by feed - back electro - optical control of the atomic state . using the same experimental set - up , a deterministic noise excess free record of unknown quantum states to continuous - variable quantum atomic memory
is proposed .
further , the memory experiment is extended by the pre - squeezing of the recorded state of light and squeezing post - correction of the recorded state . to upload a resource to the memory , as single - photon state or superposition of coherent states , the post - selection of the measurement results from homodyne detection
is suggested .
then such the probabilistic upload approaches even a lossless transfer of these highly nonclassical states into the quantum memory .
a non - classicality of quantum states is extremely fragile to disturbing effects of a noise coupled to the system .
this is a real problem for the transmission and operations with the non - classical states .
recently , the same problem arises also in a construction of quantum memory .
the quantum memories have been recently demonstrated using electromagnetically - induced transparency @xcite and off - resonant faraday effect @xcite .
it is useful to distinguish three basic steps : initial record ( or upload ) of the quantum state , its storage for a longer time inside the memory and final read - out from the memory .
in addition , quantum state written to the memory can either carry encoded information or it can be a resource for other quantum operations . to distinguish these two cases ,
the record of quantum states is used for the states carrying information and the upload of quantum state for the resources . in the former case ,
encoded information makes the state to be at least partially unknown , whereas in the latter case , the uploaded state is known completely and the upload can be optimized with a respect to that specific state . in this case , a relatively fast probabilistic upload is sufficient to transfer the resource state into the memory . the quantum memory should definitely preserve the non - classical properties of quantum states .
it is a necessary for any application of quantum memory , for example , for the quantum repeaters for the long - distance quantum key distribution .
focusing on the record / upload process , the noise during this process has to be investigated .
there is always vacuum noise which accompanies any loss in the record / upload , but even more destroying is an impact of a noise excess above this vacuum noise penalty .
therefore , an noise excess free record / upload to quantum memory should be find . in the other words , the noise excess free means that the record / upload is only pure lossy process .
such the record / upload will be not entanglement breaking channel @xcite , will not completely vanish sub - poissonian statistics of single photon state @xcite and also will not smear out interference effects in the superposition of coherent states @xcite .
also security of the continuous - variable quantum key distribution is preserved for any loss in the transfer of quantum state if there is no noise excess @xcite .
second , more complicated open problem is how to construct the lossless upload of the nonclassical state .
such the upload allows to prepare highly non - classical inside the memory .
the quantum memory for coherent states of light has been recently demonstrated using a pair of glass cells filled by cs atoms ( at room temperature ) @xcite . in this experiment ,
the coherent state is generated by the amplitude and phase modulations of the strong continuous - wave ( cw ) millisecond laser pulses .
the off - resonant interaction of cw light pulses with the collective spin atoms inside the cell ( faraday effect ) can be describe as quantum non - demolition coupling between effective quadratures of the light and atomic modes .
the coupling automatically records a single quadrature of light to the effective quadrature of collective spin of the atoms .
the complementary quadrature of light pulse passing through the memory is then measured by homodyne detection and feed - forward by the electro - optical control into the atomic memory . in the experiment @xcite , an average fidelity of recorded coherent states has been shown to be higher than a threshold corresponding to measure - prepare strategy . to approach unit fidelity in the same set - up ,
a difficult high squeezing of the collective spin state of the atoms before the uploading procedure is required .
since the average fidelity has been considered as a figure of merit , the demonstrated record of the coherent state is not the excess noise free and the recorded state has been far a way from a pure state .
there were few attempts to achieve noise excess free memory , involving several coherences of each atom in the cell @xcite or multi - pass version of the deterministic memory @xcite .
further , the scheme with two passes has been theoretically examined with an additional magnetic field to achieve the deterministic and noise excess free memory with a loss exponentially decreasing as the coupling constant is growing @xcite .
these experimentally challenging extensions are mainly orientated to approach the deterministic record / read - out of unknown quantum state carrying information into / from memory . on the other hand , to upload the known highly non - classical resource state , such as the single photon state or the superposition of coherent states into the memory , the probabilistic approach could be more simple and effective . in this paper
, the noise excess free version of deterministic record of unknown quantum state into the quantum memory is described and the probabilistic upload of highly non - classical states ( single - photon state , superposition of coherent states ) approaching lossless light - atomic transfer is proposed . in the section ii .
, is shown that the original set - up from ref .
@xcite can be directly used to obtain noise excess free deterministic record of any quantum state of light for an arbitrary weak non - demolition coupling and without any noise pre - squeezing of the atomic memory .
the recorded state is noise excess free but only up to a squeezing of the collective atomic variables .
therefore , a squeezing post - correction is suggested to approach phase - insensitive record symmetrical in both the quadratures .
further , it is demonstrated that a pre - squeezing of the state of light before entering into the memory remarkably enhances the coupling strength and subsequently , it reduces the attenuation of the uploaded state inside the memory .
unfortunately , the application of the squeezing post - correction actively is always at a cost of additional loss and for a higher squeezing of the recorded state in the memory , the additional attenuation is larger . in the result ,
the purely lossy phase - insensitive record with maximum @xmath0 of transmission can be achieved without any noise excess .
it can be restrictive for the upload of the non - classical resources , such as the single - photon state or the superposition of the coherent states . if the upload is lossy , such the single photon is uncontrollably transferred to the memory , without any trigger saying whether the photon is really uploaded .
therefore , in the section iii . , a probabilistic upload is proposed based on a post - selection of a result from the homodyne detector in the same memory set - up .
the single photon can be probabilistically uploaded into the memory without loss and with an arbitrary precision , although the coupling between the light and atoms in the memory can be weak .
the superposition of coherent states can be probabilistically uploaded with a reduced amplitude , but with the purity approaching unity .
remarkably , for the probabilistic upload the squeezing post - correction can be omitted if the light - atom coupling is sufficiently weak . to increase the success rate of the single - photon upload , the pre - squeezing of the state of light before the upload can be used .
the superposition of coherent states can be uploaded in the same way , but the pre - squeezing plays here more important role , it effectively enhances the amplitude of the coherent states in the uploaded superposition , resulting in more distinguishable interference of the coherent states . because all these proposals are within the existing experiment set - up , they can be directly implemented if the suitable non - classical sources are available .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it is wellknown that for relatively short polymer chains the standard rouse model can describe the dynamics of a melt reasonably well @xcite . on the contrary , for chain length @xmath3 exceeding a critical length , the entanglement length @xmath4 , the behavior is usually described by the reptation model @xcite . here we restrict ourselves to chain lengths @xmath5 , i.e. the entangled polymer dynamics will be beyond of our consideration . the reason why in a dense melt the rouse model provides so well dynamical description for short chains is connected with a screening of the long - range hydrodynamic as well as the excluded volume interactions . as a result the fluctuations of the chain variables are gaussian . but there are further essential questions : how does the bare monomeric friction coefficient @xmath6 and the entropic elastic modulus @xmath7 ( which are simple input parameters of the standard rouse model ) change due to the interactions of the test chain and the surrounding matix ? why does such a simple model work so well for describing short chain melts ? obviously , the corresponding answers can not be given by the rouse model , which describes only the dynamics of connected gaussian springs without further interactions . on the other hand , at relatively low temperatures close to the glass transition of the surrounding matrix the deviations from the standard rouse behavior will be definitely more pronounced . for example , monte carlo ( mc ) studies of the bond fluctuation model at low temperatures ( but still above the temperature region where possibly the glass transition mode coupling theory @xcite applies ) show that the rouse modes remain well - defined eigenmodes of the polymer chains and the chains retain their gaussian properties @xcite . nevertheless , the relaxation of the rouse modes displays a stretched exponential behavior rather than a pure exponential . it could even be expected that at temperatures below the glass transition temperature of the matrix @xmath0 the rouse modes are frozen out . in these temperature regimes the interactions between monomers take a significant role and determine the physical picture of the dynamics as will be shown below . the generalized rouse equation ( gre ) , which can be used for the investigation of the problems mentioned above , has been derived by using projection formalism methods and mode coupling approximations ( mca ) @xcite . as a result of projection operator formalism the time evolution of the test chain is expressed in terms of a frequency matrix , which is local in time , and a memory function contribution due to the inter - chain forces exerted on the test chain segments . with the assumption that the frequency matrix term has the same form as in the standard rouse model ( linear elasticity with the entropic modulus @xmath8 ) all influence of the matrix chains reduce to the memory function contribution @xcite . the projection operator methods appears to be exact but rather formal , and to derive explicit results further approximations have to be made , which can be hardly controlled often . therefore it is instructive to use another alternative theoretical method to derive the gre . recently , a non - pertubative variational method which is equivalent to a selfconsistent hartree approximation was used for the investigation of the dynamics of manifolds @xcite and sine - gordon model @xcite in a random media . as a starting point the authors employed the standard martin - siggia - rose ( msr ) functional integral technique @xcite . here we follow this approach to derive a gre and study the dynamics of a test polymer chain in a glass forming matrix . + the paper is organized as follows . in section 2 , we give a general msr - functional integral formulation for a test chain in a polymer ( or non - polymer ) matrix . under the assumption that the fluctuations of the test chain are gaussian the hartree - type approximation is applied and a gre is finally derived . the case when the fluctuation dissipation theorem ( fdt ) and the time homogenity are violated is also shortly considered . in section 3 on the basis of the gre some static and dynamical properties of the test chain are discussed . in particular the theory of the test chain ergodicity breaking ( freezing ) in a glassy matrix is formulated . section 4 gives some summary and general discussion . the appendices are devoted to some technical details of the hartree - type approximation . let us consider a polymer test chain with configurations characterized by the vector function @xmath9 with s numerating the segments of the chain , @xmath10 , and time @xmath11 . the test polymer chain moves in the melt of the other polymers ( matrix ) which positions in space are specified by the vector functions @xmath12 , where the index @xmath13 numerates the different chains of the matrix . the test chain is expected to have gaussian statistics due to the screening of the self - interactions in a melt @xcite . we consider the simultaneous dynamical evolution of the @xmath9 and @xmath12 variables assuming that the interaction between matrix and test chain is weak . + the langevin equations for the full set of variables @xmath14 has the form @xmath15 @xmath16 where @xmath17 denotes the bare friction coefficient , @xmath18 the bare elastic modulus with the length of a kuhn segment denoted by @xmath19 , @xmath20 and @xmath21 are the interaction energies of test chain - matrix and matrix - matrix respectively , and @xmath22 , @xmath23 are the random forces with the correlator @xmath24 after using the standard msr - functional integral representation @xcite for the system ( [ r]-[f ] ) , the generating functional ( gf ) takes the form @xmath25-a_1\left[{\bf r}^{(p)}(s , t),{\bf { \hat r}}^{(p)}(s , t)\right]\nonumber\\&+&\sum_{p=1}^m\int_0^n ds\int_0^n ds'\int dt\:i{\hat r}_j(s , t)\frac{\delta}{\delta r_j(s , t)}v\left[{\bf r}(s , t)-{\bf r}^{(p)}(s',t)\right]\nonumber\\&+&\sum_{p=1}^m\int_0^n ds\int_0^n ds'\int dt\:i{\hat r}^{(p)}_j(s',t)\frac{\delta}{\delta r_j^{(p)}(s',t)}v\left[{\bf r}^{(p)}(s',t)-{\bf r}(s , t)\right]\bigg\}\nonumber\end{aligned}\ ] ] where the dots represents some source fields which will be specified later and einstein s summation convention for repeated indices is used . in gf ( [ gf ] ) the msr - action of the free test chain is given by @xmath26&=&-\int_0^n ds\int dt\bigg\{i{\hat r}_j(s , t)\left[\xi_0\frac{\partial}{\partial t}r_j(s , t)-\varepsilon\frac{\partial^2}{\partial s^2}r_j(s , t)\right]\nonumber\\&+&t\xi_0\left[i{\bf { \hat r}}(s , t)\right]^2\bigg\}\qquad.\end{aligned}\ ] ] as we will realize later the explicit form of the full action of the medium @xmath27 $ ] plays no role . in principle it could have any form and in particular , for a polymer matrix , the following one @xmath28=-\sum_{p=1}^m\int_0^n ds\int dt\:i{\hat r}_j^{(p)}(s , t)\left[\xi_0\frac{\partial}{\partial t}r_j^{(p)}(s , t)-\varepsilon\frac{\partial^2}{\partial s^2}r_j^{(p)}(s , t)\right]\nonumber\\&-&\sum_{p=1}^m\int_0^n ds\int dt\:i{\hat r}_j^{(p)}(s , t)\frac{\delta}{\delta r_j^{(p)}(s , t)}\sum_{m=1}^m\int ds'{\tilde v}\left[{\bf { \hat r}}^{(p)}(s , t)-{\bf { \hat r}}^{(m)}(s',t)\right]\nonumber\\&+&\sum_{p=1}^m\int_0^n ds\int dt\:t\xi_0\left[i{\hat r}_j(s , t)\right]^2\end{aligned}\ ] ] in order to obtain an equation of motion for the test chain one should integrate over the matrix variables @xmath12 first . for this end it is reasonable to represent gf ( [ gf ] ) as @xmath29-a_{0}\left[{\bf r}(s , t),{\bf { \hat r}}(s , t)\right]\right\}\label{gf1}\end{aligned}\ ] ] where the influence functional @xmath30 is given by @xmath31=-\ln\int\prod_{p=1}^{m}d{\bf r}^{(p)}(s , t)d{\bf { \hat r}}^{(p)}(s , t)\times\label{inf}\\&\times&\exp\bigg\{-a_{1}\left[{\bf r}^{(p)},{\bf { \hat r}}^{(p)}\right]\nonumber\\ & + & \sum_{p=1}^m\int_0^n ds\int_0^n ds'\int dt\:i{\hat r}_{j}(s , t)\frac{\delta}{\delta r_{j}(s , t)}v\left[{\bf r}(s , t)-{\bf r}^{(p)}(s',t)\right]\nonumber\\&+&\sum_{p=1}^m\int_0^n ds\int_0^n ds'\int dt\:i{\hat r}_{j}^{(p)}(s',t)\frac{\delta}{\delta r_{j}(s',t)}v\left[{\bf r}^{(p)}(s',t)-{\bf r}(s , t)\right]\bigg\}\nonumber.\end{aligned}\ ] ] in the spirit of the mode coupling approximation ( mca ) @xcite the force between the test chain and the matrix should be expressed as a bilinear product of the two subsystems densities . in order to assure this we expand the influence functional ( [ inf ] with respect to the forces @xmath32 between the test chain and the matrix up to the second order . this leads to @xmath33}&=-\ln\int\prod_{p=1}^{m}d{\bf r}^{(p)}(s , t)d{\bf { \hat r}}^{(p)}(s , t)\bigg\{\exp\left\{-a_{1}\left[{\bf r}^{p},{\bf { \hat r}}^{p}\right]\right\}\nonumber\\&+&\frac{1}{2!}\int d^{3}rd^{3}r'\int ds\int ds'\int dt\:i{\hat r}_{j}(s , t)\frac{\delta}{\delta r_{j}(s , t)}v\left[{\bf r}(s , t)-{\bf r}\right]\nonumber\\&\:&\qquad\times\int dt'\:i{\hat r}_{l}(s',t')\frac{\delta}{\delta r_{l}(s',t')}v\left[{\bf r}(s',t')-{\bf r}'\right]\left<\rho({\bf r},t)\rho({\bf r}',t')\right>_{1}\nonumber\\&+&\frac{1}{2!}\int d^{3}rd^{3}r'\int ds\int ds'\int dt\int dt'v\left[{\bf r}-{\bf r}(s , t)\right]v\left[{\bf r}'-{\bf r}(s',t')\right]\nonumber\\&\;&\qquad\times\nabla_{l}\nabla_{j}'\left<\pi_{l}({\bf r},t)\pi_{j}({\bf r}',t')\right>_{1}\nonumber\\&-&\frac{1}{2!}\int d^{3}rd^{3}r'\int ds\int ds'\int dt\int_{-\infty}^{t } dt'\:i{\hat r}_{j}(s , t)\frac{\delta}{\delta r_{j}(s , t)}v\left[{\bf r}(s , t)-{\bf r}\right]\nonumber\\ & \:&\qquad\times v\left[{\bf r}'-{\bf r}(s',t')\right]\nabla_{l}'\left<\rho({\bf r},t)\pi_{l}({\bf r}',t')\right>_{1}\nonumber\\&+&(t\leftrightarrow t')+{\cal o}(f^{3})\bigg\}\label{einf}\end{aligned}\ ] ] where the matrix density @xmath34 and the response field density @xmath35 were introduced and @xmath36 denotes cumulant averaging over the full msr - action @xmath37 $ ] of the matrix . in eq . ( [ einf ] ) the term @xmath38 is the same like the previous one but with permutated time arguments . the terms which are linear with respect to @xmath39 vanishes because of the homogenity of the system . in the appendix a we show that because of causality the correlator @xmath40 equals zero @xcite . taking this into account and performing the spatial fourier transformation the expression for gf ( [ gf1 ] ) takes the form @xmath41\nonumber\\&+&\frac{1}{2}\int ds\:ds'\int dt\:dt'\:i{\hat r}_{j}(s , t)\int\frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{l}\left|v(k)\right|^{2}s({\bf k},t - t')\nonumber\\&\:&\qquad\times\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}i{\hat r}_{l}(s',t')\nonumber\\&+&\int ds\:ds'\int dt\:dt'\:i{\hat r}_{j}(s , t)\int\frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{l}\left|v(k)\right|^{2}p_{l}({\bf k},t - t')\nonumber\\&\:&\qquad\times\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}\bigg\}\label{gf2}\end{aligned}\ ] ] where the correlation function @xmath42 and the response function @xmath43 of the matrix are naturally defined . going beyond the lrt - approximation would bring us multi - point correlation and response functions . + we should stress that in contrast to the matrix with a quenched disorder which was considered in @xcite in our case the matrix has its own intrinsic dynamical evolution which is considered as given . for example , for the glass forming matrix , which is our prime interest here , the correlation and response functions are assumed to be governed by the gtze mode - coupling equations @xcite . the hartree approximation ( which is actually equivalent to the feynman variational principle ) was recently used for the replica field theory of random manifolds @xcite as well as for the dynamics of manifolds @xcite and sine - gordon model @xcite in a random media . in the hartree approximation the real msr - action is replaced by a gaussian action in such a way that all terms which include more than two fields @xmath44 or / and @xmath45 are written in all possible ways as products of pairs of @xmath44 or / and @xmath45 , coupled to selfconsistent averages of the remaining fields . as a result the hartree - action is a gaussian functional with coefficients , which could be represented in terms of correlation and response functions . after these straightforward calculations ( details can be found in the appendix b ) the gf ( [ gf2 ] ) takes the form @xmath46\nonumber\\&+&\int_{0}^{n}ds\:ds'\int_{-\infty}^{\infty } dt\int_{-\infty}^{t}dt'\:i{\hat r}_{j}(s , t)r_{j}(s',t')\lambda(s , s';t , t')\nonumber\\&-&\int_{0}^{n}ds\:ds'\int_{-\infty}^{\infty } dt\:i{\hat r}_{j}(s , t)r_{j}(s , t)\int_{-\infty}^{t}dt'\lambda(s , s';t , t')\nonumber\\&+&\frac{1}{2}\int_{0}^{n}ds\:ds'\int_{-\infty}^{\infty } dt\int_{-\infty}^{t}dt'\:i{\hat r}_{j}(s , t)i{\hat r}_{j}(s',t')\chi(s , s';t , t')\bigg\}\label{hartree}\end{aligned}\ ] ] where @xmath47 and @xmath48 in eq . ( [ lambda],[chi ] ) the response function @xmath49 the density correlator @xmath50 with @xmath51 and the longitudinal part of the matrix response function @xmath52 are defined . the pointed brackets denote the selfconsistent averaging with the hartree - type gf ( [ hartree ] ) . + up to now we considered the general off - equilibrium dynamics with the only restriction of causality @xcite . we now assume that for very large time moments @xmath11 and @xmath53 , where the difference @xmath54 is finite so that @xmath55 , time homogenity and the fluctuation - dissipation theorem ( fdt ) holds . this implies @xmath56 where @xmath57 . by using this in eq . ( [ hartree ] ) and after integration by parts in the integrals over @xmath53 the gf in hartree approximation takes the form @xmath58\frac{\partial}{\partial t'}r_{j}(s',t')\nonumber\\ & -&\int_{0}^{n}ds\:ds'\int_{-\infty}^{\infty}dt\:i{\hat r}_{j}(s , t)\bigg[\varepsilon\delta(s - s')\frac{\partial^{2}}{\partial s^{2}}+\beta\int\frac{d^{3}k}{(2\pi)^{3}}k^{2}|v(k)|^{2}s_{st}({\bf k})\nonumber\\ & \:&\:\times\left[f_{st}({\bf k};s , s')-\delta(s - s')\int_{0}^{n}ds^{''}f_{st}({\bf k};s , s^{''})\right]\bigg]r_{j}(s',t)\nonumber\\&+&t\int_{0}^{n}ds\:ds'\int_{-\infty}^{\infty } dtdt'\bigg[\xi_{0}\delta(t - t')\delta(s - s')+\theta(t - t')\beta\nonumber\\&\:&\:\int\frac{d^{3}k}{(2\pi)^{3}}k^{2}|v(k)|^{2}f({\bf k};s , s';t - t')s({\bf k};t - t')\bigg]i{\hat r}_{j}(s , t)i{\hat r}_{j}(s',t')\label{gfh}\end{aligned}\ ] ] where the subscript @xmath59 indicates the static correlation functions . this generating functional immediately leads to the following generalized rouse equation ( gre ) @xmath60 where the memory function @xmath61 and the effective elastic susceptibility @xmath62\label{omega}\end{aligned}\ ] ] are defined . the correlation function of the random force @xmath63 is given by @xmath64\label{force}\ ] ] as a result we have obtained basically the same gre as in the papers @xcite but with one additional elastic term . this term ( see the 2-nd term in eq . ( [ omega ] ) ) is mainly inversely proportional to the temperature and is , in contrast to the first term , of an energetic nature . the two factors of @xmath65 quantify the forces exerted by a pair of surrounding segments on the test chain segments @xmath66 and @xmath67 , whereas the @xmath68 and @xmath69 factors quantify the static correlations between the segments of surrounding and test chain segments , respectively . in @xcite only the entropic elastic part was taken into account . the memory function ( [ gamma ] ) has the same form as in @xcite and the relationship ( [ force ] ) is assured as soon as the fdt ( [ fdt1 ] ) and ( [ fdt ] ) is fullfilled . in this subsection we give gre s for the more general case when the time homogenity ( stationarity ) and the fdt do not hold @xcite . + by employing the standard way @xcite one can derive two coupled equations of motion for correlators @xmath70 and response functions @xmath71 @xmath72g(s , s';t , t')\nonumber\\&+&\int_{0}^{n}ds^{''}\int_{-t'}^{t}dt^{''}\lambda(s , s^{''};t , t^{''})g(s^{''},s';t^{''},t')=\delta(s - s')\delta(t - t')\label{cor}\end{aligned}\ ] ] @xmath72c(s , s';t , t')\nonumber\\&+&\int_{0}^{n}ds^{''}\int_{-\infty}^{t}dt^{''}\lambda(s , s^{''};t , t^{''})c(s^{''},s';t^{''},t')\nonumber\\ & + & \int_{0}^{n}ds^{''}\int_{-\infty}^{t}dt^{''}\chi(s , s^{''};t , t^{''})g(s',s^{''};t',t^{''})=2t\xi_{0}g(s',s;t',t)\label{response}\end{aligned}\ ] ] with the initial conditions @xmath73 and @xmath74 in the stationary case all correlators and response functions in eq . ( [ cor]-[ini2 ] ) only depend from the differences of time moments , @xmath54 . if we assume again that fdt ( [ fdt1 ] ) and ( [ fdt ] ) holds , then from eq . ( [ response ] ) after performing the integrations by parts ( in the integrals over @xmath75 ) one arrive at the gre for @xmath76 @xmath77 of course , eq . ( [ gcor ] ) could be obtained immediately from eq . ( [ gre ] ) by multiplying both sides with @xmath78 , averaging and taking into account that because of causality @xmath79 at @xmath76 . we will use the gre eq.([gcor ] ) , where the functions @xmath80 and @xmath81 are given by eqs . ( [ gamma ] , [ omega ] ) , in the next section for the investigation of the test chain ergodicity breaking ( freezing ) . the new features of the gre ( [ gcor ] ) relative to the standard rouse equation are that it contains the integral convolution with respect to the @xmath82variable in the frictional term as well as in the elastic term . the frictional term is also non - local in time . all these things together should change the statical and dynamical behaviour of the gaussian test chain in comparison to the ideal chain . + we also should stress that the gre is substantially nonlinear because the memory function ( [ gamma ] ) depends from the test chain correlator @xmath83 in such a way that a positive feedback obviously exists . that is the reason why one could expect that eq . ( [ gcor ] ) shows an ergodicity breaking in the spirit of gtze s glass transition theory @xcite . + as usual it is convenient to introduce the standard rouse mode variables @xcite : @xmath84 with the inverse transformation @xmath85 in general one also needs a 2-dimensional rouse transformation @xmath86 where functions like @xmath87 should be treated like @xmath88-matrices . for example the density correlator ( [ dc ] ) should be considered as an exponential function from a @xmath89-matrix @xmath90 and the series expansion holds : @xmath91 we also assume that matrices in the rouse mode representation are nearby diagonal @xmath92 for any @xmath2 and @xmath93 not equal zero @xcite . + then as a result of rouse mode transformation the gre for the rouse mode time correlation function , @xmath94 , takes the form ( for @xmath95 ) @xmath96 where @xmath97 - 1\right\}s({\bf k},t)\label{gammap}\ ] ] and @xmath98\label{omegap}\ ] ] for @xmath99 the gre describes the dynamics of the centre of mass @xmath100 and has the following form @xmath101 with @xmath102 and @xmath103 as a result all rouse mode variables relax independently . the conclusion that the rouse modes are still `` good eigenmodes '' even in the melt is supported by monte - carlo @xcite and molecular - dynamic @xcite simulations . + for cases where the assumption of diagonality [ 38,39,[diag ] ] can not be justified , the rouse modes do not decouple and one have to go back to eq . ( [ gcor ] ) . in the rouse mode representation it reads as @xmath104 as we have already discussed in sec.ii.b the interaction with the surrounding segments renormalizes the elastic properties of the rouse chain so that the test chains elastic susceptibility is given by eq . ( [ omegap ] ) . the additional elastic term in gre leads to the renormalized static normal modes correlator @xmath105 explicit evaluation of the @xmath106 can be done if we use for the static correlator @xmath107 the standard rouse expression @xmath108 then the calculation yields for the two limiting cases @xmath109+{\cal o}(p^{4 } ) & \frac{p\pi}{n}\ll 1\quad ( 50a ) \\ \left(\frac{p\pi}{n}\right)^{2}\varepsilon+\frac{\beta}{4\pi^{2}}\int_{0}^{l^{-1}}dk\:k^{4}|v(k)|^{2}s_{st}(k)\left(\frac{6}{l^{2}k^{2}}\right)\left(1-e^{-k^{2}r_{g}^{2}}\right ) & \frac{p\pi}{n}\simeq 1\quad\ : ( 50b)\end{array}\right.\nonumber\end{aligned}\ ] ] where @xmath110,\qquad r_{g}=\frac{nl^{2}}{6}\ ] ] and we have chosen @xmath111 as a cutting parameter . it is evident from the previous eqs . ( 50@xmath112 , 50@xmath113 ) that at small @xmath2 * the elastic modulus gains an energetic component which , in contrast to the entropic part @xmath7 , increase with the cooling of the system , * initially absolutely flexible chains acquires a stiffness because of terms of order @xmath114 and higher . at large @xmath2 the elastic behaviour reduces to the standard rouse one , as it is expected . in fig.1 is shown the result of a numerical calculation of the static correlator ( [ cst ] ) . the fourier component of the potential is taken , as it is customary e.g. in the theory of neutron scattering @xcite , in the form of a pseudo potential approximation , @xmath115 , where @xmath116 and @xmath117 have dimensions of molecular energy and distance , respectively . the static structure factor @xmath68 is chosen in the form of the percus - yevick s simple liquid model @xcite . one can see that for @xmath118 the small rouse mode index limit ( 50@xmath112 ) starts at @xmath119 whereas the opposit limit ( 50@xmath113 ) is fullfilled at @xmath120 . because the correlator @xmath121 depends mainly from @xmath122 , for relatively short test chains the high mode index limit ( 50@xmath113 ) is shifted into the window of calculations ( see fig.1 for n=20 ) . + at least qualitatively this deviation from the standard rouse behaviour have been seen by kremer and grest in their md - simulations ( see fig.3 in @xcite ) . first we consider the case @xmath95 . in the nonergodic state the rouse mode correlation functions can be represented as @xmath123 where the non - ergodicity parameter @xmath124 was introduced and @xmath125 . for the correlation function of the glassy matrix we can use the standard result of the glass transition theory @xcite @xmath126 where the proximity parameter @xmath127 is defined and @xmath0 is the temperature of the matrix ergodicity breaking ( gtze temperature ) . in eq . ( [ gg ] ) @xmath128 is the non - ergodicity parameter of the matrix , @xmath129 is the characteristic time scale , @xmath112 is the characteristic exponent , @xmath130 and @xmath131 is some amplitude . + in order to derive the equation for @xmath132 let us take the limit @xmath133 in eq . ( [ grer ] ) keeping in mind the definitions ( [ psi ] ) and ( [ gg ] ) . very close to the test chain ergodicity breaking temperature @xmath1 , @xmath132 goes to zero ( a - type transition @xcite ) and we can expand the exponential function in eq . ( [ gammap ] ) up to the first order with respect to @xmath132 . the solution of the resulting equation has the simple form @xmath134 the critical temperature @xmath1 is determined by the equation @xmath135 the numerical solution of eq . ( [ tc ] ) is given in fig.2 . it is obviously that if the entropic part of @xmath106 dominates , the critical temperature is given by @xmath136 fig.2 really shows that this law ( [ pq ] ) is well satisfied due to the fact that the critical temperatures @xmath1 are quite high . but for low temperatures the energetic contribution in @xmath106 is enhanced which leads to a deviation from this simple @xmath137-dependence . + now we consider the case for @xmath99 . the equation ( [ c.m ] ) for the velocity of the center of mass @xmath138 leads to the equation for the velocity correlator @xmath139 where @xmath140 because of causality the correlator on the r.h.s . ( [ c.m.c ] ) has the form @xmath141 where , as it comes from eq . ( [ c.m ] ) @xmath142 taking into account the definition of @xmath143 and eq . ( [ force ] ) this yields to the correlator @xmath144 because of the causality property ( 61 ) only the @xmath145-functional term on the r.h.s . ( [ force ] ) contributes to the correlator ( [ lk ] ) . therefore the resulting equation for the self - diffusion coefficient @xmath146 takes the form @xmath147}\label{d}\ ] ] which was obtained before in @xcite . + one can calculate the second term in the denominator of eq . ( [ d ] ) selfconsistently . because now the relevant times @xmath148 the approximation @xmath149 could be used in eq . ( [ c.m ] ) . then the density correlator ( [ 47 ] ) is given by @xmath150 with the use of eqs . ( [ de]),([gg ] ) and eq . ( [ gammat ] ) in the limit @xmath151 eq . ( [ d ] ) becomes @xmath152}\label{del}\ ] ] where the denominator is given by static properties only . similar statements have been suggested already in @xcite the solution of eq . ( [ del ] ) has the simple form @xmath153 where @xmath154 finally , the temperature of the ergodicity breaking ( localization ) for the mode @xmath99 of the test chain is @xmath155 fig.3 shows the results of numerical calculations of @xmath156 and @xmath157 as functions of @xmath3 . one can see that in the reasonable range of parameters @xmath158 . as a result one can say that on cooling of a test chain in a glassy matrix the mode @xmath99 is the first to be freezed . on the subsequent cooling the modes @xmath159 are freezed successively , @xmath160 it is apparent that the system studied here is a nontrivial polymeric generalization of the model introduced by sjgren @xcite . this model was used for the investigation of the @xmath161-peak in the spectrum of glass forming systems @xcite . in this paper we have derived a gre for a test polymer chain in a polymer ( or non - polymer ) matrix which has its own intrinsic dynamics , e.g. the glassy dynamics @xcite . we have used here the msr - functional integral technique which could be considered as an alternative to the projection operator formalism @xcite . one of the difficulties in this formalism is the necessity of dealing with the projected dynamic , which is difficult to handle with explicitly . on the contrary in msr - technique the dynamic of slow variables is well defined and several approximations which one have to employ could be justified . in the interaction of the test chain with the surrounding matrix only two - point correlation and response functions are involved . in terms of mca @xcite this obviously corresponds to the projection of the generalized forces only onto the bilinear variables : product of test chain density and matrix density . to handle with the action in the gf of the test chain we used the hartree - type approximation ( i.e. , equivalent to the feynman variational principle ) @xcite , which is reasonable when the fluctuations of the test chain are gaussian . in the case of a polymer melt ( high densitiy ) this is indeed the case due to the screening effects for the excluded volume @xcite . the use of the hartree - type makes the problem that we deal with analytically amenable and results in the gre s for the case when the fdt holds as well as for the case when fdt does not hold . in this paper we have restricted ourselves to the first case and have shown that the interaction with the matrix renormalizes not only the friction coefficient ( which makes the chain non - markovian ) but also the elastic modulus ( which changes the static correlator ) . the form of the static correlator for the rouse mode variables is qualitatively supported by md - simulations @xcite . as regards the dynamical behaviour , we have shown that the test chain in a glassy matrix ( with the matrix glass transition temperature @xmath0 ) undergoes the ergodicity breaking transition at a temperature @xmath162 . the critical temperature @xmath1 could be parametrized with the rouse mode index @xmath2 and is a decreasing function of @xmath2 . we have considered only the a - type transition which is assured by the bilinear term in the expansion of eq . ( [ gammap ] ) . it seems reasonable that keeping the whole exponential function in eq . ( [ gammap ] ) might lead to a b - type transition also . the results also essentially would change if the off - diagonal elements in the matrix ( [ 2rt ] ) can not be neglected ( see eq . ( [ grerq ] ) ) . in this case only one ideal transition temperature @xmath163 would be possible . the general theory of a a - type transition was discussed in @xcite . this picture of freezing here should not be mixed with a different one , the underlying glass transition by itself ( e.g. the glass transition of the matrix at @xmath164 ) . according to the present view of this phenomenon @xcite , the spontaneous arrest of the density fluctuations is driven by those of the microscopic lengthscale @xmath165 , where @xmath165 is the wave vector which corresponds to the structure factor s main maximum . the freezing of these fluctuations then arrests the others through the mode coupling . two of us gratefully acknowledge support from the deutsche forschungsgemeinschaft through the sonderforschungsbereich 262 ( v.g.rostiashvili ) and the bundesministerium fr bildung und forschung ( m.rehkopf ) for financial support . we also greatly acknowledge helpful discussions with j.baschnagel , k.binder , k.kremer , w.gtze and r.schilling . it is more convenient to handle with the spacial fourier transformation of this correlator @xmath166\right\}\right>_{1}\nonumber\\ & = & \sum_{a , b=0}^{\infty}\frac{1}{a!b!}\sum_{p , m=1}^{m}\int ds ds'\left < i{\hat r}_{l}^{(p)}(s , t)i{\hat r}_{j}^{(m)}(s',0)[i{\bf k}{\bf r}^{(p)}(s , t)]^{a}[-i{\bf k}{\bf r}^{(m)}(s',0)]^{b}\right>_{1}\nonumber\end{aligned}\ ] ] such multi - point cumulant response functions ( mrf ) were considered in @xcite . the causality condition for these functions asserts that the time argument of at least one @xmath167-variable should be the latest one , otherwise this mrf equals zero . because of the same reason self - loops of response functions vanish @xcite . mrf s which consists only of @xmath168-variables also vanish . + in the case ( [ ap ] ) all time arguments of the @xmath167-variables are equal to the corresponding time arguments of @xmath168-variables and as a result the mrf in eq . ( [ ap ] ) vanishes . in order to calculate the bilinear hartree action , we follow the way mentioned in sec.ii.b . with these strategy in mind the 2-nd term in the exponent ( [ gf2 ] ) is evaluated as @xmath169 where @xmath170\right\}i{\hat r}_{l}(s',t')\right>\nonumber\\ i_{2}(s , s';t , t')&\equiv&\left<\frac{\delta}{\delta r_{j}(s , t)}\int \frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{l}|v(k)|^{2}\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}i{\hat r}_{l}(s , t)\right>\nonumber\\ i_{3}(s , s';t , t')&\equiv&\left<\frac{\delta}{\delta r_{j}(s , t)}\int \frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{l}|v(k)|^{2}\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}i{\hat r}_{l}(s',t')\right>\nonumber\\ i_{4}(s , s';t , t')&\equiv&\left<\frac{\delta}{\delta r_{j}(s',t')}\int \frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{l}|v(k)|^{2}\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}i{\hat r}_{l}(s , t)\right>\nonumber\\ i_{5}(s , s';t , t')&\equiv&\left<\int\frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{j}|v(k)|^{2}\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}\right>\nonumber\\ i_{6}(s , s';t , t')&\equiv&\bigg < i{\hat r}_{j}(s , t)i{\hat r}_{l}(s , t)\frac{\delta^{2}}{\delta r_{n}(s , t)\delta r_{n}(s',t')}\nonumber\\ & \:&\times\int \frac{d^{3}k}{(2\pi)^{3}}k_{j}k_{l}|v(k)|^{2}\exp\left\{i{\bf k}\left[{\bf r}(s , t)-{\bf r}(s',t')\right]\right\}\bigg>.\label{il}\end{aligned}\ ] ] the pointed brackets in eq . ( [ il ] ) represent the selfconsistent averaging with the gaussian hartree action . taking this into account and using the generalized wick theorem @xcite , after straightforward algebra , we have @xmath171 where the last equation comes from the fact that the response function @xmath172 and @xmath173 . + the 3-rd term in the exponent eq . ( [ gf2 ] ) can be handled in the same way . the response function for the isotropic matrix has the form @xmath174 where @xmath175 is the longitudinal part of the matrix response function . then the hartree approximation of the 3-rd term in the exponent ( [ gf2 ] ) takes the form @xmath176 where @xmath177 taking into account eq . ( [ ap2 ] ) with eq . ( [ il ] ) and eq . ( [ j ] ) leads to the hartree - type approximation ( [ hartree ] ) . 99 m.doi and s.f.edwards , _ the theory of polymer dynamics _ , clarendon press , oxford , 1986 j.d.ferry , _ viscoelastic properties of polymers _ , wiley , n.y . , 1980 w.gtze in , _ liquids , freezing and glass transition _ , ed . by j.p.hansen , d.levesque and j.zinn-justin , amsterdam , north - holland , 1991 k.okun , m.wolfgardt , j.baschnagel and k.binder , preprint w.hess , macromolecules , 21 , 2620 ( 1988 ) k.s.schweizer , j.chem.phys . , 91 , 5802 ( 1989 ) k.s.schweizer , j.chem.phys . , 91 , 5822 ( 1989 ) h.kinzelbach and h.horner , j.phys.(france ) , 1,3 , 1329 ( 1993 ) d.cule and y.shapir , phys.rev . e 53 , 1553 ( 1996 ) c.de dominicis and l.peliti , phys.rev . b 18 , 353 ( 1978 ) r.bausch , h.k.janssen and h.wagner , z.phys . b 24,113 ( 1976 ) v.g.rostiashvili and r.schilling , z.phys . b ( 1996 ) m.mezard and g.parisi , j.phys . a 29 , 6515 ( 1996 ) 3-rd edition , clarendon press , oxford , 1996 , sec . 4.2 . l.f.cugliandolo and j. kurchan , phys.rev.lett . , 71 , 173 ( 1993 ) k.kremer and g.s.grest , j.chem.phys . 92 , 5057 ( 1990 ) s.w.lovesey , _ theory of neutron scattering _ , clarendon press , oxford , 1984 j .- p.hansen and i.r.mcdonald , _ theory of simple liquids _ , academic press , london , 1976 s.f . edwards , t.a . vilgis , physica scripta , * t13 * , 7 , ( 1986 ) t.a . vilgis in `` disorder effects on relaxational processes '' ed . a. blumen , r. richert , springer verlag , heidelberg , 1994 l. sjgren , phys.rev . a 33 , 1254 ( 1986 ) w. gtze and l. sjgren , j.phys . condensed matter 1 , 4183 ( 1989 ) j.zinn-justin , _ quantum field theory and critical phenomena _ , 3-rd edition , clarendon press , oxford , 1996 , sec . t.franosch and w.gtze , j.phys . ( cond.mat . ) 6 , 4807 ( 1994 )
|
we consider the langevin dynamics of a gaussian test polymer chain coupled with a surrounding matrix which can undergo the glass transition . the martin - siggia - rose generating functional method and the nonpertubative hartree approximation
are used to derive the generalized rouse equation for the test chain .
it is shown that the interaction of the test chain with the surrounding matrix renormalizes the bare friction and the spring constants of the test chain in such a way that the memory function as well as the bending dependent elastic modulus appear .
we find that below the glass transition temperature @xmath0 of the matrix the rouse modes of the test chain can be frozen and moreover the freezing temperatures ( or the ergodicity - nonergodicity transition temperature ) @xmath1 depends from the rouse mode index @xmath2 .
-15 mm classification : physics abstracts , 05.20 - 36.20
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the long - standing quest for a solid state realization of the kitaev honeycomb model@xcite has triggered much of the experimental and theoretical interest in 4d and 5d compounds with two- and three - dimensional tri - coordinated lattices , in which the interplay of the strong spin - orbit coupling ( soc ) and electronic correlations leads to the dominance of the strongly anisotropic kitaev - like interactions.@xcite a lot of experimental effort has been focused on iridium oxides belonging to the a@xmath0iro@xmath1 family@xcite and , more recently , to @xmath2rucl@xmath1.@xcite the kitaev honeycomb model belongs to the class of the compass models . it is intrinsically frustrated due to the bond - depended nature of the interactions . in the quantum case , this frustration leads to the appearance of the non - trivial quantum spin liquid ( qsl ) phase with fractionalized excitations , dubbed kitaev qsl.@xcite kitaev qsl is not a unique example of non - trivial ground states of the compass models , @xcite however , it is probably the only one which allows an exact analytic solution . in honeycomb iridates and ruthenates , the magnetic degree of freedom described by an effective magnetic moment @xmath5 , arises in the presence of strong soc from electrons occupying @xmath6-manifold of states of ir@xmath7 and ru@xmath8 ions . in a@xmath0iro@xmath1 compounds , edge - shared iro@xmath9 octahedra provide 90@xmath10 paths for the dominant nearest neighbor kitaev coupling between iridium magnetic moments . a similar situation takes place in the isostructural layered honeycomb material @xmath2rucl@xmath1 and three - dimensional harmonic honeycomb iridates , @xmath11li@xmath0iro@xmath1 and @xmath12li@xmath0iro@xmath1.@xcite it is believed that the sign of the kitaev interaction may be either antiferro- ( af ) or ferromagnetic ( fm ) depending on the compound.@xcite isotropic heisenberg couplings are also present in these compounds due to the octachedra edge sharing geometry and direct overlap of @xmath13 or @xmath14orbitals which , due to their extended nature , often reach beyond nearest neighbors . further anisotropies , such as the isotropic off - diagonal @xmath3 interactions , can also be present , mainly as a result of crystal field distortions . @xcite the competition between all these couplings leads to a rich variety of experimentally observed magnetic structures.@xcite here we discuss in detail the models and the mechanisms which lead to the stabilization of magnetic ordering in two compounds : na@xmath0iro@xmath1 and @xmath2rucl@xmath1 . several experiments have shown that the low - temperature phase of na@xmath0iro@xmath1 has collinear zigzag long - range magnetic order.@xcite in addition , recent diffuse magnetic x - ray scattering data have determined the spin orientation in this zigzag state and showed that it is along the 44.3@xmath15 direction with respect to the @xmath16 axis , which corresponds to approximately half way in between the cubic @xmath17 and @xmath18 axes.@xcite both of these findings are in disagreement with the original kitaev - heisenberg model,@xcite which predicts the zigzag phase only for the antiferromagnetic nearest neighbor kitaev interaction with the magnetic moments along the cubic axes , while the kitaev interaction in na@xmath0iro@xmath1 is ferromagnetic.@xcite this shows that one needs to extend the nearest neighbor model by including some additional interactions in order to explain these experimental observations . @xmath2rucl@xmath1 also shows collinear antiferromagnetic zigzag ground state.@xcite recent x - ray and neutron scattering diffraction data@xcite indicate that the best fit to the collinear structure is obtained for the antiferromagnetic nearest neighbor kitaev interaction and when the spin direction points 35@xmath15 out of the @xmath19-plane , i.e. along one of the cubic directions . this suggests that the microscopic origin of the zigzag ground state in @xmath2rucl@xmath1 might be quite different from the one in na@xmath0iro@xmath1 , and that it can be described by the nearest neighbor kitaev - heisenberg model.@xcite in both cases , the available experimental data provides an important check of the validity of any model proposed to describe the magnetic properties of na@xmath0iro@xmath1 and @xmath2rucl@xmath1 , as it should correctly predict not only the type of the magnetic order but also its orientation in space . in this work we consider two models , the nearest neighbor kitaev - heisenberg model@xcite and its more complicated counterpart , dubbed @xmath4 model,@xcite and study how the preferred directions of the mean field order parameter are selected in these models . the formal procedure which we will be using here is based on the derivation of the fluctuational part of the free energy by integrating out the leading thermal fluctuations , and by determining which orientations of the order parameter correspond to the free energy minimum . this approach is based on the hubbard - stratonovich transformation and was outlined in refs.@xcite to which we refer the reader for technical details . in both models , the thermal fluctuations select the cubic axes as the preferred directions for spins , which describes the experimental situation in @xmath2rucl@xmath1 but not in na@xmath0iro@xmath1 . we have also checked that in both models the quantum fluctuations ( taken into account either using the quantum version of hubbard - stratonovich approach or within the semiclassical spin - wave approach ) lift the accidental degeneracy of the classical solution and also select the cubic axes as the preferred directions for spins . we did not present these calculations here as they bring no new results compared with more simple analysis of thermal fluctuations . the important point which we stress in our paper is that the selection of correct `` diagonal '' direction of the spins observed in na@xmath0iro@xmath1 might happen already on the mean - field level by inclusion of small off - diagonal positive interaction @xmath3 as soon as it is larger than the energy gain of order @xmath20 due to the quantum fluctuations . this paper is organized as follows . in sec . ii we study the order by disorder mechanism of the selection of the direction of the order parameter in the nearest neighbor kitaev - heisenberg model on the honeycomb lattice . in sec . iii we extend our consideration to the @xmath4 model . in sec . iv , we discuss the role of the off - diagonal @xmath3-term and study the selection of the direction of the magnetic order in na@xmath0iro@xmath1 and in @xmath2rucl@xmath1 . we summarize our conclusions in sec . v. appendix a discusses in detail the degeneracy of the classical manifold of the kitaev - heisenberg model . appendices b and c contain some technical details . and @xmath21 are two primitive translations . the bond vectors are @xmath22 , @xmath23 and @xmath24 . , title="fig : " ] and @xmath21 are two primitive translations . the bond vectors are @xmath22 , @xmath23 and @xmath24 . , title="fig : " ] and @xmath21 are two primitive translations . the bond vectors are @xmath22 , @xmath23 and @xmath24 . , title="fig : " ] and @xmath21 are two primitive translations . the bond vectors are @xmath22 , @xmath23 and @xmath24 . , title="fig : " ] interaction in the four - sublattice transformation : @xmath3 picks up a negative sign on the solid bonds but keeps the sign from unrotated reference frame on the dashed bonds . ] the kitaev - heisenberg model on the honeycomb lattice reads @xcite @xmath25 where @xmath26 is the interaction between @xmath27-component of the pseudospin @xmath28 , on sublattices @xmath29 . hereafter , we call these pseudospins simply spins . @xmath30 and @xmath31 correspond to the heisenberg and kitaev interactions , which in the extended model can be both af and fm . @xmath32 denote the spin components in the global reference frame . the classical phase diagram of the model ( 1 ) contains four magnetic phases:@xcite the ferromagnetic phase ( fig . 1 ( a ) ) , the nel antiferromagnet ( fig . 1(b ) ) , the stripy antiferromagnet ( fig . 1 ( c ) ) and the zigzag antiferromagnet ( fig . 1 ( d ) ) . the latter two magnetic states have a four sublattice structure . all these phases have macroscopic classical degeneracy . while the classical degeneracy of the simple fm state and of the af nel state comes straightforwardly from the infinite number of degenerate collinear states , the macroscopic degeneracy of the af stripy and zigzag phases is more complex , and the degenerate ground state manifold consists of six collinear states and a set of non - collinear multi-@xmath33 states . in appendix a we discuss this question in detail and show that using the four - sublattice klein transformation for the stripy and the zigzag af states,@xcite the nature of the classical degeneracy of all four magnetically ordered states can be understood in a similar way . importantly , in all cases , the classical degeneracy is accidental and is removed by the order by disorder mechanism which selects a set of collinear states , each with a particular direction of the order parameter . following chaloupka _ et al_,@xcite we introduce four auxiliary sublattices a , b , c and d ( see fig.2 ) , fix the direction of the spins on the sublattice a and rotate the spins on the subllatices b , c , and d such that the component of spin corresponding to the bond direction ( @xmath17 for b , @xmath18 for c and @xmath34 for d ) stays the same but two other spin components change sign . this results in the transformed hamiltonian with the same form as ( [ kh ] ) but with transformed couplings . here we consider the kitaev - heisenberg model in the full parameter space . for the parameters of the model for which either stripy or zigzag are the ground states , we perform four - sublattice transformation and treat the model ( [ kh ] ) in the rotated basis , in which the stripy order maps to the fm and the zigzag order maps to the simple two - sublattice afm nel state . next , using a hubbard - stratonovich transformation of the partition function,@xcite we discuss how the preferred directions of the order parameter in all these phases are selected by thermal below the ordering temperature . the partition function of the system of classical spins is given by the integral over the boltzmann weights of the configurations @xmath35[d\mathbf{s}_{1,j}\right]\delta(\mathbf{s}_{0,i}^{2}-1)\delta(\mathbf{s}_{1,j}^{2}-1)\nonumber\\ & & \exp\left[-\beta\sum_{\langle ij \rangle_\alpha}\sum_{\gamma}j^{\alpha\gamma}s_{0,i}^{\gamma } s_{1,j}^{\gamma}\right],\end{aligned}\ ] ] where @xmath36 and @xmath37 are classical spins on sublattices @xmath38 and @xmath39 , and @xmath40 is the inverse temperature . similarly in the case of a quantum system the partition function is a trace of the boltzman weights over the spin operators , @xmath41 $ ] . + it is more convenient to perform the hubbard - stratonovich transformation by representing the hamiltonian matrix in the basis of the eigenfunctions of the exchange matrix , which can be easily obtained in the momentum space . to this end , we first introduce a six - component vector @xmath42 , with the components given by the fourier transforms of the @xmath43 components of the spins on @xmath44 and @xmath45sublattices , correspondingly . this allows us to write the hamiltonian matrix in the momentum space as @xmath46where the @xmath47 exchange matrix @xmath48 is defined as @xmath49 with matrix elements given by @xmath50 e^{\imath{\bm q}\cdot ( { \bm d } _ { \alpha } -{\bm d } _ { z } ) } = j_{{\bm q}}+k_{\bm q}^{\gamma}.\end{aligned}\ ] ] here we drop the overall phase factor @xmath51 and denote @xmath52 , @xmath53 , where @xmath54 and @xmath55 are the lattice vectors . the matrix @xmath48 is then diagonalized by a unitary transformation , @xmath56 , leading to the following form of the hamiltonian @xmath57where the normal amplitudes of spin - like variables are defined as @xmath58 note that , depending on the form of the interaction matrix , this transformation in general will mix the spin operators on different sites of the unit cell as well as different components of the spin . however , in the case of the kitaev - heisenberg model , while the two sublattices of the honeycomb lattice are mixed , the @xmath17 , @xmath18 , and @xmath34 components stay separate . the partition function ( [ z - clas ] ) then looks like : @xmath59[d\mathbf{s}_{1,j+{\bm d}_z}\right]\delta(\mathbf{s}_{0,j}^{2}-1)\delta(\mathbf{s}_{1,j+{\bm d}_z}^{2}-1)\nonumber\\ & & \exp\left[-\beta \sum_{\bm q,\nu } \kappa_{{\bm q},\nu } { \tilde s}^*_{{\bm q},\nu } { \tilde s}_{{\bm q},\nu}\right ] .\label{z - norm}\end{aligned}\ ] ] following the steps outlined in refs.@xcite , we can separate the mean - field and the fluctuational contributions to the partition function , @xmath60 . in the gaussian approximation , the fluctuation part of the partition function , @xmath61\exp\left [ -\beta{\mathcal{s}}_{\mathrm{fluct}}\right],\end{aligned}\ ] ] where @xmath62 can be obtained by integration over the fluctuation amplitudes @xmath63 . the explicit expression for the matrix elements of the fluctuation matrix @xmath64 computed for an orientation of the mean - field order parameter along arbitrary direction @xmath65 are given in appendix b. now , the fluctuation contribution to the free energy can be written as @xmath66 while the mean - field part of the free energy has the full rotational symmetry , its fluctuational part , @xmath67 , is sensitive to the direction of the mean - field order parameter . thus , by finding the minima of the fluctuational part of the free energy , we can pin the spontaneous magnetization along some preferred direction of the lattice . fig.3 ( a ) shows the angular dependence of fluctuational free energy @xmath68 computed for representative parameters @xmath69 mev and @xmath70 mev , at which the ground state order is the af zigzag . the magnitude of @xmath68 is presented as a color - coded plot on the unit sphere , where the minima and maxima of the free energy are shown by the deep blue and red colors , correspondingly . we see that the minima of @xmath68 are achieved when the magnetization is directed along one of the cubic axes . this finding shows that the contribution of the fluctuations to the free energy removes the degeneracy of the ground state found on the mean field level . the states which are selected by the thermal fluctuations are the collinear states with the order parameter pointing along one of the cubic axes , thus confirming previous results of the monte carlo simulations@xcite and spin wave analysis by chaloupka et al.@xcite mev and @xmath70 mev and ( b ) @xmath4 model computed with @xmath71 mev , @xmath72 mev , @xmath73 mev , @xmath74 mev , and @xmath75 mev . , title="fig : " ] mev and @xmath70 mev and ( b ) @xmath4 model computed with @xmath71 mev , @xmath72 mev , @xmath73 mev , @xmath74 mev , and @xmath75 mev . , title="fig : " ] we discuss the relevance of our findings for the nearest neighbor kitaev - heisenberg model for @xmath2rucl@xmath1 in sec . iv . however in the next section , we will first consider the selection of the direction of the order parameter in the extensions of the kitaev - heisenberg model relevant for na@xmath0iro@xmath1 . despite extensive efforts , no consensus concerning the minimal model for na@xmath0iro@xmath1 has been reached yet . the most natural extension of the kitaev - heisenberg model with ferromagnetic kitaev interaction which captures the zigzag magnetic order can be obtained by inclusion of farther neighbor couplings . in na@xmath0iro@xmath1 , these couplings might not be negligible due to the extended nature of the @xmath76-orbitals of the ir ions . in the early works suggesting this possible extension,@xcite second and third neighbor couplings were taken into account phenomenologically and only the isotropic part of these interactions was included . the importance of additional nearest neighbor @xmath77-symmetric anisotropic terms ( @xmath3-terms)@xcite or of the spatial anisotropy of the nearest neighbor kitaev interactions,@xcite were also discussed in the literature as a possible source for the stabilization of the zigzag phase . here we consider the @xmath4 model,@xcite which still has the same symmetry as the original kitaev - heisenberg model but contains kitaev interactions between both nearest and second nearest neighbors . the model reads @xmath78 where @xmath79 , @xmath80 , @xmath81 , @xmath82 , and @xmath83 ; @xmath84 , @xmath85 and @xmath86 denote nearest neighbor , second nearest neighbor and third nearest neighbor , respectively . @xmath32 and @xmath87 denote the three types of nearest neighbor and second nearest neighbor bonds of the honeycomb lattice , respectively . it is important to note that the second neighbor kitaev interactions do not change the space group symmetries of the original kitaev - heisenberg model . for realistic sets of the parameters describing na@xmath0iro@xmath1 , the second neighbor kitaev interaction , @xmath88 , computed from the microscopic approach based on the ab - initio calculation by foevtsova _ et al_,@xcite appeared to be the largest interaction after the nearest neighbor kitaev interaction , @xmath89 , and turn out to be antiferromagnetic . the mechanism behind the large magnitude of @xmath88 in na@xmath0iro@xmath1 is physically very clear : it originates from the large diffusive na ions that reside in the middle of the exchange pathways , and the constructive interference of a large number of pathways . moreover , the @xmath89-@xmath88 model , that only includes kitaev interactions,@xcite already stabilizes the zigzag phase for the proper signs of @xmath89 and @xmath88 . however , as we have discussed in ref.@xcite , the @xmath89-@xmath88 model is still not sufficient to comply with all available experimental data . the classical degeneracy of the zigzag state obtained within the @xmath4 model with fm @xmath89 is different from the one of the zigzag state realized in the extended kitaev - heisenberg model with afm @xmath89 interaction . to see what difference the sign of @xmath89 makes , let us consider the zigzag pattern in fig.[fig : orders ] ( d ) . with afm @xmath89 , the pattern , that minimizes the classical energy in the zigzag state with ferromagnegnetic @xmath18 and @xmath34 bonds , has the spins pointing along the @xmath90axis to take advantage of the kitaev interaction on the afm @xmath17-bonds . on the other hand the same pattern with fm @xmath89 takes advantage of the kitaev interaction on the fm @xmath91 and @xmath92 bonds by putting spins in the @xmath93plane . thus the degenerate ground state manifold for a given zigzag pattern with fm @xmath89 is one of @xmath94 , @xmath93 , or @xmath95 planes . furthermore , when the klein duality 4-sublattice transformation@xcite is applied to the @xmath4 zigzags , these states do not turn into nel afm state , and instead turn into non - collinear states , that are more difficult to work with than the original zigzag states . working with the zigzag states directly increases the magnetic unit cell to 4 sites , labeled in fig [ fig : orders](d ) . the hamiltonian matrix in the momentum space can be again written in the form of eq . ( [ hammoment ] ) , however this time due to the larger unit cell the exchange matrix @xmath48 is @xmath96 , instead of @xmath47 . its matrix elements are given in appendix c. the fluctuations matrix @xmath97 is calculated as before according to equation ( [ amatrix ] ) , with the constraint matrix @xmath98 of equation ( [ constraint ] ) now containing 4 identical blocks instead of 2 . the fluctuation matrix again contains the information on the direction of the spins and transmits this information to the free energy corrections that we plot in fig . [ fig : nogamma](b ) . since the spins are confined to a plane for a given zigzag state we have only the angle of the direction of spins in that plane . the color of the band at a given angle then gives the size of the fluctuational correction to the free energy , with violet being lowest and red highest energy states . we see that again the kitaev anisotropies prefer to align the magnetization along the cubic axes . note , however , that unlike the extended kh model , where there were 6 equivalent states , here there are 4 directions for each of the three zigzag patterns , giving a total of 12 states . model obtained with the luttinger - tisza method is shown on the first brillouin zone . we use @xmath99 mev , @xmath72 mev , @xmath100 mev , @xmath101 mev , @xmath75 mev , and ( a ) @xmath102 mev , ( b ) @xmath103 mev , ( c ) @xmath104 mev , and ( d ) @xmath105 mev . , title="fig : " ] model obtained with the luttinger - tisza method is shown on the first brillouin zone . we use @xmath99 mev , @xmath72 mev , @xmath100 mev , @xmath101 mev , @xmath75 mev , and ( a ) @xmath102 mev , ( b ) @xmath103 mev , ( c ) @xmath104 mev , and ( d ) @xmath105 mev . , title="fig : " ] model obtained with the luttinger - tisza method is shown on the first brillouin zone . we use @xmath99 mev , @xmath72 mev , @xmath100 mev , @xmath101 mev , @xmath75 mev , and ( a ) @xmath102 mev , ( b ) @xmath103 mev , ( c ) @xmath104 mev , and ( d ) @xmath105 mev . , title="fig : " ] model obtained with the luttinger - tisza method is shown on the first brillouin zone . we use @xmath99 mev , @xmath72 mev , @xmath100 mev , @xmath101 mev , @xmath75 mev , and ( a ) @xmath102 mev , ( b ) @xmath103 mev , ( c ) @xmath104 mev , and ( d ) @xmath105 mev . , title="fig : " ] the discussion above has clearly shown , that neither the original kitaev model nor the extended @xmath4 model can correctly explain the experimental data in na@xmath0iro@xmath1 . since the easy axes directions are determined solely by the anisotropy terms , only the inclusion of other types of the anisotropies can improve the situation . here we consider the off - diagonal symmetric @xmath3-terms . the role of these terms in the nearest - neighbor kitaev model has been studied in refs.@xcite . these studies have shown that the small @xmath3-terms do not immediately destabilize the zigzag phase , but lead to a deviation of the magnetic moments from the cubic axes . the origin of @xmath3-terms can be easily seen from the most general form of the bilinear exchange coupling matrix , which on the bond @xmath106 has the form given by @xmath107 while the kitaev term comes from the anisotropy of the diagonal matrix elements of @xmath108 , e.g. @xmath109 , the symmetric and antisymmetric combinations of off - diagonal elements represent other types of possible bond - anisotropies . in the absence of the trigonal distortion , the inversion symmetry prohibits the existence of antisymmetric interactions but some of the symmetric combinations are allowed , i.e. on a given @xmath27-bond , the interaction between the other two spin components , @xmath110 , where @xmath111 , is non - zero . our previous results@xcite suggest that in na@xmath0iro@xmath1 the magnitude of the strength of @xmath3 on the nearest neighbor bonds is about 2 - 3 mev and vanishes for the second neighbors . here we consider the @xmath112 model with the previous choice of heisenberg and kitaev interactions and treat @xmath3 as a free parameter . a straightforward classical minimization in momentum space using luttinger - tisza approach@xcite shows that up to very large values of @xmath113 mev the minima of the classical energy are located at the @xmath114 points corresponding to the zigzag states . this is clearly seen in fig . 4(a ) where we plot the lowest eigenvalues obtained for @xmath115 mev . at larger values of @xmath3 , the minima shift along the lines connecting @xmath114 points and the center of the brillouin zone ( see fig . 4 ( b ) for @xmath116 mev ) , indicating the transition to incommensurate order . the incommensurability of the luttinger - tisza solution increases further with larger values of @xmath3 , which is shown in figs . 4 ( c ) and ( d ) . the exact value of @xmath3 at which the transition occurs is difficult to determine due to the transition being so smooth , note , however , that the transition occurs at values of @xmath3 far beyond those predicted from our microscopic calculations at ambient pressure.@xcite after we have demonstrated that adding small @xmath3 interactions to the @xmath4 model does not destabilize the zigzag order , let us now show that in the presence of @xmath3 the mean - field degeneracy is already lifted and the preferred directions of the order parameter are selected . this is clearly seen in fig . 5 ( a ) and ( b ) , where the mean field energy of the zigzag order is computed for @xmath115 mev and @xmath117 mev , respectively . by inspection , we can see that non - zero @xmath3 selects the face diagonals as easy axes for magnetic ordering , and the sign of @xmath3 determines which of the two face diagonals are preferred . for concreteness , let us consider the zigzag with afm @xmath92bonds . as we discussed above the case for @xmath118 , the easy @xmath119-plane is selected at the mean - field level of the @xmath4 model . then , the inclusion of positive @xmath3 interaction on @xmath17 and @xmath18 bonds gives zero contribution to the energy since on these bonds it involves the spin component perpendicular to the easy plane , but it gives maximal lowering of the energy on the @xmath34-bonds if the spins point along @xmath120 $ ] and @xmath121 $ ] , @xmath122 $ ] and @xmath122 $ ] directions correspondingly for positive and negative values of @xmath3 . the estimate for the smallest @xmath3 , at which the selection of face diagonals takes place , can be done by comparing the mean - field energy gain due to @xmath3 with the energy gain due to fluctuations at @xmath118 , which at @xmath123 is equal to the zero point energy and is a function of @xmath89 and @xmath88 . at finite temperature , the contribution to the energy from the gaussian fluctuations at each @xmath124 can be computed by our method , and this energy will give the lower bound for the magnitude of @xmath3 needed to change the orientation of magnetic order from the cubic to the face diagonal . model with the contribution of ( a ) @xmath115 mev and ( b ) @xmath117 mev . , title="fig : " ] model with the contribution of ( a ) @xmath115 mev and ( b ) @xmath117 mev . , title="fig : " ] the microscopic calculations for @xmath2rucl@xmath1 emphasized the importance of the off - diagonal nearest neighbor @xmath3 interactions.@xcite the effect of adding @xmath3 interaction to the nearest neighbor kitaev - heisenberg model is easiest to understand in the rotated reference frame of the four - sublattice klein transformation.@xcite the kitaev and heisenberg interactions do not change their form and only change the value of the coupling constants under this transformation . on the other hand , @xmath3-interaction picks up a bond dependent sign as shown in fig . 2 . in fact , @xmath3 changes the sign on half of the bonds , i.e. there are just as many negative bonds as there are positive bonds for each kitaev type of bonds . since the klein transformed version of the zigzag state is the afm nel state , all the bonds are afm and involve the same pair of spins . thus the contribution of the @xmath3 interaction to the mean - field energy cancels out , and the set of states remains degenerate . this means that as long as we remain in the small window where @xmath3 does not destabilize the zigzag order found by rau et al.,@xcite we can perform our order - by - disorder approach to see what state is chosen . figs.6 ( a)-(c ) show the fluctuation free energy computed for the @xmath125 model for @xmath126 mev and @xmath127 mev , suggested by banerjee _ et al._,@xcite and @xmath128 mev , 0.8 mev and 0.9 mev , respectively . in fig . 6 ( a ) , @xmath128 mev , the minima of the fluctuational free energy are still along cubic directions . for larger @xmath3-interaction , the system prefers the states with at least two nonzero spin components and , therefore , the transition towards [ 111 ] preferred directions of the order parameter takes place . this is shown in fig . 6 ( b ) and ( c ) , in which the fluctuational energy is plotted for @xmath129 mev and 0.9 mev . while in fig . 6 ( b ) only very shallow minima are seen along [ 111 ] directions , in fig . 6 ( c ) both the pronounced minima along the cubic body diagonals and maxima along the cubic axes are very clearly seen . remember that the computation is done in the rotated reference frame . therefore , only the states with the orientation of the order parameter along the cubic axes will give the collinear states in the unrotated reference frame . the states with order parameter pointing along [ 111 ] directions in the rotated reference frame correspond to non - collinear states in the unrotated reference frame . since recent experiments by cao et al.@xcite have established that spins point along a cubic axis , by calculating the fluctuational corrections as a function of @xmath3 , we can find an upper bound on its value , such that the kitaev fluctuations dominate and keep the cubic axes as the preferred directions . from our calculations it follows that for @xmath126 mev and @xmath127 mev the upper bound for @xmath3 is about 0.8 mev . finally , for this set of parameters the transition to the 120@xmath10- afm order occurs around @xmath130 mev . note that this estimate is far smaller than the @xmath3 values resulting from _ ab initio _ calculations.@xcite . in this paper we explored how the direction of the magnetic moments in the zigzag ground state order is chosen in na@xmath0iro@xmath1 and @xmath2rucl@xmath1 . in both compounds , the kitaev interaction plays an important role . for the case of fm nearest neighbor kitaev interaction , like in na@xmath0iro@xmath1 , farther neighbor interactions are essential for stabilizing the zigzag ground state . for the afm nearest neighbor kitaev interaction , which was widely suggested to be the dominant interaction in @xmath2rucl@xmath1,@xcite the zigzag order can be stabilized already within the nearest neighbor model . we proposed that the @xmath112 model can explain all the experimental finding in na@xmath0iro@xmath1 . in this model the selection of the experimentally observed face diagonal direction of the order parameter happens already on the mean - field level due to the small bond - dependent anisotropic term @xmath3 . in @xmath2rucl@xmath1 , if the the nearest neighbor kitaev interaction is afm , the original kitaev - heisenberg model@xcite is sufficient to explain both the collinear zigzag ground state and the cubic directions of the order parameter . we studied the effect of the @xmath3-term and showed that while on the mean - field level it does nt affect the ground state degeneracy , it favors non - collinear 3-*q * states , instead of the experimentally observed zigzag state with spins along cubic axes , once the gaussian fluctuations are included . thus , it appears to be an upper bound for @xmath3-term , which can be estimated for a given set of nearest neighbor parameters . after the completion of our paper , we became aware of an independent study by winter _ et al_. @xcite of the magnetic interactions in the kitaev materials na@xmath0iro@xmath1 and @xmath2rucl@xmath1 . in this work , the authors treated all interactions up to third neighbours on equal footing by combining exact diagonalization and ab - initio techniques . one of the main findings of this work is that the third neighbor heisenberg interaction is important in all kitaev materials . let us briefly compare the results of ref . @xcite with our findings . the conclusions of the authors of ref . @xcite about the ordering in na@xmath0iro@xmath1 are in agreement with our findings , despite the fact that their estimates for @xmath88 suggest significantly smaller values than the ones that we obtained by including only the dominant superexchange processes between the second neighbors . the agreement holds because the second neighbor kitaev interaction @xmath88 and the third neighbor interaction @xmath131 favor the same type of afm zigzag ground state . for @xmath2rucl@xmath1 , the authors of ref.@xcite suggest ( _ i _ ) that there may be possible variations of in - plane interactions due to lattice distortions , and ( _ ii _ ) that the nearest neighbor kitaev interaction may be fm and the third neighbor coupling @xmath131 may be large and afm . the fm sign of the nearest neighbor kitaev interaction was also suggested by yadav _ et al _ in ref.@xcite . if this is indeed the case , the physics of @xmath2rucl@xmath1 is similar to that of na@xmath0iro@xmath1 . this , however , still needs to be verified by a detailed comparison with the experimental data . _ _ we acknowledge insightful discussions with c. batista , g. jackeli , m.garst , g. khaliullin and i. rousochatzakis . n.p . and y.s . acknowledge the support from nsf grant dmr-1511768 . n.p . acknowledges the hospitality of kitp and partial support by the national science foundation under grant no . nsf phy11 - 25915 . in this appendix we provide detailed discussion of the classical degeneracy of the extended kitaev - heisenberg model at parameters for which either the stripy or the zigzag af phases are realized as the ground state and the manifold of classically degenerate states is rather complex . to be specific , let us first consider the stripy phase . it contains six inequivalent collinear stripy states with fm bonds along either kitaev @xmath90 , @xmath91 or @xmath92bonds . it also contains infinite number of non - collinear ( coplanar and non - coplanar ) states . the spin order in the @xmath90 , @xmath91 or @xmath92 stripy states can be described either with a help of four magnetic sublattices or by a simple spiral characterized by a single-@xmath33 wave vector : @xmath132 , @xmath133 and @xmath134 . one of the stripy states with fm @xmath34-bonds is shown in fig.1 ( c ) . in each of these stripy states the spins are aligned along one of the cubic directions which is locked to the spatial orientation of a stripy pattern by the kitaev interaction , i.e. the direction of the order parameter is defined by the wave vector @xmath135 or @xmath136 determining the breaking of the translation symmetry . the structure of the manifold of the non - collinear states , which looks rather complex in the original model , can be easily understood with the help of the four - sublattice transformation ( see fig.2 ) based on the klein duality.@xcite in the new rotated basis , the stripy phase is mapped to the fm order with a unique ordering vector @xmath137 . classically , all states with arbitrary direction of the fm order have the same energy . fm states with order parameter along the cubic axes give the six stripy phases in the unrotated spin basis discussed above . arbitrary directions of the fm order parameter lead to a set of non - coplanar states in which each component of spin , @xmath138 , @xmath139 , and @xmath140 , transforms with its own @xmath141 , @xmath142 and @xmath136 wavevector , which coincide with the @xmath143 vectors describing the spatial orientation of the stripes in the respective collinear states . using these three ordering vectors , we can write the non - coplanar phase of the unrotated spins as @xmath144 where @xmath145 and @xmath146 are the polar and azimuthal angles of the fm order parameter . @xmath147 denote the spins on the sublattice @xmath38 and the spins on the sublattice @xmath39 are obtained from @xmath147 by a constant phase shift coming from the spin rotation on that bond as prescribed by the four sublattice transformation . as in fig.1 ( c ) , the sublattices 0 and 1 are connected by the @xmath34 bond , the order of the spins on the subllatice 1 is given by @xmath148 in the zigzag phase , the structure of the classical states manifold is very similar to the stripy phase . the four - sublattice transformation maps the zigzag phase onto the nel af phase . the generic state is again described by the three-@xmath143 spiral state . the only difference is that the spins on sublattice 1 have an overall phase factor of @xmath149 , @xmath150 . the matrix elements @xmath97 can be written as @xmath151 where a repeated index implies a summation over . the first term in ( [ amatrix ] ) is the contribution from the interaction term and the second term is from the constraint term.@xcite for convenience , the constraint matrix @xmath152 can be first written in the original basis , in which the interaction term is not diagonal , and then transformed to the eigenbasis of the hamiltonian with a help of the unitary transformation @xmath153 . in the original basis the constraint matrix @xmath152 consists of two blocks , one for each sublattice . the a - sublattice block has elements @xmath154 with @xmath155 and the b - sublattice block has the elements with @xmath156 . the two blocks are identical , so @xmath152 takes the following form : @xmath157 with matrix elements given by @xmath158{l}c_{{\bm q},11}= -\frac{2}{3}\bigl [ \beta_c(1-s_{\theta}^{2}c_{\phi}^{2})+3\beta r s_{\theta}^{2}c_{\phi}^{2}\bigr ] , \\ c_{{\bm q},22}= -\frac{2}{3}\bigl [ \beta_c(1-s_{\theta}^{2}s_{\phi}^{2})+3\beta r s_{\theta}^{2}s_{\phi}^{2}\bigr ] , \\ c_{{\bm q},33}= -\frac{2}{3}\bigl [ \beta_c s^2_{\theta}+3\beta r c^2_{\theta}\bigr ] , \\ c_{{\bm q},12}=c_{{\bm q},21}=-\frac{2}{3 } ( 3\beta r-\beta_c ) s^2_{\theta } c_{\phi}s_{\phi } , \\ c_{{\bm q},13}=c_{{\bm q},31}=-\frac{2}{3 } ( 3\beta r-\beta_c ) s_{\theta}s_{\theta } c_{\phi } , \\ c_{{\bm q},23}=c_{{\bm q},32}=-\frac{2}{3 } ( 3\beta r-\beta_c ) s_{\theta}s_{\phi } c_{\phi } , \end{array } \label{cmatrix}\ ] ] where , to shorten notations , we denote @xmath159 and @xmath160 . for shortness we define @xmath162 , @xmath163 , and @xmath164 . the diagonal matrix elements for @xmath165 and 10 are equal to @xmath166 , all other diagonal elements are equal to @xmath167 . the non - zero off - diagonal elements @xmath161 for @xmath168 are @xmath169 y. singh and p. gegenwart , phys . b * 82 * , 064412 ( 2010 ) . y. singh , s. manni , j. reuther , t. berlijn , r. thomale , w. ku , s. trebst , and p. gegenwart , phys . lett . * 108 * , 127203 ( 2012 ) . x. liu , t. berlijn , w .- g . yin , w. ku , a. tsvelik , young - june kim , h. gretarsson , y. singh , p. gegenwart , and j. p. hill , phys . b * 83 * , 220403 ( 2011 ) . f. ye , s. chi , h. cao , b. c. chakoumakos , j. a. fernandez - baca , r. custelcean , t. f. qi , o. b. korneta , and g. cao , phys . b * 85 * , 180403 ( 2012 ) . s. k. choi , r. coldea , a. n. kolmogorov , t. lancaster , i. i. mazin , s. j. blundell , p. g. radaelli , yogesh singh , p. gegenwart , k. r. choi , s .- w . cheong , p. j. baker , c. stock , and j. taylor , phys . rev . lett . * 108 * , 127204 ( 2012 ) . h. gretarsson , j. p. clancy , yogesh singh , p. gegenwart , j. p. hill , jungho kim , m. h. upton , a. h. said , d. casa , t. gog , and young - june kim , phys . b * 87 * , 220407(r ) s. h. chun , j .- w . kim , j. kim , h. zheng , c. c. stoumpos , c. d. malliakas , j. f. mitchell , kavita mehlawat , yogesh singh , y. choi , t. gog , a. al - zein , m. moretti sala , m. krisch , j. chaloupka , g. jackeli , g. khaliullin , b. j. kim , nature physics * 11 * , 462 ( 2015 ) . a. biffin , r. d. johnson , s. choi , f. freund , s. manni , a. bombardi , p. manuel , p. gegenwart , and r. coldea , phys . b * 90 * , 205116 ( 2014 ) . a. biffin , r.d . johnson , i. kimchi , r. morris , a. bombardi , j.g . analytis , a. vishwanath , and r. coldea , phys . lett . * 113 * , 197201 ( 2014 ) . t. takayama , a. kato , r. dinnebier , j. nuss , h. kono , l.s.i . veiga , g. fabbris , d. haskel , and h. takagi , phys . lett . * 114 * , 077202 ( 2015 ) . i. pollini , phys . b * 53 * , 12769 ( 1996 ) . k. w. plumb , j. p. clancy , l. j. sandilands , v. v. shankar , y. f. hu , k. s. burch , h .- y . kee , and y .- j . kim , phys . b * 90 * , 041112 ( 2014 ) . j. a. sears , m. songvilay , k. w. plumb , j. p. clancy , y. qiu , y. zhao , d. parshall , and young - june kim , phys . b * 91 * , 144420 ( 2015 ) . m. majumder , m. schmidt , h. rosner , a. a. tsirlin , h. yasuoka , and m. baenitz , phys . b * 91 * , 180401 ( 2015 ) . r. d. johnson , s. c. williams , a. a. haghighirad , j. singleton , v. zapf , p. manuel , i. i. mazin , y. li , h. o. jeschke , r. valenti , and r. coldea phys . b * 92 * , 235119 ( 2015 ) . a. banerjee , c. bridges , j .- q . yan , a. a. aczel , l. li , m. b. stone , g. e. granroth , m. d. lumsden , y. yiu , j. knolle et al . , nature materials * 15 * , 733 - 740 , ( 2016 ) . cao , a. banerjee , j .- q . yan , c.a . bridges , m.d . lumsden , d.g . mandrus , d.a . tennant , b.c . chakoumakos , s.e . nagler , arxiv:1602.08112 . j. chaloupka , g. jackeli , g. khaliullin , phys . lett . * 110 * , 097204 ( 2013 ) . v. m. katukuri , s. nishimoto , v. yushankhai , a. stoyanova , h. kandpal , s. choi , r. coldea , i. rousochatzakis , l. hozoi , j. van den brink , new j. phys . * 16 * , 013056 ( 2014 ) . y. sizyuk , c. price , p. wlfle , and n. b. perkins , phys . b * 90 * , 155126 ( 2014 ) . j. reuther , r. thomale and s. rachel , phys . b * 90 * , 100405 ( r ) ( 2014 ) . i. rousochatzakis , j. reuther , r. thomale , s. rachel , and n. b. perkins , phys . x * 5 * , 5 041035 ( 2015 ) . k. foyevtsova , h. o. jeschke , i. i. mazin , d. i. khomskii , and r. valenti phys . b * 88 * , 035107 ( 2013 ) . in ref . @xcite we obtained large @xmath88 interaction by considering only dominant super - exchange processes between second neighbors . the authors of recent study@xcite claim that the second neighbor kitaev interaction might be suppressed due to the interference of the various second and third order hopping processes , which we did not include in our derivation . however , as it is discussed in the text , the combined effect of @xmath88 and @xmath131 interactions leads to the same physics .
|
the magnetic orders in na@xmath0iro@xmath1 and @xmath2rucl@xmath1 , honeycomb systems with strong spin - orbit coupling and correlations , have been recently described by models with the dominant kitaev interactions . in this work
we discuss how the orientation of the magnetic order parameter is selected in this class of models .
we show that while the order - by - disorder mechanism in the models with solely kitaev anisotropies always select cubic axes as easy axes for magnetic ordering , the additional effect of other small bond - dependent anisotropies , such as , e.g. , @xmath3-terms , lead to a deviation of the order parameter from the cubic directions .
we show that both the zigzag ground state and the face - diagonal orientation of the magnetic moments in na@xmath0iro@xmath1 can be obtained within the @xmath4 model in the presence of perturbatively small @xmath3-terms .
we also show that the zigzag phase found in the nearest neighbor kitaev - heisenberg model , relevant for @xmath2rucl@xmath1 , has some stability against the @xmath3-term .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
let @xmath0 be a simple graph of order @xmath1 . for any vertex @xmath14 , the open neighborhood of @xmath15 is the set @xmath16 and the closed neighborhood is the set @xmath17=n ( v ) \cup \{v\}$ ] . for a set @xmath18 , the open neighborhood of @xmath19 is the set @xmath20 and the closed neighborhood of @xmath19 is the set @xmath21=n ( s ) \cup s$ ] . a set @xmath22 is a total dominating set if every vertex of @xmath3 is adjacent to some vertices of @xmath2 , or equivalently , @xmath23 . the total dominating number @xmath5 is the minimum cardinality of a total dominating set in @xmath4 . a total dominating set with cardinality @xmath5 is called a @xmath24-set . an @xmath8-subset of @xmath3 is a subset of @xmath3 of cardinality @xmath8 . let @xmath25 be the family of total dominating sets of @xmath4 which are @xmath8-subsets and let @xmath26 . the polynomial @xmath27 is defined as total domination polynomial of @xmath4 . graph @xmath4 obtained by point - attaching from @xmath28.,width=264 ] let @xmath4 be a connected graph constructed from pairwise disjoint connected graphs @xmath9 as follows . select a vertex of @xmath10 , a vertex of @xmath11 , and identify these two vertices . then continue in this manner inductively . note that the graph @xmath4 constructed in this way has a tree - like structure , the @xmath13 s being its building stones ( see figure [ figure1 ] ) . usually say that @xmath4 is obtained by point - attaching from @xmath28 and that @xmath13 s are the primary subgraphs of @xmath4 . a particular case of this construction is the decomposition of a connected graph into blocks ( see @xcite ) . as an example , the @xmath1-barbell graph @xmath29 with @xmath30 vertices , is formed by joining two copies of a complete graph @xmath31 by a single edge ( figure [ bar_n ] ) . actually , this graph is a specific kind of point - attaching of two complete graphs @xmath31 and the graph @xmath32 . observe that the total domination polynomial of @xmath1-barbell graph is @xmath33 this formula obtain easily from counting the total dominating sets of @xmath29 . and @xmath34 , respectively.,width=453,height=75 ] calculating the total domination polynomial of a graph @xmath4 is difficult in general , as the smallest power of a non - zero term is the domination number @xmath5 of the graph , and determining whether @xmath35 is known to be np - complete . so presenting a closed formula for the total domination polynomial of any kind of point - attaching graphs is difficult , but for certain classes of graphs , we can find a closed form expression for the total domination polynomial . in this paper , we consider some particular cases of these graphs and study their total domination polynomials . in section 2 , we consider graphs which obtain by a special point - attaching of a graph @xmath36 and @xmath37 copies of graphs @xmath38 . we prove that all graphs whose total domination polynomial have just two roots @xmath39 are in this form . also we study the total domination polynomial of some kind of generalized friendship graphs in this section . in section 3 , we investigate the total domination polynomial of cactus chains . in this section , we consider graphs constructed from primary subgraphs and study their total domination polynomial . some kind of these graphs have interesting properties . in the subsection 2.1 , we prove that a special kind of graphs from primary subgraphs have exactly two total domination roots . in subsection 2.2 we study the total domination polynomial of the generalized friendship graph . graphs whose certain polynomials have few roots can sometimes give interesting information about the structure of the graph . the characterization of graphs with few distinct roots of characteristic polynomials ( i.e. , graphs with few distinct eigenvalues ) have been the subject of many researchers @xcite . also the first authors has studied graphs with few domination roots in @xcite . let @xmath36 be an arbitrary graph of order @xmath1 and consider @xmath1 copies of graph @xmath38 . by definition , the graph @xmath40 is obtained by identifying each vertex of @xmath36 with an end vertex of a @xmath38 . see figure [ h(3 ) ] . to obtain the total domination polynomial of @xmath40 , we need the following result . .,width=139,height=120 ] @xcite[3 ] let @xmath4 be a connected graph of order @xmath41 . then @xmath42 if and only if @xmath4 is @xmath43 , @xmath44 or @xmath40 for some connected graph @xmath36 . [ hop ] for any graph @xmath36 of order @xmath1 , we have @xmath45 . * proof . * let @xmath2 be a total dominating set of @xmath40 of size @xmath46 in figure [ h(3 ) ] . obviously @xmath47 . to choose @xmath48 ( @xmath49 ) other vertices of @xmath50 , we have @xmath51 possibilities . so we have the result.@xmath52 now , we state and prove the following result . [ 2roots ] let @xmath4 be a graph . then @xmath53 if and only if @xmath54 for some graph @xmath36 of order @xmath1 . * * ( @xmath55 ) it follows from theorem [ hop ] . ( @xmath56 ) let @xmath4 be a graph with @xmath53 . thus @xmath57 and @xmath4 has no isolated vertex . since @xmath58 , by theorem [ 3 ] , every component of @xmath4 is a cycle @xmath43 , @xmath44 or a @xmath40 for some connected graph @xmath36 . since @xmath59 and @xmath60 does not divide @xmath61 , we conclude that there exists a graph @xmath36 such that @xmath62 and the proof is complete . @xmath52 the characterization of graphs whose graph polynomials have few roots have been an interesting problem and studied well in the literature ( @xcite ) . also there is a conjecture in @xcite which states that every integer total domination roots is in the set @xmath63 . so finding the graphs whose total domination polynomial have these few roots can be a good start for solving this conjecture . theorem [ 2roots ] characterize all graphs whose total domination polynomial have just two distinct roots @xmath64 and @xmath65 . now we consider another kind of point - attaching graphs and study their total domination polynomials . the generalized friendship graph @xmath66 is a collection of @xmath1 cycles ( all of order @xmath67 ) , meeting at a common vertex ( see figure [ dutch ] ) . the generalized friendship graph may also be referred to as a flower @xcite . for @xmath68 the graph @xmath66 is denoted simply by @xmath69 and is friendship graph . the total domination polynomial of @xmath69 and its roots studied in @xcite . here , first we compute the total domination number of @xmath70 . and @xmath70 , respectively.,width=453,height=113 ] for any @xmath71 , we have @xmath72 . * proof . * let @xmath73 be vertex set of @xmath70 that adjacent by @xmath74 ( common vertex in @xmath70 ) . then @xmath75 is a total dominating set for @xmath70 ( see figure [ dutch ] ) and the set @xmath76 of size less than or equal @xmath1 is not total dominating set for @xmath70 , therefore @xmath72.@xmath52 the following theorem is useful for finding the recurrence relations of the domination polynomials of graphs . the vertex contraction @xmath77 of a graph @xmath4 by a vertex @xmath78 is the operation under which all vertices in @xmath79 are joined to each other and then @xmath78 is deleted ( see @xcite ) . @xcite [ thm ] 1 . for any vertex @xmath78 in the graph @xmath4 we have + @xmath80,x)-(1+x)p_u(g)$ ] , + where @xmath81 is the polynomial counting the total dominating sets of @xmath82 which do not contain any vertex of @xmath79 in @xmath4 . 2 . let @xmath83 be two non - adjacent vertices of @xmath4 with @xmath84 . then + @xmath85,x)$ ] . 3 . if @xmath83 be two vertices of @xmath4 with @xmath17\subseteq n[u]$ ] . then + @xmath86,x)$ ] . 4 . let @xmath87 and @xmath17=n[u]$ ] . then + @xmath88,x)$ ] . now we state and prove a recurrence relation for the total domination polynomial of @xmath70 . [ fn4 ] for any @xmath89 , we have @xmath90 $ ] , with initial value @xmath91 . * proof . * consider graph @xmath70 and @xmath78 , @xmath15 as shown in figure [ f_n,4 ] . by theorem [ thm ] , we have @xmath92 \\&\hspace*{2.5cm}+x^2d_t(p_3,x)^{n-1 } \\&\hspace*{2.2cm}=(x^2 + 2x)d_t((f_{n,4}\setminus u)/v , x)+(x^3 + 2x^2)d_t(p_3,x)^{n-1 } \\&\hspace*{1.9cm}\stackrel{part ( i)}{=}(x^2 + 2x)[d_t(f_{n-1,4},x)+xd_t(f_{n-1,4},x)-(x+1)p_w((f_{n,4}\setminus u)/v , x ) ] \\&\hspace*{2.5cm}+(x^3 + 2x^2)d_t(p_3,x)^{n-1 } \\&\hspace*{2.1cm}=x(x+1)(x+2)d_t(f_{n-1,4},x)-x(x+1)(x+2)p_w((f_{n,4}\setminus u)/v , x ) \\&\hspace*{2.5cm}+(x^3 + 2x^2)d_t(p_3,x)^{n-1}. \end{aligned}\ ] ] since @xmath93 , so we have the result.@xmath52 , width=491,height=162 ] in this section , we consider another kind of point - attaching graphs and study their total domination polynomials . these kind of graphs are important in chemistry . a cactus graph is a connected graph in which no edge lies in more than one cycle . consequently , each block of a cactus graph is either an edge or a cycle . if all blocks of a cactus @xmath4 are cycles of the same size @xmath8 , the cactus is @xmath8 -uniform . a triangular cactus is a graph whose blocks are triangles , i.e. , a @xmath94 -uniform cactus . we call the number of triangles in @xmath4 the length of the chain . obviously , all chain triangular cactus of the same length are isomorphic . hence , we denote the chain triangular cactus of length @xmath1 by @xmath95 ( see figure [ gn ] ) . by replacing triangles in this definitions by cycles of length @xmath96 we obtain cacti whose every block is @xmath97 . in this section we shall study the total domination polynomial of cactus chains . in this subsection we shall study the total domination polynomial of chain triangular cactus . to do this , we consider graph @xmath98 as shown in figure [ gn ] , which is also a kind of point attaching graphs . note that @xmath98 is a point attaching of @xmath95 and @xmath32 . first we state and prove the following theorem : * proof . * consider the graph @xmath98 as shown in figure [ gn ] . since @xmath104 is isomorphic to @xmath105 , by theorem [ thm](i ) , we have @xmath106,x)-(x+1)p_u(g_n ) \\&\hspace*{1.6cm}=(x+1)d_t(t_n , x)+x^2d_t(g_{n-2},x)-(x+1)d_t(g_{n-1},x ) . \end{aligned}\ ] ] note that @xmath107 . @xmath52 * proof . * consider the graph @xmath95 and its vertex @xmath78 as shown in the figure [ gn ] . by theorem [ thm](iii ) , we have @xmath109,x)+d_t(t_{n}\setminus n[\{u , w\}],x ) ] \\&\hspace*{1.5cm}=(x+1)d_t(g_{n-1},x)+x^2[d_t(g_{n-2},x)+d_t(g_{n-3},x ) ] . \end{aligned}\ ] ] note that @xmath110 is isomorphic to @xmath111.@xmath52 1 . consider graph @xmath123 and its vertex @xmath78 in figure [ qn ] . by theorem [ thm](iii ) , @xmath124,x)+2d_t(q_n(1)\setminus n[\{u , w\}],x ) ] \\ & \hspace*{2.1cm}=xd_t(q_{n}+e , x)+x^2[d_t(q_{n-1}x)+2d_t(q^{'}_{n-1},x ) ] . \end{aligned}\ ] ] by applying theorem [ thm](iv ) on @xmath125 , we have + @xmath126 . + so we have result . 2 . consider the vertex @xmath78 as shown in figure [ qn ] . by theorem [ thm](iii ) , we have @xmath127,x)+d_t(q_n(2)\setminus n[\{u , w\}],x ) ] \\ & \hspace*{2.1cm}=xd_t(q_{n-1}(1),x)+x^2[d_t(q^{'}_{n-1},x)+d_t(q_{n-1},x ) ] . \end{aligned}\ ] ] by using part @xmath128 in the above equation , we have result . 3 . consider graph @xmath129 in figure [ qn ] . by theorem [ thm ] , @xmath130,x ) \\ & \hspace*{1.7cm}=(x+1)d_t(q_{n}(1),x)+x^2d_t(q_{n-1},x ) . \end{aligned}\ ] ] using part @xmath128 in the above equation , we have result . 4 . by theorem [ thm](iii ) , we have @xmath131,x)+d_t(q_n^\delta \setminus n[\{u , w\}],x ) ] \\ & \hspace*{1.7cm}=(x+1)d_t(q_{n}(1),x)+x^2[d_t(q^{'}_{n-1},x)+d_t(q_{n-1},x ) ] . \end{aligned}\ ] ] so by using part @xmath128 in the above equation , we have result.@xmath52 * proof . * with regards to figure [ q_n ] and theorem [ thm ] , we have @xmath135,x)+d_t(q_n\setminus n[\{u , w\}],x ) \\&\hspace*{1.6 cm } = d_t(q_{n-1}(2),x)+xd_t(q_{n-1}^\delta , x)+x^2[d_t(q_{n-2}^{'},x)+d_t(q_{n-2},x ) ] \end{aligned}\ ] ] now by lemma [ le ] results is obtained.@xmath52 e.r . van dam , _ graphs with few eigenvalues , an interplay between combinatorics and algebra , _ center dissertation series 20 , thesis , tilburg university , 1996 . van dam , _ nonregular graphs with three eigenvalues , _ j. combin . theory ser . b 73 ( 1998 ) 101 - 118 .
|
let @xmath0 be a simple graph of order @xmath1 .
the total dominating set is a subset @xmath2 of @xmath3 that every vertex of @xmath3 is adjacent to some vertices of @xmath2 .
the total domination number of @xmath4 is equal to minimum cardinality of total dominating set in @xmath4 and denoted by @xmath5 .
the total domination polynomial of @xmath4 is the polynomial @xmath6 , where @xmath7 is the number of total dominating sets of @xmath4 of size @xmath8 .
let @xmath4 be a connected graph constructed from pairwise disjoint connected graphs @xmath9 by selecting a vertex of @xmath10 , a vertex of @xmath11 , and identify these two vertices .
then continue in this manner inductively .
we say that @xmath4 is obtained by point - attaching from @xmath12 and that @xmath13 s are the primary subgraphs of @xmath4 . in this paper
, we consider some particular cases of these graphs that most of them are of importance in chemistry and study their total domination polynomials .
department of mathematics , yazd university , 89195 - 741 , yazd , iran + [email protected] + * keywords : * total domination number , total domination polynomial , total dominating set . * ams subj . class .
: * 05a18 ; 11b 73 , 05c12 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the milky way is a typical bright spiral galaxy . its disk of stars and gas is surrounded by an extended halo of old stars , globular star clusters and a few dark matter dominated old satellite galaxies . for the past 30 years two competing scenarios for the origin of galaxies and their stellar components have driven much observational and theoretical research . eggen , lynden - bell and sandage ( 1962 ) proposed a monolithic collapse of the galaxy whilst searle and zinn ( 1978 ) advocated accretion of numerous proto - galactic fragments . enormous progress has been made in understanding the structure and origin of the milky way , as well as defining a standard cosmological model for structure formation that provides us with a framework within which to understand our origins @xcite . hierarchical growth of galaxies is a key expectation within a universe whose mass is dominated by a dark and nearly cold particle ( cdm ) , yet evidence for an evolving hierarchy of merging events can be hard to find , since much of this activity took place over 10 billion years ago . the origin of the luminous galaxy depends on the complex assembly of its @xmath2 dark halo that extends beyond @xmath3 kpc , and on how stars form within the first dark matter structures massive enough to cool gas to high densities @xcite . the galactic halo contains about 100 old metal poor globular clusters ( i.e. forbes et al . 2000 ) each containing up to @xmath4 stars . their spatial distribution falls off as @xmath5 at large radii and half the globulars lie within 5 kpc from the centre of the galaxy @xcite . there is no evidence for dark matter within the globular clusters today @xcite . the old stellar halo population has a similar spatial distribution and a total luminosity of @xmath6 @xcite . the stellar populations , ages and metallicities of these components are very similar @xcite . also orbiting the galaxy are several tiny spheroidal satellite galaxies , each containing an old population of stars , some showing evidence for more recent star - formation indicating that they can hold on to gas for a hubble time @xcite . half of the dwarf satellites lie within 85 kpc , have luminosities in the range @xmath7 and are surrounded by dark haloes at least 50 - 200 times as massive as their baryonic components @xcite . cold dark matter models have had a notoriously hard time at reconciling the observed low number of satellites with the predicted steep mass function of dark haloes @xcite . we wish to explore the hypothesis that cold dark matter dominates structure formation , the haloes of galaxies and clusters are assembled via the hierarchical merging and accretion of smaller progenitors ( e.g. lacey and cole 1993 ) . this process violently causes structures to come to a new equilibrium by redistributing energy among the collision - less mass components . early stars formed in these progenitors behave as a collisionless system just like the dark matter particles in their host haloes , and they undergo the same dynamical processes during subsequent mergers and the buildup of larger systems like massive galaxies or clusters . in a recent study , diemand et al . ( 2005 ) used cosmological n - body simulations to explore the distribution and kinematics in present - day cdm haloes of dark matter particles that originally belonged to rare peaks in the matter density field . these properties are particularly relevant for the baryonic tracers of early cdm structures , for example the old stellar halo which may have originated from the early disruption of numerous dwarf proto - galaxies @xcite , the old halo globular clusters and also giant ellipticals @xcite . since rare , early haloes are strongly biased towards overdense regions ( e.g. sheth and tormen 1999 ) , i.e. towards the centers of larger scale fluctuations that have not collapsed yet , we might expect that the contribution at @xmath8 from the earliest branches of the merger tree is much more centrally concentrated than the overall halo . indeed , a `` non - linear '' peaks biasing has been discussed by previous authors @xcite . diemand et al . ( 2005 ) showed that the present - day distribution and kinematics of material depends primarily on the rareness of the peaks of the primordial density fluctuation field that the selected matter originally belonged to , i.e. when selecting rare density peaks above @xmath9 [ where @xmath10 is the linear theory rms density fluctuations smoothed with a top - hat filter of mass @xmath11 at redshift @xmath12 , their properties today depend on @xmath13 and not on the specific values of selection redshift z and minimal mass m. in the following section of this paper we discuss a model for the combined evolution of the dark and old stellar components of the galaxy within the framework of the @xmath14cdm hierarchical model @xcite . many previous studies have motivated and touched upon aspects of this work but a single formation scenario for the above components has not been explored in detail and compared with data @xcite . we assume proto - galaxies and globular clusters form within the first rare peaks above a critical mass threshold that can allow gas to cool and form stars in significant numbers ( typically at @xmath15 ) . we assume that shortly after the formation of these first systems , the universe reionises , perhaps by these first proto - galaxies , suppressing further formation of cosmic structure until later epochs . we use the n - body simulations to trace the rare peaks to @xmath8 . most of these proto - galaxies and their globular clusters merge together to create the central galactic region . in section 3 we will compare the spatial distribution and orbital kinematics of these tracer particles with the galactic halo light and old metal poor globular clusters . we will see that a small number of these first stellar systems survive as dark matter dominated galaxies . we will compare their properties with the old satellites of the galaxy in section 4 . = 15.2 cm we propose that ` ordinary ' population ii stars and globular clusters first appeared in significant numbers at redshift @xmath16 , as the gas within protogalactic haloes with virial temperatures above @xmath17k ( corresponding to masses comparable to those of present - day dwarf spheroidals ) cooled rapidly due to atomic processes and fragmented . it is this ` second generation ' of subgalactic stellar systems , aided perhaps by an earlier generation of metal - free ( population iii ) stars and by their remnant black holes , which generated enough ultraviolet radiation to reheat and reionize most of the hydrogen in the universe by a redshift @xmath18 , thus preventing further accretion of gas into the shallow potential wells that collapsed later . the impact of a high redshift uv background on structure formation has been invoked by several authors @xcite to explain the flattening of the faint end of the luminosity function and the missing satellites problem within our local group . here we use high resolution numerical simulations that follow the full non - linear hierarchical growth of galaxy mass haloes to explore the consequences and predictions of this scenario . dark matter structures will collapse at different times , depending on their mass , but also on the underlying larger scale fluctuations . at any epoch , the distribution of masses of collapsed haloes is a steep power law towards low masses with @xmath19 . to make quantitative predictions we calculate the non - linear evolution of the matter distribution within a large region of a @xmath14cdm universe . the entire well resolved region is about 10 comoving megaparsecs across and contains 61 million dark matter particles of mass @xmath20 and force resolution of 0.27 kpc . this region is embedded within a larger 90 mpc cube that is simulated at lower resolution such that the large scale tidal field is represented . figure 1 shows the high - redshift and present - day mass distribution of a single galaxy mass halo taken from this large volume . the rare peaks collapsing at high redshift that have had sufficient time to cool gas and form stars , can be identified , followed and traced to the present day . because small fluctuations are embedded within a globally larger perturbation , the small rarer peaks that collapse first are closer to the centre of the final potential and they preserve their locality in the present day galaxy . the strong correlation between initial and final position results in a system where the oldest and rarest peaks are spatially more concentrated than less rare peaks . the present day spatial clustering of the material that was in collapsed structures at a higher redshift only depends on the rarity of these peaks @xcite . our simulation contains several well resolved galactic mass haloes which we use to trace the evolution of progenitor haloes that collapse at different epochs . the first metal free population iii stars form within minihaloes already collapsed by @xmath21 , where gas can cool via roto - vibrational levels of h@xmath22 and contract . their evolution is rapid and local metal enrichment occurs from stellar evolution . metal - poor population ii stars form in large numbers in haloes above @xmath23^{-3/2}\,m_\odot$ ] ( virial temperature @xmath24k ) , where gas can cool efficiently and fragment via excitation of hydrogen ly@xmath25 . at @xmath26 , these correspond to @xmath27 peaks of the initial gaussian overdensity field : most of this material ends up within the inner few kpc of the galaxy . within the @xmath28mpc turn - around region , a few hundred such protogalaxies are assembling their stellar systems @xcite . typically 95% of these first structures merge together within a timescale of a few gyrs , creating the inner galactic dark halo and its associated old stellar population . with an efficiency of turning baryons into stars and globular clusters of the order @xmath29 we successfully reproduce the total luminosity of the old halo population and the old dwarf spheroidal satellites . the fraction of baryons in dark matter haloes above the atomic cooling mass at redshift 12 exceeds @xmath30 . a normal stellar population with a salpeter - type initial mass function emits about 4,000 hydrogen - ionizing photons per stellar baryon . a star formation efficiency of 10% therefore implies the emission of @xmath31 a few lyman - continuum photons per baryon in the universe . this may be enough to photoionize and drive to a higher adiabat vast portions of the intergalactic medium , thereby quenching gas accretion and star formation in nearby low - mass haloes . the globular clusters that were once within the merging proto - galaxies are so dense that they survive intact and will orbit freely within the galaxy . the surviving proto - galaxies may be the precursors of the old satellite galaxies , some of which host old globular clusters such as fornax , whose morphology and stellar populations are determined by ongoing gravitational and hydrodynamical interactions with the milky way ( e.g. mayer et al . 2005 ) . recent papers have attempted to address the origin of the spatial distribution of globular clusters ( e.g. parmentier and grebel 2005 , parmentier et al . 2005 ) . most compelling for this model and one of the key results in this paper , is that we naturally reproduce the spatial clustering of each of these old components of the galaxy . the radial distribution of material that formed from @xmath32 peaks at @xmath26 now falls off as @xmath33 within the galactic halo - just as the observed old halo stars and metal poor globular clusters ( cf . figure 2 ) . cosmological hydrodynamical simulations are also begining to attain the resolution to resolve the formation of the old stellar haloes of galaxies ( abadi et al . because of the steep fall off with radius , we note that we do not expect to find any isolated globular clusters beyond the virial radius of a galaxy . these first collapsing structures infall radially along filaments and end up significantly more flattened than the mean mass distribution . they also have colder velocity distributions and their orbits are isotropic in the inner halo and increasingly radially anisotropic in the outer part . material from these rare peaks has @xmath34 at our position in the milky way , in remarkable agreement with the recently measured anisotropy and velocity dispersion of halo stars @xcite . diemand et al . ( 2005 ) show that the radial distribution of rarer peaks is even more highly biased - thus the oldest population iii stars and their remnant black holes are found mainly within the inner kpc of the galaxy , falling off with radius steeper than @xmath35 . the observational evidence for tidal stellar streams from globular clusters suggests that they are not embedded within extended dark matter structures today @xcite . this does not preclude the possibility that the globular clusters formed deep within the central region of @xmath36 dark haloes which have since merged together . ( massive substructure within the inner @xmath37 of galactic mass haloes is tidally disrupted i.e. gihgna et al . this is what we expect within our model which would leave the observed globulars freely orbiting without any trace of the original dark matter component . however , it is possible that the most distance halo globulars may still reside within their original dark matter halo . if the globular cluster is located at the center of the cdm cusp , then observations of their stellar kinematics may reveal rising dispersion profiles . if the globular cluster is orbiting within a cdm mini - halo then we would expect to see symmetric tidal streams orbiting within the potential of the cdm substructure halo rather than being stripped by the galaxy . the remaining @xmath38% of the proto - galaxies form sufficiently far away from the mayhem that they fall into the assembling galaxy late ( @xmath39 , about one gyr after the formation of the inner galaxy at @xmath40 ) . this leaves time to enhance their @xmath41 element ratios from type ii supernovae @xcite . recent studies including chemical modeling of this process support this scenario ( e.g. robertson et al . 2005 , font et al . 2005 ) . the proto - galaxies highlighted with boxes in figure 1 are those few systems that survive until the present epoch - they all form on the outskirts of the collapsing region , ending up tracing the total mass distribution as is also observed within the milky way s and m31 s satellite systems . each of our four high resolution galaxies contains about ten of these surviving proto - galaxies which have a radial distribution that is slightly _ shallower _ than that of the total mass distribution but more concentrated than the distribution of all surviving ( or @xmath8 mass selected ) subhalos ( figures [ fig : z0 ] and [ nr ] ) . this is consistent with the spatial distribution of surviving satellites in the milky way and in other nearby galaxies in the 2df @xcite and deep2 samples @xcite and with galaxy groups like ngc5044 @xcite . figure 3 shows the distribution of circular velocities of the local group satellites compared with these rare proto - galaxies that survive until the present day . the local group circular velocity data are the new data from maccio et al . ( 2005b ) where velocity dispersions have been converted to peak circular velocities using the results of kazantzidis et al . the total number of dark matter substructures is over an order of magnitude larger than the observations . reionisation and photo - evaporation must play a crucial role in suppressing star formation in less rare peaks , thus keeping most of the low mass haloes that collapse later devoid of baryons . the surviving population of rare peaks had slightly higher circular velocities just before accretion ( at @xmath42 , dashed line in figure 3 - see kravtsov et al . 2004 ) , tidal stripping inside the galaxy halo then reduced their masses and circular velocities and they match the observations at @xmath8 . dissipation and tidally induced bar formation could enable satellites to survive even closer to the galactic centre ( maccio et al . 2005a ) . likewise to the radial distribution , the kinematics of the _ surviving visible _ satellite galaxies resembles closely the one of the dark matter while the same properties for all the surviving subhalos differ ( figures [ vr ] and [ vt ] ) . within the four high resolution cdm galaxy haloes our 42 satellite galaxies have average tangential and radial velocity dispersions of 0.70@xmath43 and 0.56@xmath44 respectively , i.e. @xmath45 ( the errors are one sigma poission uncertainties ) . these values are consistent with those of the dark matter particles : @xmath46 , @xmath47 and @xmath48 ; the hint of slightly larger dispersions of the satellites are consistent with their somewhat larger radial extent . in the inner part our model satellite galaxies are hotter than the dark matter background , especially in the tangential component : within 0.3 @xmath49 we find @xmath50 and @xmath51 . this is consistent with the observed radial velocities of milky way satellites . for the inner satellites also the tangential motions are know ( with large uncertainties however ) ( e.g. mateo 1998 ; wilkinson & evans 1999 ) and just as in our simple model they are larger than the typical tangential velocities of dark matter particles in the inner halo . the total ( mostly dark ) surviving subhalo population is more extended and hotter than the dark matter while the distribution of orbits ( i.e. @xmath52 ) is similar @xcite . for the 2237 subhalos within the four galaxy haloes find @xmath53 , @xmath54 and @xmath55 , i.e. there is a similar velocity bias relative to the dark matter in both the radial and tangential components and therefore a similar anisotropy . in the inner halo the differences between dark matter particles and subhaloes are most obvious : within 0.3 @xmath49 we find @xmath56 and @xmath57 . subhalos tend to avoid the inner halo and those who lie near the center at @xmath8 move faster ( both in the tangential and radial directions ) than the dark matter particles , i.e. these inner subhalos have large orbital energies and spend most of their time further away from the center ( figures [ nr ] , [ vr ] and [ vt ] , see also diemand et al . 2004 ) . [ haloes ] .present - day properties of the four simulated galaxy haloes . the columns give halo name , virial mass , virial radius , peak circular velocity , and radius to the peak of the circular velocity curve . the virial radius is defined to enclose a mean density of 98.6 times the critical density . the mass resolution is @xmath20 and the force resolution ( spline softening length ) is 0.27 kpc . for comparison with the milk way these halos were rescaled to a peak circular velocity of 195 km / s . in sph simulations of the same halos we found that this rescaling leads to a local rotation speed of 220 km / s after the baryonic contraction @xcite . the rescaled virial radii and virial masses are given in the last two columns . [ cols="<,^,^,^,^,^,^ " , ] we have a implemented a simple prescription for proto - galaxy and globular cluster formation on to a dissipationless cdm n - body simulation . this allows us to trace the kinematics and spatial distribution of these first stellar systems to the final virialised dark matter halo . we can reproduce the basic properties of the galactic metal poor globular cluster system , old satellite galaxies and galactic halo light . the spatial distribution of material within a virialised dark matter structure depends on the rarity of the peak within which the material collapses . this implies a degeneracy between collapse redshift and peak height . for example , 3 sigma peaks collapsing at redshift 18 and 10 will have the same final spatial distribution within the galaxy . however this degeneracy can be broken since the mass and number of peaks are very different at each redshift . in this example at redshift 18 a galaxy mass perturbation has 700 collapsed 3 sigma halos of mass @xmath58 , compared to 8 peaks of mass @xmath59 the best match to the spatial distribution of globular clusters and stars comes from material that formed within peaks above 2.5 @xmath60 . we can then constrain the minimum mass / redshift pair by requiring to match the observed number of satellite galaxies in the local group ( figure [ mvst ] ) . if protogalaxies form in early , low mass 2.5 @xmath60 peaks the resulting number of luminous satellites is larger as when they form later in heavier 2.5 @xmath60 peaks . we find that efficient star formation in halos above about 10@xmath61 up to a redshift @xmath62 matches these constraints . the scatter in redshift is due to the different best fit redshifts found in our individual galaxy haloes . after this epoch star formation should be suppressed in small halos otherwise a too large number of satellites and a too massive and too extended spheroid of population ii stars is produced . the minimum halo mass to form a protogalaxy inferred from these two constraints corresponds to a minimal halo virial temperature of @xmath63 ( figure [ mvst ] ) , i.e. just the temperature needed for efficient atomic cooling . this model is general for galaxy formation , but individual formation histories may reveal more complexity . soon after reionisation , infalling gas into the already massive galactic mass halo leads to the formation of the disk and the metal enriched population of globular clusters . the first and second generation of stars forming in proto - clusters of galaxies will have a similar formation path , but occurring on a more rapid timescale . we find that the mass fraction in peaks of a given @xmath60 is independent of the final halo mass , except that it rapidly goes to zero as the host halos become too small to have sufficiently high @xmath60 progenitors ( see figure [ fielddwarfs ] and table 4 in @xcite ) . therefore , if reionisation is globally coeval throughout the universe the abundance of globulars normalised to the halo mass will be roughly constant in galaxies , groups and clusters . furthermore , the radial distribution of globular clusters relative to the host halo scale radius will the same ( see diemand et al . if rarer peaks reionise galaxy clusters earlier @xcite then their final distribution of blue globulars will fall off more steeply ( relative to the scale radius of the host halo ) and they will be less abundant per virial mass @xcite . observations suggest that the numbers of old globular clusters are correlated with the luminosity of the host galaxy @xcite . wide field surveys of the spatial distribution of globulars in groups and clusters may reveal the details of how and when reionisation occurred @xcite . we thank jean brodie , andi burkert , duncan forbes and george lake for useful discussions and andrea maccio for providing the corrected local group data for figure [ fig : massfn ] prior to publication . all computations were performed on the zbox supercomputer at the university of zrich . support for this work was provided by nasa grants nag5 - 11513 and nng04gk85 g , by nsf grant ast-0205738 ( p.m. ) , and by the swiss national science foundation .
|
the milky way contains several distinct old stellar components that provide a fossil record of its formation .
we can understand their spatial distribution and kinematics in a hierarchical formation scenario by associating the proto - galactic fragments envisaged by searle and zinn ( 1978 ) with the rare peaks able to cool gas in the cold dark matter density field collapsing at redshift @xmath0 .
we use hierarchical structure formation simulations to explore the kinematics and spatial distribution of these early star - forming structures in galaxy haloes today .
most of the proto - galaxies rapidly merge , their stellar contents and dark matter becoming smoothly distributed and forming the inner galactic halo .
the metal - poor globular clusters and old halo stars become tracers of this early evolutionary phase , centrally biased and naturally reproducing the observed steep fall off with radius .
the most outlying peaks fall in late and survive to the present day as satellite galaxies . the observed radial velocity dispersion profile and the local radial velocity anisotropy of milky way halo stars
are successfully reproduced in this model .
if this epoch of structure formation coincides with a suppression of further cooling into lower sigma peaks then we can reproduce the rarity , kinematics and spatial distribution of satellite galaxies as suggested by bullock et al .
( 2000 ) .
reionisation at @xmath1 provides a natural solution to the missing satellites problem . measuring the distribution of globular clusters and halo light on scales from galaxies to clusters
could be used to constrain global versus local reionisation models .
if reionisation occurs contemporary , our model predicts a constant frequency of blue globulars relative to the host halo mass , except for dwarf galaxies where the average relative frequencies become smaller .
[ firstpage ] methods : n - body simulations methods : numerical dark matter galaxies : haloes galaxies : clusters : general , globular clusters
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
with gpu hardware and the corresponding software environments becoming mature , compute clusters with gpu - accelerated nodes establish as a new , powerful platform for high - performance computing ( hpc ) . mainly motivated by the expected boost for application performance ( i.e. reducing `` time to solution '' ) and also by energy - efficiency considerations ( i.e. reducing `` energy to solution '' ) , major research organizations and providers of hpc resources have already deployed an appreciable amount of gpu - accelerated resources worldwide @xcite . moreover , gpu - like architectures are expected to play a major role in the upcoming exascale era @xcite . it is well known , however , in the community , that the new hardware architecture together with the apparently disruptive programming models pose substantial challenges to scientific application developers ( e.g. @xcite ) . while selected algorithms and applications have in fact been demonstrated to keep up with the shiny performance promises of gpus , in some cases even at the very large scale ( e.g. @xcite ) , it remains to be seen whether a broader class of scientific applications can take advantage of gpu - accelerated systems with reasonable programming effort and in a sustainable way . often inappropriately termed `` legacy applications '' in this context , leading scientific hpc codes are typically being actively developed , comprise many tens or hundreds of thousands of lines of code achieved by a team effort of many dozens of person years , they provide state - of - the - art functionality , as well as high optimization , parallel scalability and portability . the codes gene and vertex , which have been developed in the max - planck - society with continuous support from its high - performance computing centre ( rzg ) may serve as prototypical examples in this respect . but as a matter of fact , such highly tuned codes are often reaching the limits of ( strong ) scalability . for example , due to increasing inter - node communication times , or , as in the case of vertex , due to the lack of conventionally exploitable parallelism in the code structure , the time to solution for a given setup can no more be efficiently reduced by utilizing more cpu resources . thus , a significantly increased node performance due to accelerators appears as a promising route towards further boosting application performances at scale . although both , gene and vertex are written in fortran ( with mpi and hybrid mpi / openmp parallelization , respectively ) we decided to adopt the c - based cuda programming model because it is the performance reference for nvidia gpus . while we found the commercial cuda - fortran language to deliver competitive performance on the gpu , the employed pgi compiler falls behind the intel compiler ( which is our reference for the cpu ) on the remaining cpu parts which marginalizes the overall application speedups of the heterogeneous code . for the same reason we did not yet make productive use of the openacc programming model @xcite . the performance baseline for all comparisons is defined by a highly optimized , parallel cpu implementation of the respective algorithms . rather than quoting single - core speedups ( which , in our opinion is hardly meaningful in most cases ) our comparisons are always based on the same number of gpu cards and _ multicore _ cpus ( `` sockets '' ) . specifically , we compare the run time obtained on a certain number of nodes , each equipped with two intel xeon e5 - 2670 8-core cpus and two nvidia k20x gpus , with the run time measured with the original , parallel cpu code on the same number of nodes ( without gpus ) . gene @xcite is a massively parallel code for the simulation of plasma turbulence in fusion devices . the code solves the time - dependent , five - dimensional vlasov - maxwell system of equations on a fixed phase - space grid . depending on the physical problem , typical gene simulations take between a few days and many weeks , using thousands of cores on x86_64-based hpc systems . gene is open - source @xcite and has a world - wide user base . the gene algorithm employs coordinates aligned to the magnetic field lines in a fusion device like a tokamak . in this paper we use the so called _ x - global _ version , where all physical quantities are handled in a spectral representation with respect to the @xmath0 coordinate , which is the second of the three space dimensions @xmath1 ( radial ) , @xmath0 ( binormal ) and @xmath2 ( along the field line ) . the remaining phase - space coordinates are ( in this order ) the velocity along the field line @xmath3 and the magnetic moment @xmath4 ( see @xcite for details ) . although gene is able to handle any number of ion species and the electrons in the framework of gyrokinetics , we use for this paper only a single ion species , neutralized by electrons . for all performance comparisons a problem setup with a number of @xmath5 grid points is used . the starting point of this work was a profiling of the gene code ( svn revision 3440 ) , with the times given in table [ tab : profiling_snb ] . [ cols= " < , < " , ] table [ vertex : tab1]a shows that the run time of vertex on the cpu is dominated by solving the radiative transfer equations ( item ` transport ` ) , and in particular for computing neutrino absorption and emission rates ( item ` rates ` ) . [ vertex : fig1]a identifies the positions of the individual routines in the execution flow . about 50 percent of the run time is spent in the computation of one particular interaction rate ( named ` rate kernel ` , ` c2 ` ) . the different interaction rates are often termed `` local physics '' , which expresses the fact that the computations are to a high degree independent of each other and provide a data parallelism on the grid level . different interaction processes ( ` rates ` ) can be computed independently of each other , which implies additional , coarse - grained parallelism on the function level ( see blowup in fig . [ vertex : fig1]a ) . in the following the algorithm for offloading the ` rate kernel ` to the gpu is outlined . due to its dominance in the code , high data parallelism and arithmetic intensity the suitability for the gpu shall become immediately apparent . as input for the computations a few one - dimensional arrays are needed , which represent the local thermodynamic conditions for which the interaction kernel is evaluated . all operations are performed on a five - dimensional grid representing discretized phase space . the size of this grid varies with the resolution , in a typical setup the total number of grid points is about @xmath6 . for the major part of the kernel , computations on each grid zone can be done independently of the others , which leads to a high degree of data parallelism ( up to @xmath7 threads ) . only after all grid zones are processed , a reduction ( corresponding to a phase space integral ) to a three dimensional grid is performed . this can be still done in parallel , but with much less parallelism ( @xmath8 threads ) . all computations are done twice for subsets of different input data , accounting for two possible reaction channels . the actual implementation of the part ` c2 ` is straightforward : the data is copied asynchronously to the gpu and the five - fold nested loops of the cpu version are separated in kernel calls with about 100000 threads . the kernels are scheduled in streams , in order to allow the cuda run time to overlap kernel executions corresponding to the twofold computation of the processes . the problem is compute bound , as data transfer is negligible ( 0.9 ms ) compared to gpu computations ( 40 ms ) and at least 140 double - precision floating - point operations are executed per transferred byte . for good performance results it turned out to be crucial to use shared memory for the input data and to use as much registers as possible on the device . after tuning our cuda code with the help of the nvidia profiler , we achieve an occupancy of 93% of the theoretical upper limit for the most important kernels . however , we still encounter about 10% of branch - divergence overhead and 25% of global memory replay overhead . work is still ongoing to improve on the latter performance metrics . as mentioned above , the different sub - steps ` c1 ` to ` cn ` ( see fig . [ vertex : fig1]b ) are independent of each other and can be computed in any order within one openmp thread , or ray. in the original code , however , the order across different openmp threads is always the same , e.g. when a thread computes sub - step ` c1 ` , also the other threads work on the same sub - step . an overlap of computations on the cores and the gpu was thus achieved by : a ) individually shuffling the computations of the sub - steps ` c1 ` to ` cn ` on each ray , and b ) ensuring that the sub - step ` c2 ` from each ray to the gpu is offloaded in a queue ( see fig . [ vertex : fig1]b ) . in an ideal situation where all steps ` c1 ` to ` cn ` take the same amount of execution time , work on the cpus and the gpu would be perfectly overlapped . in reality , a balancing of about 80% could be reached . the rate kernel ` c2 ` requires 2.16 s on one cpu thread ( cf . tab [ vertex : tab1 ] ) and scales almost perfectly with openmp . the same kernel can be computed on the gpu in 0.04 s. thus , with one gpu , speedups of 7 or 54 are achieved when comparing with one cpu socket or a single core , respectively . this demonstrates that a significant speedup was achieved with respect to a sandy bridge cpu . as the coarse grained openmp parallelization of vertex ( which is crucial for achieving its excellent weak scalability ) does not allow to use the threaded rate kernel on the cpu , the acceleration factor of 54 applies for production applications which effectively eliminates the rate kernel from the computing time budget and in practice accounts for a twofold acceleration ( corresponding to the original 50% share of the rate kernel , cf . tab [ vertex : tab1 ] ) of the entire application . with the specific cases of gene and vertex we have shown that complex hpc applications can successfully be ported to heterogeneous cpu - gpu clusters . besides writing fast gpu code , exploiting and balancing both the gpu _ and _ the cpu resources of the heterogeneous compute nodes turned out to be an essential prerequisite for achieving good overall `` speedups '' , which we define as the ratio of the run time obtained on a number of gpu - accelerated sockets and the run time measured with parallel code on the same number of cpu sockets . in the case of vertex we have demonstrated twofold speedups which hold for production applications on gpu - clusters with many hundreds of nodes . in particular , the excellent weak scalability of vertex @xcite is not affected by the additional acceleration due to gpus . threefold speedups appear in reach but would require at least additional porting of a linear solver for a block - tridiagonal system . limitations in the software environment ( lack of device - callable lapack functionality ) have so far impeded a successful port of this part of the algorithm . importantly , due to the specific code structure of vertex , such speedups would not have been possible with comparable programming effort by simply using more cpu cores . the performance of gene is currently limited by the data transfer between the host cpu and the gpu as we have shown by an elaborate performance - modeling analysis . after this bottleneck will have relaxed by upcoming hardware improvements ( pcie 3 ) further optimization efforts on the gpu code will increase the overall speedups on a heterogeneous cluster . the question whether the effort of several person - months , which we have invested for each code , and which we consider typical for such projects , is well justified can not be answered straightforwardly . for complex , and `` living '' scientific hpc codes , for which gene and vertex can serve as prototypical examples , achieving up to threefold speedups in overall application performance appears very competitive @xcite . also from the point of view of hardware investment ( buying gpus instead of cpus ) and operational costs ( `` energy to solution '' ) the migration of applications from pure cpu machines to gpu - accelerated clusters can be considered cost - effective if speedups of at least about two are achieved . on the other hand , while very valuable for increasing simulation throughput , twofold or threefold application speedups usually do not enable qualitatively new science objectives . for this reason we sometimes observe reluctance in the scientific community to invest significant human resources for achieving gpu - performance improvements in this range . this is further exacerbated by legitimate concerns about sustainability , maintainability and portability of gpu - kernel code . these are no serious issues for gene and vertex , where the parts we have ported to the gpu are not under heavy algorithmic development and were also carefully encapsulated by us . in general , however , the _ need _ for kernel programming , which is considered as a pain by many , currently appears as the largest hurdle for a broader adoption of gpu programming in the scientific hpc community . moreover , it may turn out necessary to port significant parts of the application code to the gpu , e.g. in cases like gene where the data transfers become a limiting factor , or even to completely reimplement the application . these concerns could be mitigated by the establishment of a high - level , directive based programming model , based e.g. on the openacc standard @xcite or a future revision of openmp @xcite , together with appropriate compiler support . also intel s xeon phi many - core coprocessor with its less disruptive programming model appears very prospective in this respect . despite serious efforts , however , we were not yet successful with gene or vertex to achieve performances on this platform which are competitive with the gpu . we attribute this mostly to a comparably lower maturity of the xeon phi software stack and we expect improvements with upcoming versions of the compiler and the openmp run time . most importantly , today s gpus ( and many - core coprocessors ) might provide a first glimpse on the architecture and the related programming challenges of future hpc architectures of the exascale era @xcite . applications need to be prepared _ in time _ for the massive simt and simd parallelism which is expected to become prevalent in such systems . even on contemporary multicore cpus with comparably moderate thread - counts and simd width , the experience we have gained with porting gene and vertex has already led to appreciable performance improvements of the cpu codes . we thank f. jenko and h .- th . janka for encouraging the development of gpu versions for gene and vertex , respectively . nvidia corp . and intel corp . are acknowledged for providing hardware samples and technical consulting . buras , r. , janka , h .- th . , rampp , m. , kifonidis , k. _ two - dimensional hydrodynamic core - collapse supernova simulations with spectral neutrino transport . i. numerical method and results for a 15 m@xmath9 star_. astronomy & astrophysics * 447 * , 1049 ( 2006 ) . cardall , c. , endeve , e. , budiardja , r. d. , marronetti , p. , mezzacappa , a. _ towards the core - collapse supernova explosion mechanism_. advances in computational astrophysics : methods , tools , and outcome . asp conference proceedings , * 453 * , 81 ( 2012 ) . shimokawabe , t. , aoki , t. et al . _ peta - scale phase - field simulation for dendritic solidification on the tsubame 2.0 supercomputer_. proceedings of 2011 international conference for high performance computing , networking , storage and analysis , sc 11 , acm new york , ny , us ( 2011 ) 3:13:11
|
we have developed gpu versions for two major high - performance - computing ( hpc ) applications originating from two different scientific domains .
gene @xcite is a plasma microturbulence code which is employed for simulations of nuclear fusion plasmas .
vertex @xcite is a neutrino - radiation hydrodynamics code for `` first principles''-simulations of core - collapse supernova explosions @xcite .
the codes are considered state of the art in their respective scientific domains , both concerning their scientific scope and functionality as well as the achievable compute performance , in particular parallel scalability on all relevant hpc platforms .
gene and vertex were ported by us to hpc cluster architectures with two nvidia _ kepler _ gpus mounted in each node in addition to two intel xeon cpus of the _ sandy bridge _ family .
on such platforms we achieve up to twofold gains in the overall application performance in the sense of a reduction of the time to solution for a given setup with respect to a pure cpu cluster .
the paper describes our basic porting strategies and benchmarking methodology , and details the main algorithmic and technical challenges we faced on the new , heterogeneous architecture . , and gpu , hpc application , gene , vertex
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
despite years of study , we still do not know how the seeds of supermassive black holes formed . few if any of the pathways in martin rees s famous flow chart ( begelman & rees 1978 ) can be ruled out , but none of the routes is particularly well understood , either . what we do know is that some very massive ( @xmath2 ) black holes had to exist by @xmath3 in order to explain early quasars ( fan 2006 ) . if the seeds of these black holes were the remnants of massive stars , then they must have grown by eddington - limited accretion for most of the time since their formation , or else much of their growth was due to mergers . a second possibility is that the seeds formed by such a rapid accumulation of matter that it may be considered to be a direct collapse . i will focus on the latter possibility in this paper . the pop iii star formation processes we have heard about at this conference result from the infall of gas at rates @xmath4 yr@xmath5 . what would happen if the infall rate were much higher ? the entropy of matter laid down by gravitational infall onto a growing central mass increases with time . at low inflow rates , however , nuclear ignition halts the contraction of the core and raises the entropy in the interior , leading to a high - entropy object a star . if the inflow rate is high enough , however , the core will be so tightly bound by the time nuclear reactions start that the energy release will be insufficient to halt core contraction . in this case , we are left with an object with a low - entropy core and a high - entropy envelope . this is the situation that can lead to the direct formation of a black hole , without a stellar precursor . this situation should apply when the infall rate exceeds a few tenths of a solar mass per year ( begelman et al . 2006 ) , although more work needs to be done to refine this estimate . the conditions under which such high inflow rates might occur are very uncertain . such rapid infall would almost certainly be driven by gravitational torques , which could be local ( gammie 2001 ) or global ( as in the bars within bars " mechanism : shlosman et al . the natural " gravitational inflow rate is given by @xmath6 yr@xmath5 , where @xmath7 represents the internal velocity dispersion ( @xmath8 turbulent or sound speed ) for a locally unstable thin disk and the orbital speed for a globally unstable ( fully self - gravitating ) system . this means that very large inflow rates are possible in dark matter haloes with velocity dispersions exceeding about 10 km s@xmath5 , which have masses exceeding @xmath9 and become common at redshifts @xmath10 . global gravitational instabilities could occur in a significant fraction of such haloes if the gas is unable to cool much below the virial temperature ( @xmath11 k ) , which requires both that they have been spared significant metal enrichment and that h@xmath12 formation is suppressed ( bromm & loeb 2003 ; begelman et al . however , recent calculations by wise & abel ( 2007 , and these proceedings ) suggest that molecular hydrogen formation is inevitable . this may have the additional effect that many more haloes formed pop iii stars at higher redshifts , making it harder to avoid metal enrichment and possibly depleting the supply of gas available for infall as haloes merge . rapid infall may be easier in more massive haloes . there will be a larger disparity between the thermal temperature of the gas and the virial temperature , but presumably the gas will form a multiphase structure with turbulent velocities ( perhaps driven by stellar energy sources ) dominating the internal energy much like the interstellar medium . simulations are beginning to elucidate these structures , but we have a long way to go before we understand the tradeoffs between large scale inflow and in situ fragmentation and star formation . inflow may be stimulated by the large - scale gravitational torques associated with mergers . a key element in the simulations of black - hole growth and fuelling ( e.g. , di matteo et al . finally , we note that such inflow rates must be possible , because they are required to power quasars . the open question is whether they can also occur when the black hole is absent , or very small . for the remainder of this article we will assume that the answer is affirmative . once the mass of accumulated gas exceeds a few solar masses , radiation pressure dominates the envelope . the core , with gas pressure comparable to radiation pressure , maintains a roughly constant mass @xmath13 . the boundary between dynamic infall and quasistatic contraction is close to the radius where infalling gas liberates energy at the eddington limit . for a constant infall rate of @xmath14 , this radius turns out to be constant , @xmath15 au . the core shrinks under the increasing pressure of the envelope , @xmath16 , where @xmath17 is the envelope mass in solar units . one can sketch the likely interior structure of the envelope using scaling arguments . since the interior is undergoing kelvin helmholtz contraction , the density profile adjusts so that the diffusion timescale @xmath18 is of order the elapsed time at all radii . this means that the specific entropy of any mass shell , @xmath19 , declines slowly ( logarithmically ) with time , and is never far below the value it had when the mass shell was added . in the radiation pressure - dominated envelope , @xmath20 , where @xmath21 and @xmath22 are the radiation and gas pressure , respectively . there are then two possible scaling laws for the envelope structure . if @xmath23 , then @xmath24 , @xmath25 ( for uniform opacity @xmath26 ) , and the envelope joins smoothly onto the core . this was the result presented in begelman et al . the other possible scaling law has @xmath27 , i.e. , nearly isothermal structure . the density profile in the envelope is then @xmath28 , and there is a large jump in density and temperature going from the envelope to the core . the latter structure seems to be the one that emerges naturally from nearly self - similar " models for the core+envelope , which neglect rotation . given the manner in which the mass accumulates , though , rotation is bound to be important . moreover , even a small amount of it can affect the structure dramatically , increasing the binding energy of the gas ( which is otherwise very weakly bound since the mean equation of state has @xmath29 ) and protecting it from pulsational instabilities . including the effects of rotation will require the modeling of angular momentum transport and dissipation , but these complications are not prohibitive . it will be interesting to see which of the scalings applies when rotation is included . fortunately , the behavior of the core is insensitive to the structure of the envelope . since the core is largely dominated by gas pressure , its temperature must track the virial temperature , @xmath30 . rapid nuclear burning starts in the core when the envelope mass is @xmath31 and the density is not that different from the cores of main sequence stars , but the system passes through this phase so quickly ( @xmath32 yr ) that the energy release has little effect . by the time the envelope mass reaches several thousand solar masses , the core temperature has climbed to a few @xmath33 k. because of the extremely deep potential well created by the envelope , the energy released by nuclear reactions at this stage is unable to unbind the gas ( especially if rotation has increased its binding energy ) and the core proceeds to the temperatures ( @xmath34 k ) at which runaway neutrino losses occur . at this point the core loses pressure support and collapses to form a @xmath35 black hole . at the time of its formation , the black hole is embedded in an envelope of more than a hundred times its mass . if it were limited to accreting at the eddington limit for the mass of the black hole , the pressure of the escaping radiation would have essentially no effect on the envelope . in fact , the pressure of the envelope should be able to drive the accretion rate up to the point where the liberated luminosity approaches the eddington limit _ for the mass of the envelope_. relative to the mass of the black hole , the accretion is super - eddington by a factor @xmath36 . accretion inside the massive envelope ( which itself continues to grow at a rate @xmath37 ) can lead to very rapid growth of the black hole . the eddington ratio for black hole growth is @xmath38 where @xmath39 is the accretion efficiency . for an initial black hole mass of @xmath40 , the e - folding time is a thousandth of the salpeter time , or only @xmath11 yr , and super - eddington growth appears possible up to black hole masses @xmath41 . however , this estimate does not take fully into account the back reaction of black hole growth on the envelope . we will see below that the era of rapid growth is limited to much smaller ( but still interesting ! ) black hole masses . the energy liberated by accretion has to escape . since the accretion flow is rotating , some of it could exhaust through a low - density funnel . but much of it presumably percolates through the accretion flow and envelope . thus , the accreting black hole provides an energy source for the envelope , which is therefore a kind of star - like object which we have dubbed a quasistar " ( begelman et al . 2006 ) . since there is a limit to the outward energy flux that the accreting gas can carry , the accretion rate is regulated at a fraction @xmath42 of the bondi rate , where @xmath43 is the sound speed at the bondi radius ( gruzinov 1998 ; blandford & begelman 1999 ; narayan et al . 2000 ; quataert & gruzinov 2000 ) . the accretion rate , and associated energy flux , can be expressed in terms of the temperature and density deep within the quasistar , but outside the black hole s sphere of influence . the energy flux must also equal the eddington limit for the entire quasistar , @xmath44 ( where we now use @xmath45 to denote the mass of the quasistar ) the interior structure adjusts so that this is satisfied . since the eddington limit is lower inside the quasistar ( where the enclosed mass is lower ) , the quasistar s interior is strongly convective and can be modeled by an @xmath46 polytrope . ( strictly speaking , one should use a loaded polytrope " model [ huntley & saslaw 1975 ] , taking into account the mass of the black hole , but the loaded and unloaded models converge in the region of interest when @xmath47 . ) the interior structure of a quasistar is shown schematically in fig . 1 . accretes gas from a massive , radiation pressure - supported envelope at a rate set by conditions outside the bondi radius . the luminosity liberated by the accretion process is transported convectively in the inner regions of the envelope , with a transition to a radiative zone where convection becomes inefficient . we illustrate rotational flattening and ongoing disk accretion at a fraction of a solar mass per year . ] according to the polytropic relations , the density and temperature outside the bondi radius are uniquely related to the mass and radius ( @xmath48 ) of the quasistar . expressing the accretion rate in terms of these quantities and setting the luminosity equal to @xmath44 , we can express @xmath48 in terms of @xmath45 and @xmath49 : @xmath50 where the masses are expressed in solar units and @xmath51 parametrizes the efficiency of angular momentum and energy transport inside the bondi radius ( begelman et al . the photospheric temperature is @xmath52 since the black hole growth rate is proportional to @xmath45 , while the quasistar mass increases linearly with time , @xmath53 at late times . this implies that the quasistar s radius grows with time , and its photosphere becomes cooler . ( the result holds even if the quasistar mass is fixed . ) the appearance of a quasistar differs dramatically from that of the pre - black hole envelope . shortly after the black hole forms the envelope expands to @xmath54 au ( from @xmath55 au ) , and the interior temperature drops to @xmath56 k , quenching all nuclear reactions . a quasistar resembles a red supergiant , except that it is radiation pressure supported and its energy source is accretion . in some respects it is reminiscent of a very massive thorne - ytkow ( 1977 ) object , but with crucial differences . besides being dominated by radiation pressure and powered exclusively by accretion , quasistars have a distributed rather than shell - like energy source , which can not be regulated by the slow settling that characterizes accretion onto a neutron star . as a result , quasistars come to grief if their photospheric temperatures get too low . , the darker shaded region is for @xmath57 , while the dashed line is for an analytic toy " opacity model . superimposed on the figure are evolutionary tracks for an accretion rate onto the envelope of @xmath58= 1 @xmath59 and @xmath60= 0.1@xmath61 . the evolution pushes the quasistar into the forbidden region of parameter space , where it evaporates . ] begelman et al . ( 2007 ) computed quasistar models using pop iii opacities from mayer & duschl ( 2005 ) . as in red giants with standard abundances , there is a minimum photospheric temperature associated with a sharp drop in opacity . because of the absence of metals , the opacity crisis " occurs at a somewhat higher temperature around 4000 k than for metal - enriched compositions . like the hayashi track ( hayashi 1961 ; hayashi & hoshi 1961 ) for red giants and convective protostars , the minimum temperature of quasistars arises from the impossibility of matching the convective interior to the radiative zone and photosphere . the details are somewhat different , however , because radiation pressure dominates in quasistars , whereas the convective zones of ordinary red giants are gas pressure - dominated and resemble @xmath62 polytropes . as a quasistar crosses into the forbidden zone , the flux escaping from the convective interior exceeds the eddington limit and prevents the quasistar from maintaining hydrostatic equilibrium . 2 shows the forbidden zone in the @xmath63 plane , along with representative evolutionary tracks . unlike ordinary red giants reaching the hayashi track , quasistar photospheres can not hover stably at close to the minimum temperature . fixing the photospheric temperature , we find that the eddington ratio scales as @xmath64 thus , growth of the black hole exacerbates the dynamical imbalance , as does partial evaporation of the envelope , rapid accretion onto the envelope also would not be able to stabilize the quasistar for very long . ] the opacity crisis is an unstable situation . we predict that quasistars entering the forbidden zone must evaporate rather quickly . there is a complication due to a bump in the opacity from bound - free transitions , which creates a narrow region in the radiative zone where the luminosity is super - eddington . a similar problem arises in models of luminous blue variables ( e.g. , owocki et al . 2004 ) , and remains unresolved . however , it does not seem likely that this feature alone will lead to catastrophic mass loss in the case of quasistars . the rapid infall of gas in galactic nuclei or pregalactic haloes provides a means for forming seed black holes and rapidly growing them into the intermediate mass regime . for pop iii quasistars with inflow rates @xmath65 yr@xmath5 , and simple assumptions about parameters like @xmath66 , the black holes could reach masses @xmath67 before the quasistar evaporates . the quasistar masses could be as large as @xmath68 . metal - rich quasistars could reach somewhat lower temperatures , but it is not clear whether this implies larger black hole masses , because the run of opacity in the radiative zone would be more complex and dust formation might lead to enhanced mass loss . are quasistars detectable ? if so , they would be seen at their most massive , shortly before they evaporate . spectrally , they would resemble 20004000 k blackbodies , depending on metallicity , and in the pop iii limit their spectra would be featureless . since they radiate at the eddington limit corresponding to the opacity at the convective radiative transition which is close to that of electron scattering their luminosities could reach @xmath69 erg s@xmath5 . however , their short lifetimes ( @xmath70 yr ) would make them fairly rare . quasistars can exist only when the envelope mass greatly exceeds the mass of the embedded black hole . therefore , they are unlikely to form in the nuclei of galaxies that already possess a supermassive black hole in this case , a period of rapid infall ( e.g. , following a merger ) would presumably trigger a quasar outburst instead . however , low - redshift quasistars might conceivably form in galactic nuclei in which the black hole had been ejected due to three - body interactions , or in which a black hole had never formed . the earliest plausible sites of quasistar formation would have been pregalactic haloes with virial temperatures exceeding @xmath71 k. these would have been most common at redshifts @xmath72 . the spectra of pop iii quasistars at these redshifts would peak at about 10@xmath73 m , but they might be marginally detectable by _ james webb space telescope _ on the wien tail at @xmath74 m . if direct collapse in @xmath71 k haloes is a principal route for forming supermassive black hole seeds , there could be as many as 110 per _ jwst _ field , but identifying them would be an extreme challenge . the work described here was done in collaboration with marta volonteri , elena rossi , martin rees , and phil armitage , and was supported in part by nsf grant ast-0307502 , nasa s beyond einstein foundation science program grant nng05gi92 g , and a university of colorado faculty fellowship . the author thanks the director of the institute of astronomy and the master and fellows of trinity college , cambridge , for their hospitality during a sabbatical visit .
|
rapid infall of gas in the nuclei of galaxies could lead to the formation of black holes by direct collapse , without first forming stars .
black holes formed in this way would have initial masses of a few @xmath0 , but would be embedded in massive envelopes that would allow them to grow at a highly super - eddington rate .
thus , seed black holes as large as @xmath1 could form very rapidly .
i will sketch the basic physics of the direct collapse process and the properties of the accreting envelopes .
address = jila , 440 ucb , university of colorado at boulder , boulder , co 80309 - 0440 usa
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
diffraction by an infinite wedge is a fundamental canonical problem in acoustic scattering . exact closed - form frequency - domain solutions for point source , line source or plane wave excitation with homogeneous dirichlet ( sound soft ) or neumann ( sound hard , or rigid ) boundary conditions are available in many different forms@xcite . for example , series expansions in terms of eigenfunctions are available for near field calculations ( e.g. for analysing edge singularities ) . contour integral representations over so - called sommerfeld - malyuzhinets contours are better suited to far field computations ( e.g. for deriving diffraction coefficients in computational methods such as the geometrical theory of diffraction @xcite ) . more recently it has been discovered that the ` diffracted ' component of these solutions ( precisely , that which remains after subtracting from the total field the geometrical acoustics terms ) can be expressed in a more physically intuitive form , namely as a line integral superposition of directional secondary sources located along the diffracting edge @xcite . ( in fact the frequency domain expressions derived in ref . @xcite had appeared already in ref . @xcite , but the interpretation in terms of secondary edge sources seems to have been first made in ref . @xcite . ) one appealing feature of the edge source interpretation is that it offers a natural way to write down approximate solutions for finite edges , simply by truncating the domain of integration . it has also led to edge integral equation formulations of scattering problems , where the integral equation is posed on the union of all the scatterer s edges @xcite . for dirichlet and neumann boundary conditions the edge source formulations are now well understood : efficient numerical evaluation of the line integrals has been considered in ref . @xcite using the method of numerical steepest descent , as has the behaviour of the line integrals near shadow boundaries @xcite and edges @xcite . note also ref . @xcite , where the corresponding time - domain case is considered . for the more difficult case of diffraction by a wedge with impedance ( absorbing ) boundary conditions , some exact solutions are also known . for example , the case of plane wave incidence on an impedance wedge can be solved using the sommerfeld - malyuzhinets technique , and converted to a series expansion using a watson - type transformation ( see ref . @xcite and the references therein ) . but the solution obtained is much more cumbersome than those for the corresponding dirichlet and neumann problems , and the technique requires the solution of a certain non - trivial functional difference equation . this increased complexity is perhaps to be expected , since the physics of the impedance problem are fundamentally more complicated than those for the dirichlet and neumann problems ; in particular , the wedge faces can under certain conditions support surface waves . however , for the special ( yet important ) case of a right - angled wedge , the solution takes a particularly simple and explicit form@xcite . in ref . @xcite , rawlins proves that the solution to the impedance problem for a right - angled wedge ( with possibly different impedances on each face ) can be obtained from that of the corresponding dirichlet problem , generalised to allow complex incident angles , by the application of a certain linear differential operator ( see eqs . below for details ) . rawlins applies this operator to the classical series and integral representations of the dirichlet solution to obtain relatively simple series and integral representations for the impedance solution . ( the solution for the case where the impedance is the same on both faces was presented previously in a similar but more complicated form in ref . @xcite . ) in this paper it will be shown that rawlins solution for the impedance wedge can be transformed into an edge source representation of the same form as those derived for rigid ( sound - hard ) wedges in ref . this appears to be the first edge source representation for diffraction by an impedance wedge . while the solution obtained is valid only for a right - angled wedge , it should be remarked that this special case is ubiquitous in many acoustical applications ( e.g. , urban acoustics @xcite ) . the edge source formulation for ideal ( dirichlet and neumann ) wedges will briefly be reviewed . for the most general setting ( illustrated in fig . 1 ) , consider a point source @xmath0 and point receiver @xmath1 in the presence of a wedge of exterior angle @xmath2 . let @xmath3 denote cylindrical coordinates with the @xmath4-axis along the edge , the propagation domain occupying the region @xmath5 , and the wedge the region @xmath6 . consider also cartesian coordinates @xmath7 with @xmath8 , @xmath9 , @xmath10 . without loss of generality it will be assumed that the receiver is located in the plane @xmath11 , at @xmath12 . for each point @xmath4 on the edge one can also introduce local spherical coordinates @xmath13 , with @xmath14 defined as before and @xmath15 , @xmath16 , @xmath17 . for consistency with ref . @xcite the time - dependence @xmath18 will be assumed throughout . then the diffracted field at @xmath1 ( i.e. the total field minus the geometrical acoustics field ) due to a monopole source at @xmath19 can be written as a line integral @xmath20 over edge positions @xmath4 , where @xmath21 is the wedge index @xcite . the integral in eq . can be interpreted as a superposition of secondary edge sources along the edge . the factor @xmath22 can be interpreted as a directivity function , and takes the following forms for dirichlet ( @xmath23 ) and neumann ( @xmath24 ) boundary conditions : @xmath25 where @xmath26 @xmath27 and the auxiliary function @xmath28 is @xmath29 the second expression for @xmath28 in eq . shows that @xmath22 is a function only of the local spherical angles @xmath30 ; this justifies the interpretation of @xmath22 as a directivity function . the formula for @xmath31 above corrects that in ref . @xcite by reversing the sign of @xmath31 . also , the first expression for @xmath28 in eq . ( [ eq : etadef ] ) corrects a sign error in the corresponding formula in ref . @xcite . a mixed wedge with dirichlet conditions on @xmath32 and neumann conditions on @xmath33 can also be treated , by summing the dirichlet solutions for a wedge angle @xmath34 and source positions @xmath35 and @xmath36 respectively . this gives the directivity factor @xmath37 where @xmath38 which , when inserted in eq . , agrees with the solution derived by buckingham in eqs . ( 62)-(63 ) of ref . @xcite using a modal expansion . the solution for a mixed wedge with neumann conditions on @xmath32 and dirichlet conditions on @xmath33 can then be obtained by replacing @xmath14 by @xmath39 and @xmath35 by @xmath40 in eq . . for the two - dimensional case of plane wave incidence perpendicular to the edge , the source is placed at @xmath41 with @xmath42 . in this case @xmath43 and , because of symmetry , the integration range in eq . ( [ eq : basicintegral ] ) can be halved . furthermore , for plane wave incidence the @xmath44 spherical attenuation factor is removed , and , if one refers the phase of the diffracted sound pressure relative to the arrival time at the edge , the phase oscillation factor @xmath45 also disappears , giving @xmath46 for plane wave incidence perpendicular to a right - angled wedge , for which @xmath47 ( i.e. @xmath48 ) , further simplification is obtained . now . is @xmath49 and using multiple angle formulas one can show that @xmath50 where @xmath51 to summarise , for plane wave incidence perpendicular to the edge of a right - angled wedge , with @xmath52 , the total field is @xmath53{{\rm e}}^{-{{\rm i}}kr\cos(\theta-\theta_0)}\notag \\ & \quad+ r_1{{\rm h}}[\pi-\theta-\theta_0]{{\rm e}}^{-{{\rm i}}kr\cos(\theta+\theta_0)}\notag \\ & \quad+ r_2{{\rm h}}[\theta+\theta_0 - 2\pi]{{\rm e}}^{{{\rm i}}kr\cos(\theta+\theta_0)}\notag \\ & \quad+ p_d(r,\theta ) , \label{eqn : ptotal}\end{aligned}\ ] ] where the diffracted field @xmath54 is given by eq . and the remaining three terms represent the geometrical acoustics field ( here @xmath55 $ ] is the heaviside step function : @xmath55=1 $ ] , @xmath56 ; @xmath55=1/2 $ ] , @xmath57 ; @xmath55=0 $ ] , @xmath58 ) . the first term in eq . represents the incident field , the second the reflected wave from the face @xmath32 and the third the reflected wave from @xmath33 . @xmath59 are reflection coefficients : for the pure dirichlet case @xmath60 ; for the pure neumann case @xmath61 ; and for the mixed dirichlet / neumann and neumann / dirichlet cases @xmath62 , @xmath63 and @xmath64 , @xmath65 , respectively . in order to understand how eq . should be generalised to the case of impedance boundary conditions , it is instructive to review its derivation from the classical sommerfeld contour integral solution . part of this derivation was presented already in ref . @xcite , but in this section the analysis of ref . @xcite will be extended to allow a complex incident angle @xmath35 , corresponding physically to diffraction by an inhomogeneous plane wave . for brevity , attention will be restricted to the dirichlet case , but the neumann and mixed problems can be analysed similarly . for the dirichlet case the sommerfeld contour integral solution is @xcite @xmath66 where the integration contour is in two parts : @xmath67 lies above all singularities of the integrand and goes from @xmath68 to @xmath69 , and @xmath70 lies below all singularities of the integrand and goes from @xmath71 to @xmath72 ( see fig . [ fig : gamma ] ) . the integral converges rapidly for all complex @xmath14 and @xmath35 , and the integrand has poles at @xmath73 for @xmath74 . as is pointed out in ref . @xcite , the formula in eq . makes sense not just for real @xmath75 , but also for all complex @xmath35 . ] one can obtain an expression suitable for far field ( large @xmath76 ) evaluation by deforming the contour in eq . onto the steepest descent contours @xmath77 and @xmath78 passing through the saddle points at @xmath79 ( see fig . [ fig : gamma ] ) . these contours are defined by @xmath80 respectively , where @xmath81 is the gudermannian function with @xmath82 , @xmath83 for @xmath84 , and @xmath85 . to achieve the deformation one draws down the contour @xmath67 onto @xmath70 , picking up residue contributions from any poles lying between @xmath77 and @xmath78 . assuming that @xmath86 and @xmath87 , one obtains@xcite @xmath88{{\rm e}}^{-{{\rm i}}k r \cos(\theta-\theta_0)}\notag\\ & \;\;-{{\rm h}}[\pi-|\theta+\re{\theta_0}+\mathrm{gd}(\im{\theta_0})|]{{\rm e}}^{-{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \;\;-{{\rm h}}[\pi-|\theta+\re{\theta_0}+\mathrm{gd}(\im{\theta_0})-3\pi|]{{\rm e}}^{{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \;\;+ p^d_d(r,\theta ) , \label{eqn : sommtotal}\end{aligned}\ ] ] where @xmath89 @xmath90 and the integrals over @xmath77 and @xmath78 have been combined into a single integral over the contour @xmath91 illustrated in fig . [ fig : contoura ] by the changes of variable @xmath92 . note that eq . corrects a sign in one of the exponents in the corresponding formula in ref . @xcite ( eq . ( 26 ) on p. 166 ) . note also that in ref . @xcite rawlins uses the equivalent representation @xmath93 the first term in eq . represents the incident wave , and the second and third terms in eq . represent the two reflected waves . note that the location of the zone boundary across which these waves are switched on / off by the heaviside function prefactors is shifted compared to the case of purely real @xmath35 . the shift agrees exactly with that derived in ref . @xcite for the case of an inhomogeneous plane wave incident on a half plane . note also that at a zone boundary the argument of one of the heaviside functions in eq . equals zero , and a pole lies on the contour of integration . in this case a principal value integral should be taken in eq . ( for consistency with the assumption that @xmath94=1/2 $ ] ) ; the same convention applies to the other decompositions in eqs . , , and . the edge source representation can then be obtained from eq . as follows . first deform the contour of integration from @xmath91 to the imaginary axis @xmath95 in fig . [ fig : contoura ] ( equivalently , deform the original contours @xmath67 and @xmath70 onto the vertical contours @xmath96 and @xmath97 in fig . [ fig : gamma ] rather than @xmath77 and @xmath78 before changing variables ) . then write the resulting integral as an integral over the positive imaginary axis only , applying the identity@xmath98 which follows from the identity @xmath99 finally , parametrise the resulting integral by @xmath100 so that @xmath101 and @xmath102 , giving @xmath103{{\rm e}}^{-{{\rm i}}k r \cos(\theta-\theta_0)}\notag\\ & \quad-{{\rm h}}[\pi-|\theta+\re{\theta_0}|]{{\rm e}}^{-{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \quad-{{\rm h}}[\pi-|\theta+\re{\theta_0}-3\pi|]{{\rm e}}^{{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \quad+ p^d_{d,{\rm edge}}(r,\theta ) , \label{eqn : sommtotaledgesource}\end{aligned}\ ] ] where @xmath104 is given by the edge source integral in eq . . clearly eq . reduces to eq . when @xmath52 . for complex @xmath35 there is a discrepancy in the arguments of the heaviside functions between eqs . and ; this is simply a consequence of the contour deformation from @xmath91 to @xmath95 . accordingly , eq . does not represent a decomposition of the field into ` geometrical acoustics ' and ` diffracted ' components , at least not in the sense understood in ref . such discrepancies will have implications for the physical interpretation of the edge source integral derived for the impedance problem in the following section . the main aim of this paper is to generalise eq . to the case of impedance boundary conditions , @xmath105 where the unit normal vector @xmath106 points into the wedge , and @xmath107 is the complex admittance ( inversely proportional to the impedance ) , which is assumed to satisfy @xmath108 , so as to prohibit energy creation at the boundary . it will be assumed that @xmath107 takes a constant value on each of the two wedge faces , but that the value of this constant may be different for the two wedge faces . following ref . @xcite , in order to simplify later formulas @xmath107 will be written @xmath109 where @xmath110 are complex angles such that @xmath111 recalling that the wedge faces are given respectively by @xmath112 , @xmath113 ( @xmath32 ) and @xmath114 , @xmath115 ( @xmath116 ) , the boundary condition in eq . can then be stated as @xmath117 \dfrac{\partial p}{\partial x } = { { \rm i}}k \cos{\theta_2}p , & \textrm{on } \theta=3\pi/2 . \end{cases}\end{aligned}\ ] ] note that when @xmath107 takes the same value on both faces the angles @xmath118 and @xmath119 are related by @xmath120 , since @xmath121 . rawlins shows that the total field @xmath122 for the impedance problem can be written as @xcite @xmath123 where @xmath124 and @xmath125 where @xmath126 , @xmath127 and @xmath128 denotes the dirichlet solution for incident angle @xmath129 ( this follows from eqs . ( 28)-(31 ) in ref . @xcite combined with standard trigonometric identities ) . for completeness note that in eq . ( 29 ) on p. 167 of ref . @xcite , @xmath130 in the numerator should be @xmath131 . rawlins derivation of eq . in ref . @xcite is based on a trick first introduced by williams in ref . @xcite to solve the analogous problem for a mixed neumann / impedance wedge . to evaluate the formula in eq . , rawlins uses the representation for @xmath128 given in eq . , but translates @xmath132 to give @xmath133 where @xmath134 is the translated version of @xmath91 passing through the saddle point at @xmath135 ( see fig . [ fig : contourb ] . this transformation greatly simplifies the application of the differential operator @xmath136 because the spatial variables @xmath137 and @xmath138 occur only in the exponential factor in the integrand in eq . , with @xmath139 . hence eq . can be evaluated as@xcite @xmath140{{\rm e}}^{-{{\rm i}}k r \cos(\theta-\theta_0)}\notag\\ & \quad+r_1^i{{\rm h}}[\pi-\theta-\theta_0]{{\rm e}}^{-{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \quad+r_2^i{{\rm h}}[\theta+\theta_0 - 2\pi]{{\rm e}}^{{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \quad+t_1^i{{\rm h}}[\pi-\theta-\re{\theta_1}-\operatorname{gd}(\im{\theta_1})]{{\rm e}}^{-{{\rm i}}k r \cos(\theta+\theta_1)}\notag\\ & \quad+t_2^i{{\rm h}}[\theta+\re{\theta_2}+\operatorname{gd}(\im{\theta_2})-2\pi]{{\rm e}}^{{{\rm i}}k r \cos(\theta+\theta_2)}\notag\\ & \quad+ p^i_d(r,\theta ) , \label{eqn : sommtotalimp}\end{aligned}\ ] ] where @xmath141 and ( changing variable back to @xmath142 ) @xmath143 the first term in eq . represents the incident field ; the second and third terms represent reflected waves from the two wedge faces ; the fourth and fifth terms represent surface waves propagating along the two wedge faces ; and the final term @xmath144 represents the diffracted field . a surface wave associated with the face @xmath32 is excited if @xmath145 , and in this case is confined to the angular region @xmath146 . similarly , a surface wave associated with the face @xmath116 is excited if @xmath147 , and in this case is confined to the angular region @xmath148 . in terms of the admittance parameter @xmath107 , recalling eq . one finds that surface waves are excited if @xmath149 the surface waves ( when they exist ) decay exponentially with increasing distance both perpendicular to and along the face with which they are associated ( unless @xmath107 is pure negative imaginary , in which case they maintain a constant amplitude along the face itself , decaying in the perpendicular direction ) . the diffracted field @xmath150 can be approximated in the far field using the method of steepest descent , giving@xcite @xmath151 as @xmath152 , where the diffraction coefficient @xmath153 note that eq . corrects a typographical error in ref . @xcite : in eq . ( 36 ) of ref . @xcite , @xmath154 in the denominator should be @xmath155 . note also that the approximation in eq . breaks down near zone boundaries ( i.e. , at values of @xmath14 for which the argument of one of the heaviside functions in eq . equals zero ) . a more sophisticated far field approximation , valid uniformly across the zone boundaries , is given in ref . @xcite . an edge source representation for @xmath144 can be derived by closely following the procedure outlined in section [ sec : dirderivation ] for the dirichlet case . first deform the contour of integration from @xmath91 to @xmath95 ( recall fig . [ fig : contoura ] ) . then write the resulting integral as an integral over the positive imaginary axis only . simplifying the resulting expression requires slightly more work than in the dirichlet case because the part of the integrand in eq . not involving @xmath156 is no longer an even function of @xmath142 as it was in the dirichlet case . to deal with this , first decompose @xmath157 into a sum of even and odd parts ( with respect to @xmath142 ) @xmath158 then deal with the contribution from the even part using eq . , and that from the odd part using the identity @xmath159 where @xmath160 with @xmath161 eq . follows from the identity @xmath162 finally , parametrising the contour using eq . gives an edge source representation for the impedance solution : @xmath140{{\rm e}}^{-{{\rm i}}k r \cos(\theta-\theta_0)}\notag\\ & \quad+r_1^i{{\rm h}}[\pi-\theta-\theta_0]{{\rm e}}^{-{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \quad+r_2^i{{\rm h}}[\theta+\theta_0 - 2\pi]{{\rm e}}^{{{\rm i}}k r \cos(\theta+\theta_0)}\notag\\ & \quad+t_1^i{{\rm h}}[\pi-\theta-\re{\theta_1}]{{\rm e}}^{-{{\rm i}}k r \cos(\theta+\theta_1)}\notag\\ & \quad+t_2^i{{\rm h}}[\theta+\re{\theta_2}-2\pi]{{\rm e}}^{{{\rm i}}k r \cos(\theta+\theta_2)}\notag\\ & \quad+ p^i_{d,{\rm edge}}(r,\theta ) , \label{eqn : sommtotalimpedgesource}\end{aligned}\ ] ] where @xmath163 and @xmath164 where @xmath165 and , for @xmath166 , @xmath167 and @xmath168 denote the functions @xmath31 and @xmath169 evaluated at incidence angle @xmath129 . with regard to surface waves , note that the arguments of the heaviside functions multiplying the fourth and fifth terms on the right - hand - side of eq . are always equal to zero except in the degenerate cases ( i ) @xmath170 and @xmath32 and ( ii ) @xmath171 and @xmath172 , respectively . thus , recalling the discussion at the end of section [ sec : impcontint ] , one finds that if either @xmath118 or @xmath119 is such that surface waves are present , these surface waves must form part of the edge source integral in eq . . in this case , in the region where the surface waves exist it holds that @xmath173 so that the edge source integral in eq . can not be associated solely with the diffracted field , as is the case for ideal ( dirichlet or neumann ) boundary conditions with a real incidence angle . nonetheless , one can check that by applying the method of stationary phase to the integral in eq . , the far - field diffraction coefficient approximation in eq . is recovered . this does not contradict the above remarks ( in particular eq . ) since the surface wave contributions ( when present ) are exponentially small with respect to increasing @xmath76 , and hence are not picked up by the method of stationary phase . it should be remarked that the existence of the edge source representation in eq . is of mainly theoretical interest ( for example in the development of approximate solutions for finite edges and of edge integral equation formulations of scattering problems - see , e.g. , ref . @xcite ) . for numerical computations of the infinite wedge solution at medium to high frequencies the expression in eq . should be used rather than that in eq . , because of the faster convergence of the integral in eq . compared to that in eq . . a secondary edge source representation has been presented ( in eq . ) for the exact solution of scattering of a plane wave at perpendicular incidence on a right - angled impedance wedge . when the impedance parameters are such that surface waves are present , the edge source integral can not be associated solely with the diffracted field , as in the case of ideal ( dirichlet or neumann ) boundary conditions , because it also incorporates the surface waves . a similar edge source representation should also be possible for general wedge angles , starting from the contour integral solutions in refs . @xcite and @xcite . but the analysis , and the resulting edge integral , are expected to be significantly more complicated than in the right - angled case considered here , for which one has the particularly simple contour integral solution provided by ref . another interesting problem would be the derivation of edge source representations for more general incident waves , for example line source or point source excitation . however , to the present author s knowledge no convenient contour integral solution exists for these cases . certainly the expressions obtained would be significantly more complicated than those obtained here for plane wave incidence ; for a start , the green s function for a line source or point source above an impedance boundary can not be obtained by the method of images , as it can in the plane wave case . these generalisations are left for future work .
|
this paper concerns the frequency domain problem of diffraction of a plane wave incident on an infinite right - angled wedge on which impedance ( absorbing ) boundary conditions are imposed .
it is demonstrated that the exact sommerfeld - malyuzhinets contour integral solution for the diffracted field can be transformed to a line integral over a physical variable along the diffracting edge .
this integral can be interpreted as a superposition of secondary point sources ( with directivity ) positioned along the edge , in the spirit of the edge source formulations for rigid ( sound - hard ) wedges derived in [ u. p. svensson , p. t. calamia and s. nakanishi , acta acustica / acustica 95 , 2009 , pp .
568 - 572 ] .
however , when surface waves are present the physical interpretation of the edge source integral must be altered : it no longer represents solely the diffracted field , but rather includes surface wave contributions .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
we describe the calculation of the hadronic matrix elements that are required for the extraction of the @xmath1 and @xmath2 ckm matrix elements from experimental data @xcite . to reach the bottom quark mass our strategy is to interpolate between results from relativistic quarks with @xmath3 and results from lattice hqet @xcite . here we discuss only the hqet simulations , as our clover form factor simulations have only just started . all of our simulations use @xmath4 dynamical staggered configurations with a volume @xmath5 and @xmath6 . the isgur - wise function is the qcd matrix element required in the extraction of @xmath1 from experimental data . experimental measurements of the slope of the isgur - wise function vary from @xmath7 to @xmath8 , and the variations in theoretical predictions are nearly as large @xcite . initial attempts to calculate the isgur - wise function in lattice hqet had problems either with the signal to noise ratio @xcite or the renormalization factors @xcite . the first complete calculation has been done recently by the kentucky group @xcite . we use the same method as the kentucky group ( see also @xcite ) . we ran at all permutations of the following velocities : @xmath9 , @xmath10 , @xmath11 and @xmath12 . our sample size is @xmath13 configurations , and our wilson @xmath14 values are @xmath15 and @xmath16 . a relative smearing function of @xmath17 was used between the quarks in the @xmath18 meson . -9 mm -8 mm in fig . [ fig : isgurwisefunc ] we plot the bare isgur - wise function for various time separations between the current and the @xmath18 source . if the ground state has been isolated , then the isgur - wise function should be independent of this separation . the data for @xmath19 are consistent within present errors . it is traditional to report the slope of the isgur - wise function as a function of the dot product of the two meson velocities ( @xmath20 ) . in table [ tb : rholinear ] we report fits to @xmath21 .preliminary fits to bare isgur - wise function data [ cols="^,^,^,^,^ " , ] the observation of the decay @xmath0 allows a determination of @xmath2 , if the relevant qcd form factors can be calculated . there have been a number of lattice qcd calculations of the required form factors ( see @xcite for reviews ) . however , previous approaches suffer from the drawback that calculations are done at large @xmath23 , thus requiring a large extrapolation to @xmath24 , where measurements are currently made . to reach a low @xmath23 requires large meson velocities not easily achieved for heavy mesons in the nrqcd- or propagating - quark - approaches . as we have shown @xcite , a good signal can be obtained for the hqet @xmath18 meson with a large velocity ( @xmath25 ) , so we propose the use of hqet light simulations to explore lower @xmath23 ( see @xcite for similar ideas ) . it is not clear that hqet will be a good approximation to the dynamics of the @xmath18 meson , at these values of @xmath23 , nor that a sufficiently good signal will be obtained . however , results should be very useful in helping to reduce the heavy quark extrapolation errors over simulations that only use clover quarks . because @xmath26 light meson form factors have never been studied before using lattice hqet ( although the static limit was studied in @xcite ) , we have computed the matrix element for @xmath0 using hqet to check for a signal . we use the setup described in @xcite with the heavy clover quark replaced by a hqet quark . in fig . [ fig : hqetlight ] we plot the ratio of three point functions to two point functions @xmath27 that is proportional to the @xmath28 matrix element , as a function of the operator time @xmath29 . the @xmath18 source is fixed at @xmath30 , and the light meson source is fixed at @xmath31 . this work is supported by the doe and the nsf . the computations were carried out at ccs ( ornl ) . this presentation was prepared with the kind additional support and assistance of the university of bielefeld and its zentrum fr interdisziplinre forschung .
|
we use simulations of heavy quark effective field theory to calculate the isgur - wise function , and we demonstrate the feasibility of calculating the matrix element for the @xmath0 decay in the lattice heavy quark effective theory ( hqet ) .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
while abstract polytopes are a combinatorial generalisation of classical polyhedra and polytopes , maniplexes generalise maps on surfaces and ( the flag graph of ) abstract polytopes . the combinatorial structure of maniplexes , maps and polytopes is completely determined by a edge - coloured @xmath2-valent graph with chromatic index @xmath2 , often called the flag graph . the symmetry type graph of a map is the quotient of its flag graph under the action of the automorphism group . in this paper we extend the notion of symmetry type graphs of maps to that of maniplexes ( and polytopes ) . given a maniplex , its symmetry type graph encapsulates all the information of the local configuration of the flag orbits under the action of the automorphism group of the maniplex . traditionally , the main focus of the study of maps and polytopes has been that of their symmetries . regular and chiral ones have been extensively studied . these are maps and polytopes with either maximum degree of symmetry or maximum degree of symmetry by rotation . edge - transitive maps were studied in @xcite by siran , tucker and watkins . such maps have either 1 , 2 or 4 orbits of flags under the action of the automorphism group . more recently orbani , pellicer and weiss extend this study and classify @xmath0-orbit maps ( maps with @xmath0 orbits of flags under the automorphism group ) up to @xmath3 in @xcite . little is known about polytopes that are neither regular nor chiral . in @xcite hubard gives a complete characterisation of the automorphism group of 2-orbit and fully - transitive polyhedra ( i.e. polyhedra transitive on vertices , edges and faces ) in terms of distinguished generators of them . moreover , she finds generators of the automorphism group of a 2-orbit polytope of any given rank . symmetry type graphs of the platonic and archimedean solids were determined in @xcite . in @xcite del ro - francos , hubard , orbani and pisanski determine symmetry type graphs of up to 5 vertices and give , for up to 7 vertices , the possible symmetry type graphs that a properly self - dual , an improperly self - dual and a medial map might have . the possible symmetry type graphs that a truncation of a map can have is determined in @xcite . one can find in @xcite a strategy to generate symmetry type graphs . by making use of symmetry type graphs , in this paper we classify 3-orbit polytopes and give generators of their automorphism groups . in particular , we show that 3-orbit polytopes are never fully - transitive , but they are @xmath4-face - transitive for all @xmath4 but one or two , depending on the class . we extend further the study of symmetry type graphs to show that if a 4-orbit polytope is not fully - transitive , then it is @xmath4-face - transitive for all @xmath4 but at most three ranks . moreover , we show that a fully - transitive 3-maniplex ( or 4-polytope ) that is not regular can not have an odd number of orbits of flags , under the action of the automorphism group . the main result of the paper is stated in theorem [ auto ] . given a maniplex @xmath5 in theorem [ auto ] we give generators for the automorphism group of @xmath5 with respect to some base flag . the paper is divided into six sections , organised in the following way . in section [ sec : polymani ] , we review some basic theory of polytopes and maniplexes , and describe their respective flag graphs . in section [ sec : stg ] , we define and give some properties of the symmetry type graphs of polytopes and maniplexes , extending the concept of symmetry type graphs of maps . in section [ sec : stg - highly ] , we study symmetry type graphs of highly symmetric maniplexes . in particular , we classify symmetry type graphs with 3 vertices , determine the possible transitivities that a 4-orbit mainplex can have and study some properties of fully - transitive maniplexes of rank 3 . in section [ gen - autg ] we give generators of the automorphism group of a polytope or a maniplex . in the last section of the paper we define oriented and orientable maniplexes . further on , we define the oriented flag di - graph which emerge from a flag graph if this is bipartite . the oriented symmetry type di - graph of an oriented maniplex is then a quotient of the oriented flag di - graph , just as the symmetry type graph was a quotient of the flag graph . using these graphs we classify oriented 2-orbit maniplexes and give generators for their orientation preserving automorphism group . in this section we briefly review the basic theory of abstract polytopes and their monodromy groups ( for details we refer the reader to @xcite and @xcite ) . an ( _ abstract _ ) _ polytope of rank _ @xmath2 , or simply an _ @xmath2-polytope _ , is a partially ordered set @xmath6 with a strictly monotone rank function with range @xmath7 . an element of rank @xmath8 is called a _ @xmath8-face _ of @xmath6 , and a face of rank @xmath9 , @xmath10 or @xmath11 is called a _ vertex _ , _ edge _ or _ facet _ , respectively . a _ chain of @xmath12 _ is a totally ordered subset of @xmath12 . the maximal chains , or _ flags _ , all contain exactly @xmath13 faces , including a unique least face @xmath14 ( of rank @xmath15 ) and a unique greatest face @xmath16 ( of rank @xmath2 ) . a polytope @xmath6 has the following homogeneity property ( diamond condition ) : whenever @xmath17 , with @xmath18 a @xmath19-face and @xmath20 a @xmath21-face for some @xmath8 , then there are exactly two @xmath8-faces @xmath22 with @xmath23 . two flags are said to be _ adjacent _ ( @xmath4-_adjacent _ ) if they differ in a single face ( just their @xmath4-face , respectively ) . the diamond condition can be rephrased by saying that every flag @xmath24 of @xmath12 has a unique @xmath4-adjacent flag , denoted @xmath25 , for each @xmath26 . finally , @xmath6 is _ strongly flag - connected _ , in the sense that , if @xmath24 and @xmath27 are two flags , then they can be joined by a sequence of successively adjacent flags , each containing @xmath28 . let @xmath6 be an abstract @xmath2-polytope . universal _ string coxeter group @xmath29 $ ] of rank @xmath2 , with distinguished involutory generators @xmath30 @xmath31 , acts transitively on the set of flags @xmath32 of @xmath12 in such a way that @xmath33 , the @xmath4-adjacent flag of @xmath27 , for each @xmath34 and each @xmath27 in @xmath32 . in particular , if @xmath35 then @xmath36 the _ monodromy or connection group _ of @xmath12 ( see for example @xcite ) , denoted @xmath37 , is the quotient of @xmath38 by the normal subgroup @xmath39 of @xmath38 consisting of those elements of @xmath38 that act trivially on @xmath32 ( that is , fix every flag of @xmath40 . let @xmath41 denote the canonical epimorphism . clearly , @xmath37 acts on @xmath32 in such a way that @xmath42 for each @xmath43 in @xmath38 and each @xmath27 in @xmath32 , so in particular @xmath44 for each @xmath4 . we slightly abuse notation and also let @xmath45 denote the @xmath4-th generator @xmath46 of @xmath37 . we shall refer to these @xmath45 as the _ distinguished _ generators of @xmath37 . since the action of @xmath38 is transitive on the flags , the action of @xmath37 on the flags of @xmath12 is also transitive ; moreover , this action is faithful , since only the trivial element of @xmath37 fixes every flag . thus @xmath37 can be viewed as a subgroup of the symmetric group on @xmath32 . note that for every flag @xmath24 of @xmath12 and @xmath47 such that @xmath48 , we have that @xmath49 . since the action of @xmath37 is faithful in @xmath32 , this implies that @xmath50 , whenever @xmath48 . an _ automorphism _ of a polytope @xmath12 is a bijection of @xmath12 that preserves the order . we shall denote by @xmath51 the group of automorphisms of @xmath12 . note that any automorphism of @xmath12 induces a bijection of its flags that preserves the @xmath4-adjacencies , for every @xmath52 . a polytope @xmath12 is said to be _ regular _ if the action of @xmath51 is regular on @xmath32 . if @xmath51 has exactly 2 orbits on @xmath32 in such a way that adjacent flags belong to different orbits , @xmath12 is called a _ chiral polytope_. we say that a polytope is a _ @xmath0-orbit polytope _ if the action of @xmath51 has exactly @xmath0 orbits on @xmath32 . hence , regular polytopes are 1-orbit polytopes and chiral polytopes are 2-orbit polytopes . given an @xmath2-polytope @xmath12 , we define the _ graph of flags _ @xmath53 of @xmath12 as follows . the vertices of @xmath53 are the flags of @xmath12 , and we put an edge between two of them whenever the corresponding flags are adjacent . hence @xmath53 is @xmath2-valent ( i.e. every vertex of @xmath53 has exactly @xmath2 incident edges ; to reduce confusion we avoid the alternative terminology ` @xmath2-regular ' ) . furthermore , we can colour the edges of @xmath53 with @xmath2 different colours as determined by the adjacencies of the flags of @xmath12 . that is , an edge of @xmath53 has colour @xmath4 , if the corresponding flags of @xmath12 are @xmath4-adjacent . in this way every vertex of @xmath53 has exactly one edge of each colour ( see figure [ fig : baricentic ] ) . it is straightforward to see that each automorphism of @xmath12 induces an automorphism of the flag graph @xmath53 that preserves the colours . conversely , every automorphism of @xmath53 that preserves the colours is a bijection of the flags that preserves all the adjacencies , inducing an automorphism of @xmath12 . that is , the automorphism group @xmath51 of @xmath12 is the colour preserving automorphism group @xmath54 of @xmath53 . note that the connectivity of @xmath12 implies that the action of @xmath51 on @xmath32 is free ( or semiregular ) . hence , the action of @xmath54 is free on the vertices of the graph @xmath53 . one can re - label the edges of @xmath53 and assign to them the generators of @xmath37 . in fact , since for each flag @xmath24 , the action of @xmath45 takes @xmath24 to @xmath55 , by thinking of the edge of colour @xmath4 of @xmath53 as the generator @xmath45 , one can regard a walk along the edges of @xmath53 as an element of @xmath37 . that is , if @xmath43 is a walk along the edges of @xmath37 that starts at @xmath24 and finishes at @xmath27 , then we have that @xmath56 . hence , the connectivity of @xmath12 also implies that the action of @xmath37 is transitive on the vertices of @xmath53 . furthermore , since the @xmath4-faces of @xmath12 can be regarded as the orbits of flags under the action of the subgroup @xmath57 , the @xmath4-faces of @xmath12 can be also regarded as the connected components of the subgraph of @xmath53 obtained by deleting all the edges of colour @xmath4 . maniplexes were first introduced by steve wilson in @xcite , aiming to unify the notion of maps and polytopes . in this section we review the basic theory of them . an _ @xmath2-complex _ @xmath58 is defined by a set of _ flags _ @xmath59 and a sequence @xmath60 @xmath61 , such that each @xmath45 partitions the set @xmath59 into sets of size 2 and the partitions defined by @xmath45 and @xmath62 are disjoint when @xmath63 . furthermore , we ask for @xmath5 to be _ connected _ in the following way . thinking of the @xmath2-complex @xmath58 as the graph @xmath64 with vertex set @xmath59 , and with edges of colour @xmath4 corresponding to the matching @xmath45 , we ask for the graph @xmath64 indexed by @xmath5 to be connected . an _ @xmath2-maniplex _ is an @xmath2-complex such that the elements in the sequence @xmath65 correspond to the distinguished involutory generators of a coxeter string group . in terms of the graph @xmath64 , this means that the connected components of the induced subgraph with edges of colours @xmath4 and @xmath8 , with @xmath66 are 4-cyles . we shall refer to the _ rank _ of an @xmath2-maniplex , precisely to @xmath2 . a 0-maniplex must be a graph with two vertices joined by an edge of colour 0 . a 1-maniplex is associated to a 2-polytope or @xmath67-gon , which graph contains @xmath68 vertices joined by a perfect matching of colours 0 and 1 , and each of size @xmath67 . a 2-maniplex can be considered as a map and vice versa , so that maniplexes generalise the notion of maps to higher rank . regarding polytopes , the flag graph of any @xmath69-polytope can be associated to an @xmath2-maniplex , generalising in such way the notion of polytopes . one can think of the sequence @xmath65 of a maniplex @xmath5 as permutations of the flags . in fact , if @xmath70 are flags of @xmath5 belonging to the same part of the partition induced by @xmath45 , for some @xmath4 , we say that @xmath71 and @xmath72 . in this way each @xmath45 acts as a involutory permutation of @xmath59 . in analogy with polytopes , we let @xmath73 and define the _ connection group _ @xmath74 of @xmath5 as the quotient of @xmath75 over @xmath39 . as before , we abuse notation and say that @xmath74 is generated by @xmath76 and define the action of @xmath74 on the flags inductively , induced by the action of the sequence @xmath65 . in this way , the action of @xmath74 on @xmath59 is faithful and transitive . note further that since the sequence @xmath65 induces a string coxeter group , then , as elements of @xmath74 , @xmath50 whenever @xmath48 . this implies that given a flag @xmath24 of @xmath5 and @xmath77 such that @xmath48 , we have that @xmath78 . an _ automorphism _ @xmath79 of an @xmath2-maniplex is a colour - preserving automorphism of the graph @xmath64 . in a similar way as it happens for polytopes , the connectivity of the graph @xmath80 implies that the action of the automorphism group @xmath81 ) of @xmath5 is free on the vertices of @xmath80 . hence , @xmath79 can be seen as a permutation of the flags in @xmath59 that commutes with each of the permutations in the connection group . to have consistent concepts and notation between polytopes and maniplexes , we shall say that an @xmath4-face ( or a face of rank @xmath4 ) of a maniplex is a connected component of the subgraph of @xmath80 obtained by removing the @xmath4-edges of @xmath80 . furthermore , we say that two flags @xmath24 and @xmath27 are @xmath4-adjacent if @xmath82 ( note that since @xmath45 is an involution , @xmath82 implies that @xmath83 , so the concept is symmetric ) . to each @xmath4-face @xmath18 of @xmath5 , we can associate an @xmath84-maniplex @xmath85 by identifying two flags of @xmath18 whenever there is a @xmath8-edge between them , with @xmath86 . equivalently , we can remove from @xmath18 all edges of colours @xmath87 , and then take one of the connected components . in fact , since @xmath88 commutes with @xmath89 , the connected components of this subgraph of @xmath18 are all isomorphic , so it does not matter which one we pick . if @xmath24 is a flag of @xmath5 that contains the @xmath4-face @xmath18 , then it naturally induces a flag @xmath90 in @xmath85 . similarly , if @xmath91 fixes @xmath18 , then @xmath92 induces an automorphism @xmath93 , defined by @xmath94 . to check that this is well - defined , suppose that @xmath95 ; we want to show that @xmath96 . since @xmath95 , it follows that @xmath97 for some @xmath98 . then @xmath99 , so that @xmath100 . by definition , the edges of @xmath80 of one given colour form a perfect matching . the 2-factors of the graph @xmath80 are the subgraphs spanned by the edges of two different colours of edges . since the automorphisms of @xmath5 preserve the adjacencies between the flags , it is not difficult to see that the following lemma holds . [ orbittoorbit ] let @xmath24 be a flag of @xmath5 and let @xmath101 . if @xmath102 and @xmath103 denote the flag orbits of @xmath24 and @xmath104 ( under @xmath105 ) , respectively , then @xmath106 if and only if @xmath107 . we say that a maniplex @xmath5 is _ @xmath4-face - transitive _ if @xmath105 is transitive on the faces of rank @xmath4 . we say that @xmath5 is _ fully - transitive _ if it is @xmath4-face - transitive for every @xmath108 . if @xmath105 has @xmath0 orbits on the flags of @xmath5 , we say that @xmath5 is a @xmath0-orbit maniplex . a 1-orbit maniplex is also called a _ reflexible _ maniplex . a 2-orbit maniplex with adjacent flags belonging to different orbits is a _ chiral _ maniplex . if a maniplex has at most 2 orbits of flags and @xmath109 is a bipartite graph , then the maniplex is said to be _ in this section we shall define the symmetry type graph of a polytope or a maniplex . to this end , we shall make use of quotient of graphs . therefore , we now consider pregraphs ; that is , graphs that allow multiple edges and semi - edges . as it should be clear , it makes no difference whether we consider an abstract @xmath2-polytope or an @xmath110-maniplex . hence , though we will consider maniplexes throughout the paper , similar results will apply to polytopes . given an edge - coloured graph @xmath80 , and a partition @xmath111 of its vertex set @xmath112 , the _ coloured quotient with respect to @xmath111 _ , @xmath113 , is defined as the pregraph with vertex set @xmath111 , such that for any two vertices @xmath114 , there is a dart of colour @xmath115 from @xmath116 to @xmath117 if and only if there exists @xmath118 and @xmath119 such that there is a dart of colour @xmath115 from @xmath120 to @xmath121 . edges between vertices in the same part of the partition @xmath111 quotient into semi - edges . throughout the remainder of this section , let @xmath5 be an @xmath110-maniplex and @xmath109 its coloured flag graph . as we discussed in the previous section , @xmath105 acts semiregularly on the vertices of @xmath109 . we shall consider the orbits of the vertices of @xmath109 under the action of @xmath105 as our partition @xmath122 , and denote @xmath123 . note that since the action is semiregular , every two orbits @xmath124 have the same number of elements . the _ symmetry type graph _ @xmath125 of @xmath5 is the coloured quotient graph of @xmath109 with respect to @xmath126 . since the flag graph @xmath109 is an undirected graph , then @xmath125 is a pre - graph without loops or directed edges . furthermore , as we are taking the coloured quotient , and @xmath109 is edge - coloured with @xmath2 colours , then @xmath125 is an @xmath2-valent pre - graph , with one edge or semi - edge of each colour at each vertex . it is hence not difficult to see that if @xmath5 is a reflexible maniplex , then @xmath125 is a graph consisting of only one vertex and @xmath2 semi - edges , all of them of different colours . in fact , the symmetry type graph of a @xmath0-orbit maniplex has precisely @xmath0 vertices . figure [ stgrank3a ] shows the symmetry type graph of a reflexible 2-maniplex ( on the left ) , and the symmetry type graph of the cuboctahedron : the quotient graph of the flag graph in figure [ fig : baricentic ] with respect to the automorphism group of the cubocahedron . note that by the definition of @xmath125 , there exists a surjective function @xmath127 that assigns , to each vertex of @xmath128 its corresponding orbit in @xmath125 . hence , given @xmath129 , we have that @xmath130 if and only if @xmath24 and @xmath27 are in the same orbit under @xmath105 . given vertices @xmath131 of @xmath125 , if there is an @xmath4-edge joining them , we shall denote such edge as @xmath132 . similarly , @xmath133 shall denote the semi - edge of colour @xmath4 incident to the vertex @xmath121 . because of lemma [ orbittoorbit ] , we can define the action of @xmath74 on the vertices of @xmath125 . in fact , given @xmath134 and @xmath101 , then @xmath135 , where @xmath136 . note that the definition of the action does not depend on the choice of @xmath136 ; in fact , we have that @xmath137 if and only if @xmath130 and this in turn is true if and only if @xmath24 and @xmath27 are in the same orbit under @xmath105 . by lemma [ orbittoorbit ] , the fact that @xmath24 and @xmath27 are in the same orbit under @xmath105 implies that , for any @xmath101 , the flags @xmath104 and @xmath138 are also in the same orbit under @xmath105 . hence @xmath139 and therefore the definition of @xmath140 does not depend on the choice of the element @xmath136 . since @xmath74 is transitive on the vertices of @xmath109 , then it is also transitive on the vertices of @xmath125 , implying that @xmath125 is a connected graph . furthermore , the action of each generator @xmath45 of @xmath74 on a vertex @xmath121 of @xmath125 corresponds precisely to the ( semi-)edge of colour @xmath4 incident to @xmath121 . hence , the orbit @xmath141 corresponds to the orbit under @xmath105 of an @xmath4-face @xmath18 of @xmath5 such that @xmath142 , for some @xmath143 ( as before , different choices of flag @xmath144 induce the same orbit of @xmath4-faces ) . therefore , the connected components of the subgraph @xmath145 of @xmath125 with edges of colours @xmath146 correspond to the orbits of the @xmath4-faces under @xmath105 . in particular this implies the following proposition . let @xmath5 be a maniplex , @xmath125 its symmetry type graph and let @xmath145 be the subgraph resulting by erasing the @xmath4-edges of @xmath125 . then @xmath5 is @xmath4-face - transitive if and only if @xmath145 is connected . we shall say that a symmetry type graph @xmath147 is @xmath4-face - transitive if @xmath148 is connected , and that @xmath147 is a fully - transitive symmetry type graph if it is @xmath4-face - transitive for all @xmath4 . recall that to each @xmath4-face @xmath18 of @xmath5 , there is an associated @xmath84-maniplex @xmath85 . the symmetry type graph @xmath149 is related in a natural way to the connected component of @xmath145 that corresponds to @xmath18 : [ stgoffaces ] let @xmath18 be an @xmath4-face of the maniplex @xmath5 , and let @xmath85 be the corresponding @xmath84-maniplex . let @xmath150 be the connected component of @xmath145 corresponding to @xmath18 . then there is a surjective function @xmath151 . furthermore , if @xmath152 then each @xmath8-edge @xmath153 of @xmath150 yields a @xmath8-edge @xmath154 in @xmath149 , and if @xmath86 , then @xmath155 . first , let @xmath24 and @xmath27 be flags of @xmath5 that are both in the connected component @xmath18 , and suppose that they lie in the same flag orbit , so that @xmath156 for some @xmath91 . then the induced automorphism @xmath157 of @xmath85 sends @xmath90 to @xmath158 , and therefore @xmath90 and @xmath158 lie in the same orbit . furthermore , every flag of @xmath85 is of the form @xmath90 for some @xmath24 in @xmath18 . thus , each orbit of @xmath5 that intersects @xmath18 induces an orbit of @xmath85 , and it follows that there is a surjective function @xmath151 . consider an edge @xmath153 in @xmath150 . then @xmath159 for some flag @xmath24 in @xmath18 , and we can take @xmath160 . both @xmath24 and @xmath161 induce flags in @xmath85 . if @xmath152 , then @xmath162 . therefore , there must be a @xmath8-edge from the orbit of @xmath90 to the orbit of @xmath163 ; in other words , a @xmath8-edge from @xmath164 to @xmath165 . on the other hand , if @xmath86 , then @xmath166 , and so @xmath90 and @xmath163 lie in the same orbit and thus @xmath167 . note that the edges of a given colour @xmath4 of @xmath125 form a perfect matching ( where , of course , we are allowing to match a vertex with itself by a semi - edge ) . given two colours @xmath4 and @xmath8 , the subgraph of @xmath125 consisting of all the vertices of @xmath125 and only the @xmath4- and @xmath8-edges shall be called a @xmath168 2-factor of @xmath125 . because @xmath50 whenever @xmath169 , in @xmath109 , the alternating cycles of colours @xmath4 and @xmath8 have length 4 . by lemma [ orbittoorbit ] each of these 4-cycles should then factor , in @xmath125 , into one of the five graphs in figure [ 4cyclequotient ] . hence , if @xmath169 , then the connected components of the @xmath168 2-factors of @xmath125 are precisely one of these graphs . coloured 4-cycles.,width=302 ] in light of the above observations we state the following lemma . [ 2factors4vertices ] let @xmath125 be the symmetry type graph of a maniplex . if there are vertices @xmath170 such that @xmath171 with @xmath66 , then the connected component of the @xmath168 2-factor that contains @xmath121 has four vertices one can classify maniplexes with small number of flag orbits ( under the action of the automorphism group of the maniplex ) in terms of their symmetry type graphs . the number of distinct possible symmetry types of a @xmath0-orbit @xmath110-maniplex is the number of connected pre - graphs on @xmath0 vertices that are @xmath2-valent and that can be edge - coloured with exactly @xmath2 colours . furthermore , given a symmetry type graph , one can read from the appropriate coloured subgraphs the different types of face transitivities that the maniplex has . as pointed out before , the symmetry type graph of a reflexible @xmath110-maniplex consists of one vertex and @xmath2 semi - edges . the classification of two - orbit maniplexes ( see @xcite ) in terms of the local configuration of their flags follows immediately from considering symmetry type graphs . in fact , for each @xmath2 , there are @xmath172 symmetry type graphs with 2 vertices and @xmath2 ( semi)-edges , since given any proper subset @xmath173 of the colours @xmath174 , there is a symmetry type graph with two vertices , @xmath175 semi - edges corresponding to the colours of @xmath173 , and where all the edges between the two vertices use the colours not in @xmath173 ( see figure [ 2orbitstg ] ) . this symmetry type graph corresponds precisely to polytopes in class @xmath176 , see @xcite . .,width=302 ] highly symmetric maniplexes can be regarded as those with few flag orbits or those with many ( or all ) face transitivities . in @xcite one can find the complete list of symmetry type graphs of @xmath177-maniplexes with at most 5 vertices . in this section we classify symmetry type graphs with 3 vertices and study some properties of symmetry type graphs of 4-orbit maniplexes and fully - transitive 3-maniplexes . [ stg_3-orbit ] there are exactly @xmath178 different possible symmetry type graphs of 3-orbit maniplexes of rank @xmath11 . let @xmath5 be a 3-orbit @xmath110-maniplex and @xmath125 its symmetry type graph . then , @xmath125 is an @xmath2-valent well edge - coloured graph with vertices @xmath179 and @xmath180 . recall that the set of colours @xmath174 correspond to the distinguished generators @xmath181 of the connection group of @xmath5 , and that by @xmath132 we mean the edge between vertices @xmath120 and @xmath121 of colour @xmath4 . since @xmath125 is a connected graph , without loss of generality , we can suppose that there is at least one edge joining @xmath182 and @xmath183 and another joining @xmath183 and @xmath184 . let @xmath185 be the colours of these edges , respectively . that is , without loss of generality we may assume that @xmath186 and @xmath187 are edges of @xmath125 . by lemma [ 2factors4vertices ] , we must have that @xmath188 , as otherwise @xmath125 would have to have at least 4 vertices . this implies that the only edges of @xmath125 are either @xmath186 and @xmath189 , @xmath186 and @xmath190 or @xmath186 , @xmath189 and @xmath190 , with @xmath191 . ( see figure [ i - transitive3 ] ) . -maniplexes with edges of colours @xmath192 , @xmath8 , and @xmath193 , with @xmath191.,width=453 ] an easy computation now shows that there are @xmath178 possible different symmetry type graphs of 3-orbit maniplexes of rank @xmath11 . given a 3-orbit @xmath110-maniplex @xmath5 with symmetry type graph having exactly two edges @xmath194 and @xmath195 of colours @xmath8 and @xmath193 , respectively , for some @xmath196 , we shall say that @xmath5 is in class @xmath197 . if , on the other hand , the symmetry type graph of @xmath5 has one edge of colour @xmath8 and parallel edges of colours @xmath192 and @xmath193 , for some @xmath198 , then we say that @xmath5 is in class @xmath199 . from figure [ i - transitive3 ] we observe that a maniplex in class @xmath200 is @xmath4-face - transitive whenever @xmath201 , while a maniplex in class @xmath199 if @xmath4-face - transitive for every @xmath202 . a @xmath203-orbit maniplex is @xmath8-face - transitive if and only if it does not belong to any of the classes @xmath199 , @xmath197 or @xmath204 . [ no3-orbitfully ] there are no fully - transitive 3-orbit maniplexes . using proposition [ stgoffaces ] , we get some information about the number of flag orbits that the @xmath8-faces have : a @xmath203-orbit maniplex in class @xmath199 or @xmath197 has reflexible @xmath8-faces . if @xmath5 is a @xmath203-orbit maniplex , then the orbits of the @xmath8-faces correspond to the connected components of @xmath205 . assuming that @xmath5 is in class @xmath199 or @xmath197 , the graph @xmath205 has two connected components ; an isolated vertex , and two vertices that are connected by a @xmath21-edge ( and a @xmath19-edge , if @xmath5 is in class @xmath197 . then by proposition [ stgoffaces ] , the @xmath8-faces that correspond to the isolated vertex are reflexible ( that is , 1-orbit ) , and the edge with label @xmath193 forces an identification between the two vertices of the second component , so the @xmath8-faces in that component are also reflexible . it does not take long to realise that counting the number of symmetry type graphs with @xmath206 vertices , and perhaps classifying them in a similar fashion as was done for 2 and 3 vertices , becomes considerably more difficult . in this section , we shall analyse symmetry type graphs with 4 vertices and determine how far a 4-orbit maniplex can be from being fully - transitive . the following lemma is a consequence of the fact that by taking away the @xmath4-edges of a symmetry type graph @xmath207 ) , the resulting @xmath145 can not have too many components . [ 4orbmi - faceorb ] let @xmath5 be a 4-orbit @xmath110-maniplex and let @xmath208 . then @xmath5 has one , two or three orbits of @xmath4-faces . if an @xmath110-maniplex @xmath5 is not fully - transitive , there exists at least one @xmath208 such that @xmath145 is disconnected . we shall divide the analysis of the types in three parts : when @xmath209 has three connected components ( two of them of one vertex and one with two vertices ) , when @xmath145 has a connected component with one vertex and another connected component with three vertices , and finally when @xmath209 has two connected components with two vertices each . before we start the case analysis , we let @xmath210 be the vertices of @xmath125 . suppose that @xmath209 has three connected components with @xmath183 and @xmath184 in the same component . without loss of generality we may assume that @xmath125 has edges @xmath211 and @xmath212 . let @xmath213 be the colour of an edge between @xmath183 and @xmath184 . since there is no edge of @xmath125 between @xmath182 and @xmath214 , lemma [ 2factors4vertices ] implies that there are at most two such possible @xmath0 , namely @xmath215 and @xmath216 . if @xmath217 , @xmath125 can have either both edges or exactly one of them , while if @xmath218 there is one possible edge ( see figure [ 3components4 ] ) . -maniplex @xmath5 with four orbits on its flags , and three orbits on its @xmath4-faces.,width=377 ] let us now assume that @xmath209 has two connected components , one consisting of the vertex @xmath182 and the other one containing vertices @xmath219 and @xmath214 . this means that the @xmath4-edge incident to @xmath182 is the unique edge that connects this vertex with the rest of the graph and , without loss of generality , @xmath125 has the edge @xmath211 . as with the previous case , lemma [ 2factors4vertices ] implies that an edge between @xmath183 and @xmath184 has colour either @xmath220 or @xmath221 . first observe that having either @xmath222 or @xmath223 in @xmath125 immediately implies ( by lemma [ 2factors4vertices ] ) that there is no edge between @xmath183 and @xmath214 . now , if both edges @xmath222 and @xmath223 are in @xmath125 , then an edge between @xmath184 and @xmath214 would have to have colour @xmath4 , contradicting the fact that @xmath145 has two connected components . hence , there is exactly one edge between @xmath183 and @xmath184 . it is now straightforward to see that @xmath125 should be as one of the graphs in figure [ 2components4 ] , implying that there are exactly four symmetry type graphs with these conditions for each @xmath224 , but only two symmetry type graph of this kind when @xmath225 or @xmath11 . -maniplexes with four orbits on its flags , and two orbits on its @xmath4-faces such that one contains three flag orbits and the other contains a single flag orbit.,width=264 ] it is straightforward to see from figure [ 2components4 ] that the next lemma follows . let @xmath5 be a 4-orbit @xmath110-maniplex with two orbits of @xmath4-faces such that @xmath209 has a connected component consisting of one vertex , and another one consisting of three vertices . then either @xmath226 or @xmath227 has two connected components , each with two vertices . finally , we turn out our attention to the case where @xmath209 has two connected components , with two vertices each . suppose that @xmath182 and @xmath183 belong to one component , while @xmath184 and @xmath214 belong to the other . as the two components must be connected by the edges of colour @xmath4 , we may assume that @xmath228 is an edge of @xmath125 . if the vertices @xmath183 and @xmath214 have semi - edges of colour @xmath4 , lemma [ 2factors4vertices ] implies that @xmath125 is one of the graphs shown in figure [ 2 - 2components4 ] . -maniplexs with four orbits on its flags , and two orbits on its @xmath4-faces such that each contains two flag orbits.,width=377 ] on the other hand , if @xmath228 and @xmath229 are both edges of @xmath125 , given @xmath230 , we use again lemma [ 2factors4vertices ] to see that @xmath186 is an edge of @xmath125 if and only if @xmath231 is also an edge of @xmath125 . by contrast , @xmath125 can have either two edges of colour @xmath232 ( each joining the vertices of each connected component of @xmath209 ) , four semi - edges or an edge and two semi - edges of colour @xmath232 . hence , if @xmath217 , for each @xmath233 there are ten symmetry type graph with semi - edges of colours in @xmath234 and edges of colours not in @xmath234 , as shown in figures [ 3(n-2)_2compa ] and [ 3(n-2)_2compb ] , while for @xmath235 there are six such graphs ( shown in figure [ 3(n-2)_2compb ] ) . on the other hand if @xmath236 , for each @xmath233 there are two graphs as in figure [ 3(n-2)_2compa ] and one as in figure [ 3(n-2)_2compb ] , while for @xmath235 , there is only one of the graphs in figure [ 3(n-2)_2compb ] . -maniplexes with four orbits on its flags , and two orbits on its @xmath4-faces such that each contains two flag orbits.,width=264 ] -maniplexes with four orbits on its flags , and two orbits on its @xmath4-faces such that each contains two flag orbits.,width=453 ] we summarize our analysis of the transitivity of 4-orbit maniplexes below . let @xmath5 be a 4-orbit maniplex . then , one of the following holds . 1 . @xmath5 is fully - transitive . there exists @xmath208 such that @xmath5 is @xmath8-face - transitive for all @xmath237 . there exist @xmath238 , @xmath239 , such that @xmath5 is @xmath8-face - transitive for all @xmath240 . there exists @xmath208 such that @xmath5 is @xmath8-face - transitive for all @xmath241 . every 1-maniplex is reflexible and hence fully - transitive . fully - transitive 2-maniplexes correspond to fully - transitive maps . it is well - known ( and easy to see from the symmetry type graph ) that if a map is edge - transitive , then it should have one , two or four orbits . moreover , a fully - transitive map should be regular , a two - orbit map in class 2 , @xmath242 , @xmath243 or @xmath244 , or a four - orbit map in class @xmath245 or @xmath246 ( see , for example , @xcite ) . when considering fully - transitive @xmath2-maniplexes , @xmath247 , the analysis becomes considerably more complicated . in @xcite hubard shows that there are @xmath248 classes of fully - transitive two - orbit @xmath2-maniplexes . by theorem [ no3-orbitfully ] , there are no 3-orbit fully - transitive @xmath2-maniplexes . we note that there are 20 symmetry type graphs of 4-orbit 3-maniplexes that are fully transitive ( see figure [ 4orbitfully ] ) . the following theorem shall be of great use to show that a fully - transitive 3-maniplex must have an even number of flag orbits unless it is reflexible . let @xmath5 be a fully - transitive 3-maniplex and let @xmath125 be its symmetry type graph . then either @xmath5 is reflexible or @xmath125 has an even number of vertices . on the contrary suppose that @xmath125 has an odd number of vertices , different than 1 . whenever @xmath249 , the connected components of the @xmath168 2-factor of a symmetry type graph are as in figure [ 4cyclequotient ] . hence , there is a connected component of the @xmath250 2-factor of @xmath125 with exactly one vertex @xmath121 ( and , hence , semi - edges of colours 0 and 2 ) . the connectivity of @xmath125 implies that there is a vertex @xmath182 adjacent to @xmath121 in @xmath125 . if @xmath182 is the only neighbour of @xmath121 , then @xmath125 has the edges @xmath251 and @xmath252 as otherwise @xmath5 is not fully - transitive . since the connected components of the @xmath253 2-factor of @xmath125 are as in figure [ 4cyclequotient ] , @xmath182 has a 0 coloured semi - edge . because @xmath125 has more than two vertices , the edge of @xmath182 of colour 2 joins @xmath182 to another vertex , say @xmath120 . but removing the edge @xmath254 disconnects the graph contradicting the fact that @xmath5 is 2-face - transitive . on the other hand , if @xmath121 has more than one neighbour it has exactly two , say @xmath182 and @xmath120 and @xmath125 has the two edges @xmath251 and @xmath255 . this implies that the connected component of the @xmath256 2-factor containing @xmath121 has four vertices : @xmath257 and @xmath183 . ( therefore @xmath258 and @xmath259 are edges of @xmath125 . ) using the @xmath253 2-factor one sees that @xmath120 has a semi - edge of colour 0 . now , if @xmath260 is an edge of @xmath125 , then the vertices @xmath261 and @xmath120 are joined to the rest of @xmath125 by the edges of colour 2 , implying that removing them shall disconnect @xmath125 ( there exists at least another vertex in @xmath125 as it has an odd number of vertices ) , which is again a contradiction . on the other hand , if @xmath182 ( or @xmath183 ) has an edge of colour 0 to a vertex @xmath184 , then by lemma [ 2factors4vertices ] @xmath183 ( or @xmath182 ) has a 0-edge to a vertex @xmath214 . again , if @xmath262 is an edge of @xmath125 , since the number of vertices of the graph is odd , removing the edges of colour 2 will leave only the vertices @xmath263 in one component , which is a contradiction . proceeding now by induction on the number of vertices one can conclude that @xmath125 can not have an odd number of vertices it is well - known among polytopists that the automorphism group of a regular @xmath2-polytope can be generated by @xmath2 involutions . in fact , given a base flag @xmath264 , the distinguished generators of @xmath105 with respect to @xmath24 are involutions @xmath265 such that @xmath266 . generators for the automorphism group of a two - orbit @xmath2-polytope can also be given in terms of a base flag ( see @xcite ) . in this section we give a set of distinguished generators ( with respect to some base flag ) for the automorphism group of a @xmath0-orbit @xmath110-maniplex in terms of the symmetry type graph @xmath125 , provided that @xmath125 has a hamiltonian path . given two walks @xmath267 and @xmath268 along the edges and semi - edges of @xmath269 such that the final vertex of @xmath267 is the starting vertex of @xmath268 , we define the sequence @xmath270 as the walk that traces all the edges of @xmath267 and then all the edges of @xmath268 in the same order ; the inverse of @xmath267 , denoted by @xmath271 , is the walk which has the final vertex of @xmath267 as its starting vertex , and traces all the edges of @xmath267 in reversed order . since each of the elements of @xmath272 associated to the edges of @xmath269 is its own inverse , we shall forbid walks that trace the same edge two times consecutively ( or just remove the edge from such walk , shortening its length by two ) . given a set of walks in @xmath269 , we say that a subset @xmath273 is _ a generating set of @xmath274 _ if each @xmath275 can be expressed as a sequence of elements of @xmath276 and their inverses . now , let @xmath274 be the set of closed walks along the edges and semi - edges of @xmath269 starting at a distinguished vertex @xmath277 . recall that the walks along the edges and semi - edges of @xmath269 correspond to permutations of the flags of @xmath58 ; moreover , each closed walk of @xmath274 corresponds to an automorphism of @xmath58 . thus , by finding a generating set of @xmath274 , we will find a set of automorphisms of @xmath58 that generates @xmath278 . ( however , the converse is not true , as an automorphism of @xmath5 may be described in more than one way as a closed walk of @xmath125 . ) given @xmath269 , we may easily find such generating set . the construction goes as follows : let @xmath58 be a @xmath0-orbit maniplex of rank @xmath11 such that @xmath279 is a walk of minimal length that visits all the vertices of @xmath125 . the sets of vertices and edges ( and semi - edges ) of @xmath269 will be denoted by @xmath112 and @xmath280 , respectively . the set of edges visited by @xmath281 will be denoted by @xmath282 . in this section , the edges joining two vertices @xmath283 and @xmath284 will be denoted by @xmath285 , @xmath286 , @xmath287, ... ,@xmath288 ; if @xmath289 then @xmath290 . ( note that in order to not start carrying many subindices , we modify the notation of the edges of @xmath125 that we had used throughout the paper . if one wants to be consistent with the notation of the edges used in the previous sections , one would have to say that the edges between @xmath291 and @xmath292 are @xmath293 , @xmath294 ) . similarly , we denote all semi - edges incident to a vertex @xmath283 by @xmath295 , @xmath296 , @xmath297, ... ,@xmath298 . for the sake of simplicity , @xmath285 will be just called @xmath299 . let @xmath274 be the set of all closed walks in @xmath269 with @xmath300 as its starting vertex . we shall now construct @xmath301 , a generating set of @xmath274 . for each edge @xmath302 we shall define the walk @xmath303 that is , we walk from @xmath277 to @xmath291 in @xmath282 , and then we take the edge @xmath304 , and then we walk back from @xmath292 to @xmath277 in @xmath282 . let @xmath305 be the set of all such walks . for each semi - edge @xmath306 we shall define the walk @xmath307 . that is , we walk from @xmath277 to @xmath291 in @xmath282 , and then we take the semi - edge @xmath298 , and then we walk back from @xmath291 to @xmath277 in @xmath282 . let @xmath308 be the set of all such walks . we define @xmath309 . [ generatingwalks ] with the notation from above , @xmath310 is a generating set for @xmath274 . we shall prove that any @xmath275 can be expressed as a sequence of elements of @xmath310 and their inverses . let @xmath275 be a closed walk among the edges and semi - edges of @xmath269 starting at @xmath277 . from now on , semi - edges will be referred to simply as `` edges '' . we shall proceed by induction over @xmath2 , the number of edges in @xmath311 visited by @xmath43 . if @xmath43 visits only one edge in @xmath311 , then @xmath312 or @xmath313 . let us suppose that , if a closed walk among the edges of @xmath269 visits @xmath314 different edges in @xmath311 , with @xmath315 , then it can be expressed as a sequence of elements of @xmath310 and their inverses . let @xmath275 be a walk that visits exactly @xmath2 edges in @xmath311 . let @xmath316 be the last edge of @xmath311 visited by @xmath43 . without loss of generality we may assume that the vertex @xmath317 was visited after @xmath318 , so let @xmath319 be the edge that @xmath43 visits just before @xmath320 ( note that @xmath319 may or may not be in @xmath282 ) . let @xmath321 be the closed walk that traces the same edges ( in the same order ) as @xmath43 until reaching @xmath322 and then traces the edges @xmath323 , @xmath324 , ... , @xmath325 , and let @xmath326 be the closed walk that traces the edges @xmath327 and then traces @xmath320 and continues the way @xmath43 does to return to @xmath277 . it is clear that @xmath267 visits exactly @xmath11 edges in @xmath311 and that @xmath268 visits only one . by inductive hypothesis both @xmath267 and @xmath268 can be expressed as a sequence of elements of @xmath310 , and therefore so does @xmath43 since @xmath328 . let @xmath24 be a base flag of @xmath5 that projects to the initial vertex of a walk that contains all vertices of @xmath125 of a symmetry type graph . following the notation of @xcite , given @xmath329 such that @xmath330 is in the same orbit as @xmath24 ( that is , @xmath331 ) , we denote by @xmath332 the automorphism taking @xmath24 to @xmath330 . moreover , if @xmath333 for some @xmath334 , then we may also denote @xmath332 by @xmath335 . the following theorem gives distinguished generators ( with respect to some base flag ) of the automorphism group of a maniplex @xmath5 in terms of a distinguished walk of @xmath125 , that travels through all the vertices of @xmath125 . its proof is a consequence of the previous lemma . [ auto ] let @xmath5 be a @xmath0-orbit @xmath2-maniplex and let @xmath125 its symmetry type graph . suppose that @xmath336 is a distinguished walk that visits every vertex of @xmath125 , with the edge @xmath337 having colour @xmath338 , for each @xmath339 . let @xmath340 be such that @xmath291 has a semi - edge of colour @xmath341 if and only if @xmath342 . let @xmath343 be the set of colours of the edges between the vertices @xmath291 and @xmath292 ( with @xmath344 ) that are not in the distinguished walk and let @xmath345 be a base flag of @xmath5 such that @xmath24 projects to @xmath182 in @xmath125 . then , the automorphism group of @xmath5 is generated by the union of the sets @xmath346 and @xmath347 we note that , in general , a set of generators of @xmath105 obtained from theorem [ auto ] can be reduced since there might be more than one element of @xmath310 representing the same automorphism . for example , the closed walk @xmath43 through an edge of colour @xmath177 , then a @xmath9-semi - edge and finally a 2-edge corresponds to the element @xmath348 of @xmath74 . hence , the group generator induced by the walk @xmath43 is the same as that induced by the closed walk consisting only of the semi - edge of colour @xmath9 . the following two corollaries give a set of generators for 2- and 3-orbit polytopes , respectively , in a given class . the notation follows that of theorem [ auto ] , where if the indices of some @xmath79 do not fit into the parameters of the set , we understand that such automorphism is the identity . * @xcite * let @xmath5 be a 2-orbit @xmath110-maniplex in class @xmath176 , for some @xmath349 and let @xmath350 . then @xmath351 is a generating set for @xmath105 . let @xmath5 be a 3-orbit @xmath110-maniplex . 1 . if @xmath5 is in class @xmath352 , for some @xmath353 , then @xmath354 is a generating set for @xmath105 . if @xmath5 is in class @xmath355 , for some @xmath356 , then @xmath357 is a generating set for @xmath105 . a maniplex @xmath5 is said to be _ orientable _ if its flag graph @xmath109 is a bipartite graph . since a subgraph of a bipartite graph is also bipartite , all the sections of an orientable maniplex are orientable maniplexes themselves . an _ orientation _ of an orientable maniplex is a colouring of the parts of @xmath109 , with exactly two colours , say black and white . oriented maniplex _ is an orientable maniplex with a given orientation . note that any oriented maniplex @xmath5 has an enantiomorphic maniplex ( or mirror image ) @xmath358 . one can think of the enantiomorphic form of an oriented maniplex simply as the orientable maniplex with the opposite orientation . if the connection groups @xmath74 of @xmath5 is generated by @xmath181 , for each @xmath359 let us define the element @xmath360 . then , @xmath361 , for @xmath362 . the subgroup @xmath363 of @xmath74 generated by @xmath364 is called _ even connection group of @xmath5_. note that @xmath363 has index at most two in @xmath74 . in fact @xmath365 . it should be clear then that any maniplex and its enantiomorphic form are in fact isomorphic as maniplexes . an _ oriented flag di - graph _ @xmath366 of an oriented maniplex @xmath5 is constructed in the following way . the vertex set of @xmath366 consists of one of the parts of the bipartition of @xmath109 . that is , the black ( or white ) vertices of the flag graph of @xmath5 . the darts of @xmath366 will be the 2-arcs of @xmath109 of colours @xmath367 , for each @xmath368 . we then identify two darts to obtain an edge if they have the same vertices , but go in opposite directions . note that for @xmath369 and each flag @xmath24 of @xmath5 , the 2-arc starting at @xmath24 and with edges coloured @xmath11 and @xmath4 has the same end vertex than the 2-arc starting at @xmath24 and with edges coloured @xmath4 and @xmath11 . hence , all the darts corresponding to 2-arcs of colours @xmath370 and @xmath4 , with @xmath362 will have both directions in @xmath366 giving us , at each vertex , @xmath371 different edges . on the other hand , the 2-arcs on edges of two colours @xmath372 will in general be directed darts of @xmath366 . an example of an oriented flag di - graph is shown in firgure [ orientedgraphflag ] . we note that the oriented flag di - graph of @xmath358 can be obtained from @xmath366 by reversing the directions of the @xmath373 darts . note that the 2-arcs of colours @xmath374 correspond to the generators @xmath375 of @xmath363 . in fact , as @xmath363 consists precisely of the even words of @xmath74 , a maniplex is orientable if and only if the index of @xmath363 in @xmath74 is exactly two . we can then colour the edges and darts of @xmath366 with the elements @xmath375 . the fact that @xmath361 for every @xmath376 indeed implies that the edges of @xmath366 are labelled by these first @xmath371 elements , while the darts are labelled by @xmath377 . we can see now that for each @xmath359 , the @xmath4-faces of @xmath5 are in correspondence with the connected components of the subgraph of @xmath366 with edges of colours @xmath378 . to identify the facets of @xmath5 as subgraphs of @xmath366 , we first consider some oriented paths on the edges of @xmath366 . we shall say that an oriented path on the edges of @xmath366 is _ facet - admissible _ if no two darts of colour @xmath377 are consecutive on the path . then , two vertices of @xmath366 are in the same facet of @xmath5 if there exists a facet admissible oriented path from one of the vertices to the other . for the remainder of this section , by a maniplex we shall mean an oriented maniplex , with one part of the flags coloured with black and the other one in white . an _ orientation preserving automorphism _ of an ( oriented ) maniplex @xmath5 is an automorphism of @xmath5 that sends black flags to black flags and white flags to white flags . an _ orientation reversing automorphism _ is an automorphism that interchanges black and white flags . a _ reflection _ is an orientation reversing involutory automorphism . the group of orientation preserving automorphisms of @xmath5 shall be denoted by @xmath379 . the orientation preserving automorphism @xmath379 of a maniplex @xmath5 is a subgroup of index at most two in @xmath105 . in fact , the index is exactly two if and only if @xmath105 contains an orientation reversing automorphism . note that in this case , there exists an orientation reversing automorphism that sends @xmath5 to its enantiomorphic form @xmath358 . pisanski @xcite defines a maniplex to be _ chiral - a - la - conway _ if @xmath380 . if a maniplex @xmath5 is chiral - a - la - conway , then its enantiomorphic maniplex @xmath358 is isomorphic to @xmath5 , but there is no automorphism of the maniplex sending one to the other . it follows from the definition that @xmath5 is chiral - a - la - conway if and only if the automorphisms of @xmath5 preserve the bipartition of @xmath109 and therefore we have the following proposition . [ oddcycles ] let @xmath5 be an oriented maniplex and let @xmath125 its symmetry type graph . then , @xmath5 is chiral - a - la - conway if and only if @xmath125 has no odd cycles . similarly as before , the orientation preserving automorphisms of a maniplex @xmath5 correspond to colour preserving automorphism of the bipartite graph @xmath109 that preserves the two parts . but these correspond to colour preserving automorphisms of the di - graph @xmath366 , implying that @xmath381 . note that the action of @xmath379 on the set @xmath382 of all the black flags of @xmath5 is semiregular , and hence , the action on @xmath383 is semiregular on the vertices of @xmath366 . an oriented maniplex @xmath5 is said to be _ rotary ( or orientably regular ) _ if the action of @xmath379 is regular on @xmath384 . equivalently , @xmath5 is rotary if the action of @xmath383 is regular on its vertices . we say that @xmath5 is _ orientably @xmath0-orbit _ if the action of @xmath383 has exactly @xmath0 orbits on the vertices of @xmath366 . the following lemma is straightforward . [ orientablekorbit ] let @xmath5 be a chiral - a - la - conway maniplex . then @xmath125 has no semi - edges and if @xmath5 is an orientably @xmath0-orbit maniplex , then @xmath5 is a @xmath385-orbit maniplex . we now consider the semiregular action of @xmath379 on the vertices of @xmath366 , and let @xmath386 be the partition of the vertex set of @xmath366 into the orbits with respect to the action of @xmath379 . ( as before , since the action is semiregular , all orbits are of the same size . ) the _ oriented symmetry type di - graph _ @xmath387 of @xmath5 is the quotient colour di - graph with respect to @xmath388 . similarly as before , if @xmath5 is rotary , then the oriented symmetry type di - graph of @xmath5 consists of one vertex with one loop and @xmath371 semi - edges . note that for oriented symmetry type di - graphs we shall not identify two darts with the same vertices , but different directions . if we now turn our attention to oriented symmetry type di - graphs with two vertices , one can see that for each @xmath389 , there is an oriented symmetry type di - graphs with two vertices having semi - edges ( or loops ) of colours @xmath4 at each vertex for every @xmath390 , and having edges ( or both darts ) of colour @xmath8 , for each @xmath391 . an oriented maniplex with such oriented symmetry type di - graph shall be say to be in class @xmath392 . hence , there are @xmath393 classes of oriented 2-orbit @xmath110-maniplexes . note that if @xmath5 is a @xmath0-orbit maniplex , then @xmath387 has either @xmath0 or @xmath394 vertices . the next result follows from proposition [ oddcycles ] and lemma [ orientablekorbit ] . let @xmath5 be an oriented maniplex . then , @xmath125 and @xmath387 have the same number of vertices if and only if @xmath125 has a semi - edge or an odd cycle . it is not difficult to see that if we are to consider for a moment an oriented symmetry type di - graph @xmath395 with an ( undirected ) hamiltonian path , then the construction of section [ gen - autg ] gives us a way to construct a generating set of the closed walks based at the starting vertex of the path ( and lemma [ generatingwalks ] implies that the set actually generates . ) hence , one can find generators for the group of orientation preserving automorphisms of an oriented maniplex , provided that it has an ( undirected ) hamiltonian path . in particular we have the following theorem . let @xmath5 be an oriented 2-orbit @xmath110-maniplex in class @xmath392 , for some @xmath389 . then 1 . if @xmath396 , let @xmath350 , then @xmath397 is a generating set for @xmath398 . 2 . if @xmath399 but there exists @xmath350 , @xmath400 , then @xmath401 is a generating set for @xmath398 . if @xmath402 , then @xmath403 is a generating set for @xmath398 . given an oriented maniplex @xmath5 and its symmetry type graph @xmath125 , we shall say that @xmath387 is the associated oriented symmetry type di - graph of @xmath125 . hence , given a symmetry type graph @xmath147 one can find its associated oriented symmetry type di - graph @xmath395 by erasing all edges of @xmath147 and replacing them by the @xmath367 paths of @xmath147 . note that this replacement of the edges may disconnect the new graph . if that is the case , we take @xmath395 to be one of the connected components . in a similar way as one can classify maniplexes with small number of flag orbits ( under the action of the automorphism group of the maniplex ) in terms of their symmetry type graph , one can classify oriented maniplexes with small number of flags ( under the action of the orientation preserving automorphism group of the maniplex ) in terms of their oriented symmetry type di - graph . let @xmath58 be a @xmath404-orbit chiral - a - la - conway @xmath110-maniplex , with @xmath405 . let @xmath269 be its symmetry type graph and @xmath406 be its oriented symmetry type di - graph . recall that @xmath269 is a graph with 6 vertices and no semi - edges or odd cycles , and that @xmath406 is a di - graph with 3 vertices . let @xmath407 be the vertex set of @xmath269 . we may label the vertices of @xmath125 in such a way that the edges @xmath408 , @xmath409 , @xmath410 are coloured with the colour @xmath110 , and that no two vertices of the set @xmath411 are adjacent . let @xmath412 be the vertex set of @xmath413 . each @xmath414 corresponds to the vertex @xmath415 , @xmath416 . in what follows , in the same way as in section [ sec : stg ] , @xmath417 denotes the @xmath0-coloured edge joining the vertices @xmath283 and @xmath284 , @xmath418 , @xmath419 ; and @xmath420 denotes the @xmath421-coloured edge joining the vertices @xmath422 and @xmath423 , @xmath424 and @xmath425 . since there are no semi - edges in @xmath125 , for each colour @xmath426 there is one edge ( and one semi - edge ) of colour @xmath427 in @xmath413 if and only if the 2-factor of @xmath125 of colours @xmath4 and @xmath110 consists of one 4-cycle and one 2-cycle of alternating colours . likewise , there are three semi - edges of colour @xmath427 in @xmath413 if and only if the 2-factor of @xmath125 of colours @xmath4 and @xmath110 consist of three 2-cycles . it is straightforward to see that there are two consecutive edges of colour @xmath427 and @xmath428 @xmath63 , in @xmath413 if and only if the 2-factor of colours @xmath4 and @xmath8 consists of a single 6-cycle . it follows that if there are two consecutive edges of colour @xmath427 and @xmath429 in @xmath413 , then @xmath430 . notice that the possible 2-factors of colour @xmath110 and @xmath431 in @xmath125 are either a single 6-cycle of alternating colours , a 4-cycle along with a 2-cycle , or three separate 2-cycles . hence , the darts in @xmath413 are arranged in either a 3-cycle , a 2-cycle along with a loop , or three separate loops . we proceed case by case . consider the case when there are three loops in @xmath413 . since oriented symmetry type di - graphs are connected , then without loss of generality @xmath432 and @xmath433 must be edges of @xmath413 . we may suppose that @xmath432 is the only edge joining @xmath267 and @xmath434 . if there is a third edge in @xmath413 , then it is necessarily @xmath435 . note that , since the edges coloured by @xmath110 and @xmath431 do not lie on a 6-cycle in @xmath125 , there are no restrictions on the semi - edges of @xmath413 . thus , there is one oriented symmetry type di - graph for each pair of colours @xmath4 and @xmath221 , with @xmath426 and one for each triple @xmath220 , @xmath4 and @xmath221 , @xmath436 . therefore , there are @xmath437 oriented symmetry type di - graphs with 3 loops . consider the case when @xmath413 has only one loop . we may suppose that the loop is in @xmath438 and the vertices @xmath267 or @xmath434 are joined by darts . this implies that @xmath439 , @xmath440 and @xmath441 are edges of @xmath125 . as @xmath413 is connected , there must be an edge joining @xmath434 and @xmath438 of colour @xmath442 necessarily @xmath443 , since the edges @xmath444 , @xmath440 , @xmath445 , @xmath446 , @xmath447 , @xmath448 form a 6-cycle in @xmath125 . notice that there are no restrictions on the semi - edges of @xmath413 . hence , there are exactly two oriented symmetry type di - graph with a single loop : one with a single edge of colour @xmath449 between @xmath450 and @xmath451 , and one with two edges of colours @xmath449 and @xmath452 between them . consider the case when the darts in @xmath413 are arranged in a 3-cycle . it is clear that the 2-factor of @xmath125 of colours @xmath431 and @xmath110 is a single 6-cycle . therefore , if @xmath453 , the 2-factor of @xmath125 of colours @xmath4 and @xmath110 can not consist of three 2-cycles , as this implies the existence of a 6-cycle of alternating colours @xmath4 and @xmath431 such that @xmath454 . that is , @xmath413 has one edge ( and one semi - edge ) of colour @xmath427 for each @xmath453 and either one edge and a semi - edge , or three semi - edges for colour @xmath455 note that if @xmath456 , the set @xmath457 has more than three elements and thus all edges of colour @xmath427 in @xmath413 , @xmath453 , must be joining the same pair of vertices . otherwise , there would be at least two consecutive edges of colours @xmath427 and @xmath429 , with @xmath458 . figure [ 3orient ] below shows the only four possible oriented symmetry type di - graphs with a 3-cycle of darts and at least two consecutive edges . two correpond to 4-maniplexes , one to 3-maniplexes and one to 5-maniplexes . these will be treated as special cases . we may suppose that @xmath413 has no consecutive edges . it follows that here are exactly two oriented symmetry type di - graph with a 3-cycle of darts : one with an edge joining the same pair of vertices for each colour @xmath459 and one with three semi - edges of colour @xmath449 and an edge joining the same pair of vertices for each colour @xmath453 . considering all the cases above , there are @xmath460 oriented symmetry type graphs with three vertices for oriented maniplexes of rank @xmath461 ; @xmath462 for oriented maniplexes of rank 3 ; @xmath463 for oriented maniplexes of rank 4 ; and @xmath464 for oriented maniplexes of rank 5 . this work was done with the support of programa de apoyo a proyectos de investigacin e innovacin tecnolgica ( papiit ) de la unam , ib101412 _ grupos y grficas asociados a politopos abstractos _ `` . the second author was partially supported by slovenian research agency ( arrs ) and the third author was partially supported by conacyt under project 166951 and by the program para las mujeres en la ciencia loreal - unesco - amc 2012 ''
|
a @xmath0-orbit maniplex is one that has @xmath0 orbits of flags under the action of its automorphism group . in this paper
we extend the notion of symmetry type graphs of maps to that of maniplexes and polytopes and make use of them to study @xmath0-orbit maniplexes , as well as fully - transitive 3-maniplexes .
in particular , we show that there are no fully - transtive @xmath0-orbit 3-mainplexes with @xmath1 an odd number , we classify 3-orbit mainplexes and determine all face transitivities for 3- and 4-orbit maniplexes .
moreover , we give generators of the automorphism group of a polytope or a maniplex , given its symmetry type graph . finally , we extend these notions to oriented polytopes , in particular we classify oriented 2-orbit maniplexes and give generators for their orientation preserving automorphism group .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in many wireless sensor applications , temperature increase caused by sensor operation has to be carefully managed . for example , wireless sensors implanted in the human body have to be designed such that the temperature due to their operation does not cause any threat for the metabolism . a line of medical research started by pennes in 1948 @xcite explores the temperature dynamics due to electromagnetic radiation in conjunction with heat losses to the environment and dissipation of heat in the tissue . in the context of sensors that communicate data , temperature sensitivity varies depending on the type of tissue . for a given specific tissue , it is recommended that the temperature does not exceed a critical level , in order to prevent damage to the tissue . this necessitates careful scheduling of data transmission @xcite . this problem arises in various types of body area sensor networks , see e.g. , @xcite and references therein . finally , temperature increase in a sensor is a threat for the proper operation of the hardware itself @xcite . in this context , the electric power that feeds the amplifier circuitry has to be carefully scheduled so as to avoid permanent damage in the circuit . in order to obtain design principles with regard to temperature sensitivity of such systems , determining transmission schemes under a safe temperature threshold @xmath0 is a useful objective . in this paper , we consider data transmission with energy harvesting sensors under such temperature constraints . data transmission with energy harvesting transmitters has been the topic of recent research @xcite . in particular , throughput maximization under offline and online knowledge of the energy arrivals is considered in these references for single - user and multi - user energy harvesting communication systems . in @xcite , this problem is investigated under imperfections such as battery energy leakage , charge / discharge inefficiency , and presence of processing costs . in the current paper , we aim to bridge physical heat dissipation with data transmission in energy harvesting communication systems . when the sole purpose is to maximize the throughput , the transmitter may generate excessive heat while utilizing the energy resource . in a temperature sensitive application , the heat accumulation caused by the transmission power policy has to be explicitly taken into account . in such a case , heat generated in the transmitter circuitry causes a form of information - friction " @xcite . we study the effect of this friction " in a deadline constrained communication of an energy harvesting transmitter over an awgn channel . for simplicity , we use transmit power as a proxy for hardware power . that is , we assume that the energy dissipated by the power amplifier dominates other energy sinks in the circuitry . more work is needed to understand full implications of communication circuitry s energy in this context . our formulation also relates to @xcite in that the cumulative effect of heat generated in the hardware affects the communication performance . we determine the throughput optimal offline power scheduling policy under energy harvesting and temperature constraints . our thermal model is based on a view of the transmitter s circuitry as a linear heat system where transmit power is an input as in @xcite , @xcite , @xcite . we impose that the temperature does not exceed a critical level @xmath0 . consequently , we obtain a convex optimization problem . we solve this problem using a lagrangian framework and kkt optimality conditions . we first derive the structural properties of the solution for the general case of multiple energy arrivals . then , we obtain closed form solutions under a single energy arrival . for the general case , we observe that the optimal power policy may make jumps at the energy arrival instants , generalizing the optimal policies in @xcite . between energy harvests , the optimal power is monotonically decreasing . we establish for the case of a single energy arrival that the optimal power policy monotonically decreases , corresponding temperature monotonically increases , and both remain constant when the critical temperature is reached . then , we consider the case of multiple energy arrivals . we observe that the properties of the solution for the single energy arrival case are guaranteed to hold only in the last epoch of the multiple energy arrival case . in the remaining epochs , the temperature may not be monotone and the transmitter may need to cool down to create a temperature margin for the future , if the energy harvested in the future is large . we illustrate possible cases and obtain insights regarding the optimal temperature pattern in the multiple energy arrival case . we consider an energy harvesting transmitter node placed in an environment as depicted in fig . [ model ] . the node harvests energy to run its circuitry and wirelessly send data to a receiver . the received signal @xmath1 , the input @xmath2 , fading level @xmath3 and noise @xmath4 are related as @xmath5 where @xmath4 is additive white gaussian noise with zero - mean and unit - variance . in this paper , the channel is non - fading , i.e. , @xmath6 . we use a continuous time model : a scheduling interval has a short duration with respect to the duration of transmission and we approximate it as @xmath7 $ ] where @xmath8 denotes infinitesimal time . in @xmath7 $ ] , the transmitter decides a feasible transmit power level @xmath9 and @xmath10 bits are sent to the receiver , where the base of @xmath11 is @xmath12 . to be precise , the underlying physical signaling is in discrete time and the scalings in snr and rate due to bandwidth and the base of the logarithm are inconsequential for the analysis . . ] as shown in fig . [ model2 ] , the initial energy available in the battery at time zero is @xmath13 . energy arrivals occur at times @xmath14 in amounts @xmath15 with @xmath16 . we call the time interval between two consecutive energy arrivals an _ epoch_. @xmath17 is the deadline . @xmath18 and @xmath19 are known offline and are not affected by the heat due to transmission . let @xmath20 and @xmath21 be the number of energy arrivals in the interval @xmath22 and by convention we let @xmath23 . power scheduling policy @xmath9 is subject to energy causality constraints as : @xmath24\end{aligned}\ ] ] in our thermal model , we use the transmit power as a measure of heat dissipated to the environment . in particular , we model the temperature dynamics of the system as follows : @xmath25 where @xmath9 is the transmit power policy and @xmath26 is the temperature at time @xmath27 . @xmath28 is the constant temperature of the environment that is not affected by the heating effect due to the transmit power level @xmath9 . @xmath29 and @xmath30 are non - negative constants . @xmath31 represents the cumulative effect of additional heat sources and sinks and it can take both positive and negative values . in the following , we consider the case of no extra heat source or sink , i.e. , @xmath32 . our thermal model in ( [ thermal ] ) is intimately related to the thermal model in @xcite where hardware heating is modeled as a first order @xmath33 heat circuit . in particular , thermal dynamics of a power controlled transmitter due to its amplifier power consumption ( see e.g. , @xcite ) could be modeled as in ( [ thermal ] ) . we also refer the reader to @xcite for a related heating model . our thermal model is also related to the well - known pennes bioheat equation @xcite . we assume , for simplicity , that the spatial variation in temperature is not significant and leave the general case of spatial temperature variations as future work . becomes available for data transmission at time @xmath19 . @xmath17 is the deadline . ] from ( [ thermal ] ) , the solution of @xmath26 for any given @xmath9 with the initial condition @xmath34 at time @xmath35 is : @xmath36 by inserting @xmath37 in ( [ four ] ) , we get ( c.f . * eq . ( 3 ) ) ) : @xmath38 the temperature should remain below a critical temperature @xmath0 , i.e. , @xmath39 , where we assume that @xmath40 . let us define @xmath41 , which is the largest allowed temperature deviation from the environment temperature . typically , initial temperature is @xmath28 , i.e. , initially the temperature is stabilized at the constant environment temperature @xmath28 . from ( [ ths ] ) , using @xmath42 and @xmath43 , we get the following equivalent condition for the temperature constraint : @xmath44\end{aligned}\ ] ] note that the temperature constraints in ( [ gg ] ) and the energy causality constraints in ( [ causality ] ) do not interact . due to the heat generation dynamics governed by ( [ thermal ] ) , we observe in ( [ gg ] ) that the cost of power increases exponentially in time ( i.e. , the multiplier in front of @xmath45 is exponential in @xmath46 ) while the heat budget also increases exponentially in time ( i.e. , the upper bound on the right hand side of ( [ gg ] ) is exponential in @xmath27 ) . offline throughput maximization problem over the interval @xmath47 $ ] under energy causality and temperature constraints with initial temperature @xmath43 is : @xmath48 } \quad & \int_0^d \frac{1}{2}\log\left(1 + p(\tau)\right)d\tau \\ \nonumber \mbox{s.t . } \quad & \int_0^{t } a e^{b \tau } p(\tau ) d \tau \leq t_{\delta}e^{bt } , \quad \forall t \\ \qquad & \int_{0}^{t } p(\tau ) d\tau \leq \sum_{i=0}^{h(t ) } e_i , \quad \forall t \label{opt_prob1}\end{aligned}\ ] ] where the space of actions is the set of measurable functions @xmath9 defined over the interval @xmath47 $ ] . note that ( [ opt_prob1 ] ) is a convex functional optimization problem . the lagrangian for ( [ opt_prob1 ] ) is : @xmath49 taking the derivative of the lagrangian with respect to @xmath9 and equating to zero : @xmath50 which gives @xmath51^+\end{aligned}\ ] ] in addition , the complementary slackness conditions are : @xmath52 in ( [ rf ] ) and ( [ slck1])-([slck2 ] ) , @xmath53 and @xmath54 are distributions that are allowed to have impulses and their total measure over @xmath47 $ ] interval are not both zero , i.e. , @xmath55 or @xmath56 , in order to prohibit @xmath9 from being unbounded . we note that ( [ rf ] ) and ( [ slck1])-([slck2 ] ) are necessary and sufficient conditions since the problem is convex . the solution is unique almost everywhere as the objective function is strictly concave . we note that the problem in ( [ opt_prob1 ] ) could be solved by using calculus of variations . see @xcite for application of calculus of variations for a similar problem to ( [ opt_prob1 ] ) . as another alternative , we note that ( [ opt_prob1 ] ) could equivalently be solved by using a hamiltonian approach from optimal control theory . in particular , we can cast the problem in ( [ opt_prob1 ] ) as an optimal control problem with pure state constraints @xcite . in this case , the state of the system is the tuple @xmath57 $ ] where @xmath58 is the total energy expenditure by the time @xmath27 . the input is @xmath9 for @xmath59 . this problem is in the following form : @xmath48 } \quad & \int_0^d \frac{1}{2}\log\left(1 + p(\tau)\right)d\tau \\ \nonumber \mbox{s.t . } \quad & \frac{d}{dt}t(t ) = f_1(t , b , p ) , \ \frac{d}{dt}b(t)= f_2(t , b , p ) \\ \qquad & g_1(t , b , t ) \leq 0 , \ g_2(t , b , t ) \leq 0 \label{opt_prob2}\end{aligned}\ ] ] where @xmath60 and @xmath61 while @xmath62 and @xmath63 . note that @xmath64 and @xmath65 do not depend on the input @xmath66 . with these selections , optimization problem ( [ opt_prob2 ] ) is in the same form as that stated in ( * ? ? ? ( 2.1)-(2.6 ) ) . in this case , hamiltonian is @xmath67 and the corresponding lagrangian is @xmath68 where @xmath69 and @xmath70 are the co - state trajectories ; @xmath71 and @xmath72 are multiplier functions . we note that pontryagin s maximum principle is necessary and sufficient in this case since ( [ opt_prob2 ] ) is a concave maximization problem . one can derive the equivalence of necessary and sufficient conditions for this optimal control problem to those in ( [ rf ] ) and ( [ slck1])-([slck2 ] ) . in the following , we proceed with the lagrangian formulation in ( [ lgrng ] ) and the corresponding optimality conditions in ( [ rf ] ) and ( [ slck1])-([slck2 ] ) . in this section , we obtain the structural properties of the optimal power scheduling policy using the optimality conditions . in the following lemmas , @xmath9 refers to the optimal policy and @xmath26 is the resulting temperature unless otherwise stated . we first note that the temperature level never drops below @xmath28 . in particular , if the initial temperature is between @xmath28 and @xmath0 , the temperature at all times will remain between @xmath28 and @xmath0 . [ temp ] @xmath73 whenever the initial temperature is @xmath74 . from ( [ thermal ] ) , since @xmath75 we have @xmath76 whenever @xmath77 . the constraint @xmath39 is satisfied by any feasible policy in ( [ opt_prob1 ] ) . the following lemma states that if the temperature @xmath26 is constant , then the power @xmath9 is constant also ( while it is not true the other way around , see lemma [ piecewise ] ) , and that if the temperature hits the maximum allowed level @xmath0 , then the power must be below a threshold . [ constant ] whenever @xmath26 is constant over an interval @xmath78 $ ] , @xmath9 is also constant over that interval . if the temperature hits the level @xmath0 at @xmath79 , then @xmath80 for all sufficiently small @xmath81 . if @xmath26 is constant in @xmath82 , @xmath83 and from ( [ thermal ] ) , @xmath9 is also constant in the same interval . if @xmath84 for some @xmath85 , then @xmath86 and from ( [ thermal ] ) , @xmath87 . the following lemma shows that if the power @xmath9 is a monotone increasing function , then so is the temperature @xmath26 . we first prove this result for piecewise constant functions and then generalize it to arbitrary functions . we note that a particular instance of a monotone increasing piecewise constant power is observed in the solution of the throughput maximization problem without temperature constraints @xcite . [ piecewise ] if @xmath9 is a monotone increasing piecewise constant function , then @xmath26 is monotone increasing . more generally , if @xmath9 is a monotone increasing function , so is @xmath26 . we first prove the first statement of the lemma which is concerned with piecewise constant functions . let us start with the case of a single constant power value for the entire duration of communication , i.e. , @xmath88 for @xmath89 $ ] . from ( [ ths ] ) , we have : @xmath90 for @xmath43 , ( [ fin ] ) is a monotone increasing function of @xmath27 . in particular , @xmath91 . now , let us consider the case of @xmath92 constant power levels for the duration of communication , i.e. , @xmath93 over the interval @xmath94 where @xmath95 for all @xmath96 and @xmath97 where @xmath98 is the number of intervals . in this case , we have for @xmath99 : @xmath100 where @xmath101 . hence , the coefficient of @xmath102 in ( [ fin2 ] ) has a negative sign as @xmath103 . this proves that @xmath26 is monotone increasing . to generalize this result for any monotone increasing function @xmath9 , we obtain any monotone increasing simple approximation @xcite of @xmath9 , denoted as @xmath104 , such that @xmath105 for all @xmath89 $ ] and @xmath106 pointwise . for example , one can select @xmath107 for @xmath108 and @xmath109 for @xmath110 . let us call the resulting temperature @xmath111 . hence , @xmath112 for all @xmath89 $ ] and @xmath113 pointwise . by monotone convergence theorem @xcite , we have @xmath114\end{aligned}\ ] ] accordingly , @xmath115 pointwise and we have @xmath116\end{aligned}\ ] ] since @xmath104 is a monotone increasing piecewise constant function , from the first part of the proof , @xmath111 is monotone increasing , i.e. , @xmath117 . since @xmath118 pointwise , this implies @xmath119 , i.e. , @xmath26 is monotone increasing as well . the next lemma shows that if the temperature remains constant over an interval , then that level could only be @xmath28 or @xmath0 , i.e. , any other temperature can not be a stable temperature . [ const_temp ] if @xmath26 is constant over an interval @xmath78 $ ] , then that constant level could only be @xmath28 or @xmath0 . assume @xmath26 is constant over @xmath82 . without loss of generality , assume that there is no energy arrival in the interval @xmath82 , and otherwise let @xmath82 be the portion of the interval without any energy arrivals . by lemma [ constant ] , @xmath9 is constant over @xmath82 . if @xmath120 over @xmath82 , then @xmath77 from ( [ thermal ] ) . if @xmath121 , we have from ( [ nn ] ) @xmath122 where @xmath123 over the interval @xmath82 by ( [ slck2 ] ) since @xmath124 implies energy constraint is tight and @xmath120 . therefore , @xmath125 is constant over @xmath82 . if @xmath126 , then by ( [ slck1 ] ) , @xmath127 over @xmath82 and hence @xmath128 is constant over @xmath82 . however , this makes ( [ nn2 ] ) a time varying function of @xmath27 because of the @xmath129 term in the denominator , and this contradicts the fact that @xmath9 is constant . finally , if @xmath130 , this means that the temperature constraint is never tight . in this case , the piecewise constant power policy in @xcite is optimum , and the temperature is monotonically increasing from lemma [ piecewise ] , and therefore , can not be a constant over an interval . the following lemma states that at the end of the communication session either the harvested energy is exhausted or the critical temperature is reached . [ lm12 ] at @xmath131 , either the temperature constraint or the energy causality constraint or both are tight . if neither of the constraints are tight , then the power policy @xmath9 could be increased over a set of non - zero lebesgue measure in the last epoch . this strictly increases the throughput , contradicting the optimality . the following lemma shows that the optimal power should be monotonically decreasing between energy harvests . [ new_lm ] @xmath9 is piecewise monotone decreasing except possibly at the energy arrival instants . in particular , it is monotone decreasing between consecutive energy harvests . we prove the statement by contradiction . assume that for some interval @xmath132 $ ] , @xmath9 is strictly monotone increasing , and that the interval @xmath132 $ ] does not contain an energy arrival instant . define a new power policy as @xmath133 over @xmath134 $ ] and @xmath135 otherwise . @xmath136 satisfies the energy causality constraint in ( [ opt_prob1 ] ) since @xmath136 uses the same amount of energy as @xmath9 over @xmath132 $ ] and the energy constraint for @xmath9 is not tight in this interval . @xmath136 also satisfies the temperature constraint . to see this , we first note that @xmath136 satisfies the following inequality ( see ( * ? ? ? * theorem on p. 207 ) ) : @xmath137 as both @xmath9 and @xmath129 are monotone increasing . in addition , since @xmath9 is temperature feasible : @xmath138 combining ( [ cnt1 ] ) and ( [ cnt3 ] ) , we conclude that @xmath136 satisfies the temperature constraint at @xmath139 : @xmath140 additionally , the temperature constraint is satisfied for @xmath141 since @xmath136 and @xmath9 are identical for @xmath141 and @xmath9 is temperature feasible . hence , we need to show that @xmath136 satisfies the temperature constraint for all @xmath142 to establish the temperature feasibility of @xmath136 . that is , we need to show : @xmath143 since @xmath144 is constant over @xmath132 $ ] , we have : @xmath145\end{aligned}\ ] ] using ( [ corin ] ) in ( [ gggg ] ) and since @xmath146 , ( [ gggg ] ) takes the following equivalent form : @xmath147 note that the left hand side of ( [ ffgg ] ) is either monotone increasing or monotone decreasing in @xmath27 as it is a linear function of @xmath148 . since the inequality ( [ ffgg ] ) holds at @xmath149 and @xmath139 as @xmath136 satisfies the temperature constraint at those points , we conclude that @xmath136 satisfies the temperature constraint for all @xmath134 $ ] . in addition , @xmath136 yields higher throughput than @xmath9 due to the concavity of logarithm . this contradicts the optimality of @xmath9 . the proof holds even when @xmath132 $ ] includes an energy arrival instant provided that the energy causality constraint is not tight at that instant . next , we show that discontinuities in the power level could only occur in the form of positive jumps , and only at the instances of energy harvests . [ lm1 ] if there is a discontinuity in @xmath9 , it is a positive jump and it occurs only at the energy arrival instants . the temperature @xmath26 is continuous throughout the @xmath47 $ ] interval . since @xmath129 is a continuous function of @xmath27 , @xmath53 and @xmath150 , any jump in @xmath9 has to be positive due to ( [ nn ] ) . any positive jump at instants other than @xmath151 violates monotonicity of @xmath9 within each epoch due to lemma [ new_lm ] . due to ( [ ths ] ) , the resulting temperature @xmath26 is continuous throughout the @xmath47 $ ] interval . by lemma [ lm1 ] , we can take @xmath152 in the form @xmath153 without loss of optimality , where @xmath154 , @xmath155 , are finitely many lagrange multipliers corresponding to the energy causality constraints at the energy harvesting instants @xmath156 and the deadline , @xmath23 . the next lemma shows , for an arbitrary feasible policy @xmath9 , that if the temperature reaches the critical level @xmath0 at some @xmath157 , then the power just before @xmath157 must be larger than a threshold . [ lm2 ] if @xmath84 for some @xmath85 , then @xmath158 for all sufficiently small @xmath81 . since @xmath84 , we have : @xmath159 we combine ( [ gg ] ) with ( [ kk ] ) to get @xmath160\end{aligned}\ ] ] which implies in view of the continuity of @xmath9 ( except for the finitely many energy arrival instants ) proved in lemma [ lm1 ] that @xmath158 for all sufficiently small @xmath81 . we next state the continuity of the optimal power policy @xmath9 at points when it hits the critical temperature @xmath0 . [ kkk ] if @xmath84 for some @xmath85 then @xmath9 is continuous at @xmath157 and @xmath161 . the proof follows from lemma [ constant ] and lemma [ lm2 ] and the fact that negative jumps in @xmath9 are not allowed due to lemma [ lm1 ] . next , we show that when the temperature hits the boundary @xmath0 , it has to return to @xmath0 . [ extend ] whenever @xmath84 for some @xmath162 , there exists @xmath163 such that @xmath164 . assume that @xmath84 for some @xmath162 and @xmath126 for all @xmath165 . by lemma [ kkk ] , @xmath166 . from ( [ four ] ) with @xmath84 , the constraint @xmath39 becomes : @xmath167 since @xmath126 in @xmath165 , only energy causality constraint is active and thus @xmath9 for @xmath165 is the piecewise constant monotone power allocation in @xcite . on the other hand , @xmath168 satisfies ( [ reef_r2 ] ) with equality for all @xmath27 . therefore , we must have @xmath169 for all @xmath170 for some @xmath171 . however , this contradicts @xmath166 since there can not be a negative jump in @xmath9 by lemma [ lm1 ] . the following lemma identifies the exact conditions where the power @xmath9 makes a jump . if there is a jump in @xmath9 , it occurs only at an energy arrival instant , when the battery is empty and the temperature is strictly below @xmath0 . due to the slackness conditions in ( [ slck1])-([slck2 ] ) , a jump occurs if either the battery is empty or the temperature constraint is tight , i.e. , @xmath164 . by lemma [ kkk ] , @xmath9 is continuous whenever @xmath164 . therefore , a jump in @xmath9 occurs at an energy arrival instant , when the battery is empty and @xmath126 . we finally remark that energy may have to be wasted as aggressive use of energy may cause temperature to rise above the critical level . in this section , we consider a single epoch where @xmath172 units of energy is available at the transmitter at the beginning . we first develop further structural properties for the optimal power control policy in this specific case and then obtain the solution . the next lemma shows that , if the power falls below a certain threshold at an intermediate point and remains under that threshold until the deadline , then it should remain constant throughout . [ lmn ] if @xmath173 for @xmath174 $ ] , then @xmath9 is constant over @xmath175 $ ] . assume @xmath9 is not constant over @xmath175 $ ] . let @xmath176 . define a new policy @xmath177 for @xmath178 $ ] and @xmath135 otherwise . @xmath136 is both energy and temperature feasible . energy feasibility holds by construction as @xmath179 and @xmath66 have the same energy over @xmath180 $ ] . temperature feasibility also holds : @xmath181 since @xmath9 is temperature feasible and as @xmath182 , we have @xmath42 for all @xmath183 from ( [ gg ] ) . now , by jensen s inequality @xmath136 achieves strictly larger throughput since @xmath11 is strictly concave . this contradicts the optimality of @xmath9 . hence , @xmath184 for @xmath178 $ ] . the following lemma states that the power has to remain constant at the level @xmath185 when the temperature reaches the critical level @xmath0 . [ multiple_r ] let @xmath186 $ ] denote @xmath187 : t(t)=t_c\}$ ] . if @xmath188 , then @xmath189 for all @xmath190 $ ] . by lemma [ kkk ] , @xmath191 . by lemma [ new_lm ] , @xmath9 is monotone decreasing , and thus @xmath192 for @xmath193 . by lemma [ lmn ] , @xmath194 for all @xmath190 $ ] . by lemma [ lm1 ] , @xmath9 is continuous and therefore , @xmath189 for all @xmath190 $ ] . the following lemma states that the optimal power is always larger than a constant value determined by the fixed system parameters . [ corl ] the optimal policy @xmath9 satisfies : @xmath195\end{aligned}\ ] ] if the temperature constraint is not tight , then the problem reduces to the energy constrained problem in which case @xmath196 . if the temperature constraint is tight , @xmath9 is monotone decreasing by lemma [ new_lm ] and when the temperature level reaches @xmath0 , @xmath9 remains at @xmath185 by lemma [ multiple_r ] . hence , @xmath197 . the following lemma shows that , since the power is always larger than a constant value , battery energy level is never zero , except possibly at the deadline . [ cor1 ] in an optimal policy , energy in the battery is non - zero except possibly at @xmath131 . by lemma [ corl ] , the optimal power is always larger than a positive constant . thus , the battery energy does not drop to zero . the following lemma shows that the temperature is monotone increasing throughout the transmission duration , and also is a concave function of time . [ temperature ] the temperature with the optimal power policy is monotone increasing and concave . if the temperature constraint is never tight , then the optimal power level is @xmath198 , and from lemma [ piecewise ] , the temperature is monotone increasing . concavity in this case follows from the concavity of the explicit expression in ( [ fin ] ) with @xmath43 . now , assume that the temperature constraint is tight at @xmath131 . by lemma [ corl ] , @xmath197 . from ( [ thermal ] ) , we have : @xmath199 as @xmath39 by the temperature constraint . since @xmath9 is monotone decreasing by lemma [ new_lm ] and @xmath26 is monotone increasing , from ( [ sxtn ] ) , @xmath200 is monotone decreasing , proving the concavity of @xmath26 in this case . in view of lemma [ cor1 ] , the energy constraint can be tight only at @xmath131 . therefore , the corresponding lagrange multiplier is a single variable @xmath201 . from lemma [ temperature ] , @xmath26 is monotone increasing . due to lemma [ multiple_r ] , when @xmath26 reaches @xmath0 , power level has to remain at @xmath185 . accordingly , we denote the instant when the temperature reaches @xmath0 as @xmath202 . in this case , the energy constraint is never tight , and @xmath203 . in view of lemma [ lm12 ] , the temperature constraint is tight at @xmath131 . first , consider the case that @xmath17 is sufficiently large so that there exists @xmath204 such that @xmath205 . for @xmath206 , @xmath126 and from ( [ slck1 ] ) , @xmath127 . from ( [ nn ] ) , when @xmath206 we have @xmath207 where @xmath208 . since at @xmath209 the temperature reaches @xmath0 , from lemma [ multiple_r ] , we have @xmath189 for @xmath210 $ ] . then , the optimal power has the form : @xmath211 where @xmath212 is the unit step function . now , from lemma [ kkk ] , @xmath9 is continuous at @xmath202 and @xmath213 should be chosen accordingly . in particular , @xmath214 . the following lagrange multiplier @xmath215 verifies ( [ kkl ] ) : @xmath216 the corresponding optimal temperature pattern for @xmath217 is : @xmath218 and @xmath164 for @xmath219 . we note that @xmath202 satisfies : @xmath220 so that @xmath205 . hence , @xmath26 monotonically increases till it reaches @xmath0 , which is consistent with lemma [ temperature ] . next , consider the case that @xmath221 . in this case , @xmath222 where @xmath223 and @xmath224 . therefore , the optimal @xmath9 in this case is @xmath225 we also remark that @xmath202 level that satisfies ( [ sol ] ) monotonically increases with @xmath226 . to see this , we rearrange ( [ sol ] ) as follows : @xmath227 let us define a multi - variable real function @xmath228 as the left hand side of ( [ sol2 ] ) and denote a specific solution as @xmath229 for fixed @xmath226 . it is easy to see that ( [ sol2 ] ) always has a solution @xmath202 for fixed @xmath226 . to see this , we evaluate the derivative with respect to @xmath202 as : @xmath230 that is , @xmath228 is monotone decreasing with @xmath202 . at @xmath231 , @xmath232 while @xmath233 as @xmath202 grows . in view of the continuity of @xmath228 , there exists a @xmath202 such that @xmath234 . additionally , we observe in ( [ sol2 ] ) that for fixed @xmath202 , @xmath228 monotonically increases with @xmath226 . therefore , if @xmath235 , then , due to monotone increasing property with respect to @xmath226 , @xmath236 for @xmath237 . hence , for @xmath238 such that @xmath239 , we have @xmath240 due to monotone decreasing property with respect to @xmath202 . note that the optimal power policies in the energy unconstrained cases in ( [ kkl ] ) and ( [ halfway ] ) have finite energies . if the available energy @xmath172 is larger than the corresponding energy level in ( [ kkl ] ) and ( [ halfway ] ) , then the solution is as in ( [ kkl ] ) and ( [ halfway ] ) . otherwise , the energy constraint is active and the lagrange multiplier is @xmath241 . from ( [ nn ] ) , we have : @xmath242 we first note that there is a critical energy level @xmath243 such that if @xmath244 , then constant power policy @xmath196 is optimal . this critical level is : @xmath245 this is the critical level below which the temperature constraint is not tight by the constant power allocation @xmath196 . the expression in ( [ thrtn ] ) is evaluated from ( [ fin ] ) by inserting @xmath43 , and requiring @xmath246 . when @xmath247 , @xmath127 since temperature constraint is never tight . in this case , @xmath243 is the maximum energy level for which a constant power level is optimal . if @xmath249 , @xmath26 is monotone increasing over @xmath47 $ ] and reaches @xmath0 at @xmath131 . if @xmath250 , the constant power level @xmath251 does not satisfy the temperature constraint . we note from ( [ thrtn ] ) that @xmath243 increases with the deadline @xmath17 . therefore , there exists a deadline level @xmath252 for which @xmath253 implies @xmath254 and hence constant power policy is optimal . an alternative way of observing the behavior of the optimal policy is to fix the available energy @xmath172 and @xmath28 and vary the critical temperature @xmath0 . in this case , there is a critical temperature limit @xmath255 for which @xmath196 is optimal whenever @xmath256 : @xmath257 which again is evaluated from ( [ fin ] ) with @xmath43 . in the following , we consider @xmath250 or @xmath258 so that both energy and temperature constraints are tight at the end of the communication session . again , we consider two possibilities : temperature constraint becomes tight at a @xmath204 , and temperature constraint becomes tight at @xmath131 . in both cases , the energy constraint becomes tight at @xmath131 . first , consider the case that @xmath204 : due to ( [ slck1 ] ) , @xmath127 for @xmath206 and from ( [ nn ] ) , we get : @xmath259 where @xmath208 . additionally , @xmath189 for the remaining portion of the epoch in view of lemma [ multiple_r ] . @xmath202 is such that for @xmath260 , @xmath189 and @xmath205 . since @xmath261 we have : @xmath262 similarly , for @xmath205 , we have from ( [ ths ] ) with @xmath43 : @xmath263 finally , the energy constraint has to be satisfied at @xmath131 : @xmath264 if there exists @xmath265 for ( [ bir])-([uc ] ) , then @xmath9 is : @xmath266 in this case , the corresponding lagrange multiplier is : @xmath267 otherwise , when no such @xmath204 exists , the temperature constraint is tight only at @xmath268 . in this case , @xmath9 is as in ( [ yyy ] ) for @xmath89 $ ] where @xmath269 and @xmath213 have to satisfy : @xmath270 the corresponding lagrange multiplier is @xmath224 . depending on the energy @xmath172 and the critical temperature @xmath0 , the optimal power scheduling policy @xmath9 varies according to the plots in fig . [ reg_o ] . for small @xmath172 and fixed @xmath0 or for large @xmath0 and fixed @xmath172 , a constant power policy is optimal . for moderate and large @xmath172 , the optimal power policy is exponentially decreasing and may hit the power level @xmath271 . note that @xmath202 level at which temperature touches the critical level decreases as @xmath0 is decreased and as @xmath172 is increased . in particular , for fixed @xmath0 , the level of @xmath202 is bounded below by the solution for @xmath272 whereas for fixed @xmath172 , @xmath202 goes to @xmath273 as @xmath0 approaches @xmath28 . in this section , we extend the solution to the case of multiple energy arrivals . we start with extending the properties observed for the single energy arrival case when initial temperature @xmath274 is different from @xmath28 . the following lemma generalizes lemmas [ new_lm ] , [ multiple_r ] and [ temperature ] for the case of an arbitrary @xmath274 . [ extension ] assume that the initial temperature @xmath274 is in the range @xmath275 instead of @xmath43 and consider the single energy arrival case : @xmath9 is monotone decreasing . let @xmath276 $ ] denote @xmath187 : t(t)=t_c\}$ ] . if @xmath277 , then @xmath189 for all @xmath278 $ ] and the temperature is monotone increasing and concave . if @xmath279 , then @xmath280 . if @xmath274 is in the range @xmath281 then , instead of ( [ gg ] ) , we have the following temperature constraint : @xmath282\end{aligned}\ ] ] where @xmath283 . note that @xmath284 for all @xmath89 $ ] , i.e. , the right hand side of ( [ gg2 ] ) is always non - negative . the argument in lemma [ new_lm ] is valid in the presence of the additional term @xmath285 in ( [ gg2 ] ) , and therefore @xmath9 is monotone decreasing . the second claim follows from the argument in lemma [ multiple_r ] . in particular , in addition to lemma [ new_lm ] , lemma [ lmn ] directly extends with the constraint in ( [ gg2 ] ) . hence , the result follows by applying the argument in lemma [ multiple_r ] . finally , @xmath26 is monotone increasing and concave due to the steps followed in lemma [ temperature ] . in particular , if the temperature constraint is tight at @xmath131 , @xmath197 . hence , ( [ sxtn])-([svnt ] ) hold and the temperature is monotone increasing and concave . if @xmath279 , then @xmath280 due to the energy constraint . note that the temperature decreases in case @xmath286 . as in the single epoch case , we will investigate the solution under special cases . in particular , we will investigate the solution according to the time when the temperature hits the critical level . to this end , we specialize in an interval @xmath132 $ ] such that @xmath126 for all @xmath287 and @xmath288 where @xmath289 . note that the temperature @xmath26 is a continuous function of @xmath27 and hence there exist such intervals . we assume that the solution is known in @xmath290 $ ] and we let @xmath291 . in this case , the solution of ( [ opt_prob1 ] ) over the interval @xmath132 $ ] is equal to the solution of the following problem obtained by restricting the temperature constraint to be satisfied at @xmath139 only : @xmath292 } \quad & \int_{t_1}^{t_2 } \frac{1}{2}\log\left(1 + p(\tau)\right)d\tau \\ \nonumber \mbox{s.t . } \quad & \int_{t_1}^{t_2 } a e^{b \tau } p(\tau ) d \tau = t_{\delta}e^{bt_2 } - t_g \\ \qquad & \int_{0}^{t } p(\tau ) d\tau \leq \sum_{i=0}^{\tilde{h}(t ) } \tilde{e}_i , \qquad \forall t \in [ t_1,t_2 ] \label{opt_prob_relax}\end{aligned}\ ] ] where @xmath293 . in ( [ opt_prob_relax ] ) , @xmath294 is determined as follows : @xmath295 is the available energy in the battery at time @xmath149 . @xmath294 for @xmath296 are the energy arrivals at instants @xmath297 . @xmath298 is defined accordingly . while the times @xmath299 are exactly those in the original problem , the amounts @xmath294 may be different from the original amounts as some energy may be left for use in the @xmath300 $ ] interval . for the following argument , whether @xmath294 equals the original energy arrival amount is not relevant and we leave @xmath294 as arbitrary amounts . to obtain the solution of ( [ opt_prob_relax ] ) using this lagrangian framework , it is necessary and sufficient to find @xmath301 variables @xmath302 , @xmath303 and @xmath304 such that @xmath305^+ , \quad t \in [ \tilde{s}_{i-1},\tilde{s}_{i } ) , \ i=1,\ldots , \tilde{n}+1\end{aligned}\ ] ] with the corresponding slackness conditions . therefore , for the @xmath132 $ ] interval , the solution has the structure in ( [ pow ] ) , which is parameterized by finitely many lagrange multipliers . in particular , throughout an epoch over which @xmath126 , power level satisfies @xmath306^{+}$ ] for some @xmath307 and @xmath304 not both equal to zero . this also holds in a subinterval of an epoch over which @xmath126 . in the following lemma , we show that in such an epoch , the temperature @xmath26 is unimodal . [ unimodal ] if @xmath308^{+}$ ] for @xmath134 $ ] for some @xmath309 and @xmath310 , the resulting @xmath26 is unimodal over @xmath132 $ ] . from ( [ four ] ) , we have for @xmath134 $ ] , @xmath311^+ + bt_e\right ) d \tau + t(t_1 ) \right)\end{aligned}\ ] ] first , we note that when @xmath120 , @xmath312 from ( [ thermal ] ) . hence , it suffices to show that @xmath26 is unimodal when @xmath313 . by evaluating the integral , we get @xmath314 we claim that @xmath26 in ( [ lbbb ] ) is unimodal for @xmath315 . note that the derivative of @xmath26 is : @xmath316 we let @xmath317 , @xmath318 and concentrate on @xmath319 for @xmath320 . we note that @xmath319 is a strictly monotone decreasing function of @xmath321 for @xmath322 . in particular , we have : @xmath323 thus , @xmath324 is strictly monotone decreasing in @xmath27 . as @xmath325 at @xmath149 , we conclude that the factor in ( [ dfsa ] ) that multiplies @xmath148 can take value @xmath273 at most once . in particular , @xmath326 can take positive or negative values at @xmath149 . if it is positive at @xmath149 , it hits value @xmath273 at most once for @xmath315 . if it is negative at @xmath149 , it stays negative throughout @xmath315 . this proves that @xmath26 is unimodal over @xmath132 $ ] . in the following lemma , we show that , in an epoch @xmath327 $ ] , the temperature can not return to @xmath0 if it hits and falls below @xmath0 . [ new ] if @xmath84 and @xmath328 for some @xmath329 where both @xmath157 and @xmath330 are in @xmath327 $ ] , then @xmath126 for all @xmath331 $ ] . by lemma [ kkk ] , @xmath161 . by lemma [ new_lm ] , power is monotone decreasing in an epoch . therefore , if @xmath328 , then @xmath332 and hence @xmath333 for all @xmath331 $ ] . this , in turn , means that @xmath126 for all @xmath331 $ ] . next , we complete the unimodal structure of the temperature by showing that it has to be monotone decreasing if it hits and falls below @xmath0 . [ new2 ] in an epoch @xmath327 $ ] , if the temperature touches @xmath0 at @xmath157 and falls below it , then the temperature is monotone decreasing in @xmath334 $ ] . by lemma [ new ] , if @xmath328 , then @xmath126 for all @xmath335 $ ] . therefore , we have @xmath336^{+ } , \quad t \in [ t_h+\delta , s_{i+1}]\end{aligned}\ ] ] for some @xmath337 and @xmath310 . by lemma [ unimodal ] , @xmath26 is unimodal over @xmath335 $ ] . therefore , @xmath26 is monotone decreasing . we next consider epochs @xmath327 $ ] and its subintervals over which @xmath126 and @xmath164 . by lemma [ unimodal ] and in view of the discussion around ( [ opt_prob_relax ] ) , whenever @xmath126 over an epoch , @xmath26 reaches its peak level over that epoch at only one instance . consequently , if @xmath126 for all @xmath338 $ ] , there are three possible cases . the first two possibilities are that @xmath26 is monotone increasing or monotone decreasing throughout the epoch . the third possible case is that @xmath26 is monotone increasing in @xmath339 $ ] and monotone decreasing in @xmath340 $ ] for some @xmath341 . otherwise , @xmath26 hits @xmath0 and @xmath26 does not return to @xmath0 if it falls below it due to lemma [ new ] . therefore , if @xmath26 hits @xmath0 in an epoch @xmath342 $ ] , then that epoch is divided into three successive subintervals @xmath343 with @xmath344 , @xmath345 and @xmath346 $ ] for some @xmath347 . @xmath26 is monotone increasing over @xmath348 , remains at @xmath0 over @xmath349 and is monotone decreasing over @xmath350 . we finally note that if @xmath126 at @xmath131 , then @xmath126 for all @xmath89 $ ] . this follows from lemma [ extend ] . in this case , the temperature constraint is never tight and the optimal power policy is identical to the one in @xcite . is decreased , the energy is spent faster subject to energy causality . ] in fig . [ exp2 ] , we plot the optimal energy expenditure for different values of critical temperature level @xmath0 . we observe that as @xmath0 is decreased , the temperature budget shrinks and the temperature constraint becomes more likely to be tight . in this case , energy is spent faster not to create high amounts of heat in the system . in general , there is a tension between causing unnecessary heat in the system and maximizing the throughput . while we have fully characterized this tension in the single energy arrival case , it needs to be further explored in the multiple energy arrivals case . in particular , when a high amount of energy arrives into the system during the progression of communication , the transmitter has to accommodate it by cooling down and creating a temperature margin for future use . while maximizing the throughput generally requires using the energy in the system to the fullest extent , the transmitter may have to waste energy due to the temperature limit . we investigate this tension in numerical examples in the next section . in this section , we provide numerical examples to illustrate the optimal power policy and the resulting temperature profile . for plots in figs . [ model4 ] , [ model5 ] , [ model6 ] and [ model7 ] , we set @xmath351 , @xmath352 , @xmath353 and @xmath354 . therefore , the critical power level is @xmath355 . in figs . [ model4 ] and [ model5 ] , we consider the energy unlimited scenario . in this case , the solution of ( [ sol ] ) is found as @xmath356 . in fig . [ model4 ] , we set @xmath357 and we observe that the optimal power policy is always above the level @xmath185 . in this case , power strictly monotonically decreases while temperature strictly monotonically increases with temperature touching the critical level @xmath0 at the deadline . in fig . [ model5 ] , we set the deadline as @xmath358 . we calculate that the energy needed to have the power policy in fig . [ model5 ] is @xmath359 . in other words , if the initial energy is @xmath360 then the power policy in fig . [ model5 ] is optimal . we observe that the optimal power level monotonically decreases to the level @xmath185 and remains at that level afterwards . similarly , the temperature level rises to @xmath0 and remains at that level afterwards . note that the throughput and the energy consumption in fig . [ model5 ] are higher with respect to those in fig . [ model4 ] . parallel to this observation , the monotone decrease is sharper in the power policy in fig . [ model5 ] compared to that in fig . [ model4 ] . since the power level has to be stabilized at @xmath185 , the temperature increase cost paid for achieving certain throughput is minimized if energy consumption starts faster and drops later . in fig . [ model6 ] , we set the deadline to @xmath361 and the energy limit to @xmath362 . note that this energy level is slightly less than the energy of the power policy in fig . [ model5 ] , which translates into a right shift of the point @xmath202 . in particular , we calculate @xmath363 as the solution of ( [ bir])-([uc ] ) in this case . similar to the effect of decreasing the deadline observed in the comparison of figs . [ model4 ] and [ model5 ] , we observe that decreasing the energy level yields a _ smoother _ power policy . power level drops to @xmath185 and the temperature hits @xmath0 at a later time @xmath202 and both remain constant afterwards . for the single epoch case . ] in fig . [ model7 ] , we consider the same system as in previous figures with two energy arrivals instead of one and with @xmath364 . in particular , @xmath365 is available initially and @xmath366 arrives at time @xmath367 . in this case , we calculate @xmath368 . the energy causality constraint is tight and the power level makes a jump at the energy arrival instant . note that the temperature is continuous at the energy arrival instant even though its first derivative is not . while the power level has a smooth start , a sharper decrease is observed towards the end since the harvested energy has to be fully utilized . in particular , the temperature increase before the energy arrival is kept to a minimum level so as to have a higher heat budget for the larger energy that arrives later . the temperature hits @xmath0 at @xmath369 after which the power and temperature both remain constant . for the single epoch case . ] finally in fig . [ model8 ] , we illustrate a curious behavior in the optimal policy . for this example , we set @xmath351 , @xmath370 , @xmath371 and @xmath372 . initial energy is @xmath373 and energy arrives at @xmath374 with amount @xmath375 and the deadline is @xmath361 . we observe that energy causality constraint is tight at @xmath374 whereas it is not tight at @xmath131 meaning that some energy is wasted in order not to cause excessive heat . the temperature generated in this throughput optimal power policy first monotonically increases , hits @xmath0 at @xmath376 , remains there till @xmath377 and drops below @xmath0 . we interpret the drop in the temperature in the first epoch as an effort to create temperature margin for the high energy arrival in the next epoch . we calculate @xmath378 as the time after which power level remains at @xmath379 and the temperature remains at @xmath0 . note that under unlimited energy , temperature would hit @xmath0 at @xmath380 . due to the energy scarcity in the first epoch , temperature hits @xmath0 later and drops below @xmath0 . a common behavior we observe in each numerical example is that temperature ultimately increases between two epochs where energy causality constraint is tight . further research is needed to quantify the relations between the amount of temperature generated while performing optimally in terms of throughput . while monotonicity of the temperature is lost when multiple energy harvests exist , we note that monotonicity of the temperature is guaranteed in the last epoch due to lemma [ extension ] . and @xmath381 for the single epoch case . ] and @xmath366 at @xmath382 and @xmath364 . ] at @xmath383 and @xmath375 at @xmath374 and @xmath361 . ] we considered throughput maximization for an energy harvesting transmitter over an awgn channel under temperature constraints . we used a linear system model for the heat dynamics and determined the throughput optimal power scheduling policy under a maximum temperature constraint by using a lagrangian framework and the kkt optimality conditions . we determined for the single energy arrival case that the optimal power policy is monotone decreasing whereas the temperature is monotone increasing and both remain constant after the temperature hits the critical level . we then generalized the solution for the case of multiple energy arrivals . while monotonicity of the temperature is lost when multiple energy harvests exist , we observed that the temperature ultimately increases while maximizing the throughput . we also observed that the main impact of the temperature constraints is to facilitate faster energy expenditure subject to energy causality constraints . additionally , even though using all of the available energy is optimal for throughput maximization only , with temperature constraints , energy may have to be wasted in order not to exceed the critical temperature . q. tang , n. tummala , s. gupta , and l. schwiebert , `` communication scheduling to minimize thermal effects of implanted biosensor networks in homogeneous tissue , '' _ ieee trans . on biomed . _ , vol . 52 , pp . 12851294 , july 2005 . s. ullah , h. higgins , b. braem , b. latre , c. blondia , i. moerman , s. saleem , z. rahman , and k. kwak , `` a comprehensive survey of wireless body area networks , '' _ jour . medical . _ , vol . 36 , pp . 10651094 , june 2012 . a. mutapcic , s. boyd , s. murali , d. atienza , g. de micheli , and r. gupta , `` processor speed control with thermal constraints , '' _ ieee trans . on circ . and sys .- i _ , vol . 56 , pp . 19942008 , september 2009 . o. ozel , k. tutuncuoglu , j. yang , s. ulukus , and a. yener , `` transmission with energy harvesting nodes in fading wireless channels : optimal policies , '' _ ieee jour . on selected areas in commun . _ , vol . 29 , pp . 17321743 , september 2011 . o. ozel , j. yang , and s. ulukus , `` optimal broadcast scheduling for an energy harvesting rechargeable transmitter with a finite capacity battery , '' _ ieee trans . on wireless comm . _ , vol . 11 , pp . 21932203 , june 2012 . b. devillers and d. gunduz , `` a general framework for the optimization of energy harvesting communication systems with battery imperfections , '' _ jour . of comm . and netw . _ , vol . 14 , pp . 130 139 , april 2012 . k. tutuncuoglu , a. yener , and s. ulukus , `` optimum policies for an energy harvesting transmitter under energy storage losses , '' _ ieee jour . on selected areas in commun . _ , vol . 33 , pp . 467481 , march 2015 . j. xu and r. zhang , `` throughput optimal policies for energy harvesting wireless transmitters with non - ideal circuit power , '' _ ieee jour . on selected areas in commun . _ , vol . 32 , pp . 322332 , february 2014 . o. ozel , k. shahzad , and s. ulukus , `` optimal energy allocation for energy harvesting transmitters with hybrid energy storage and processing cost , '' _ ieee trans . on signal proc . _ , vol . 62 , pp . 32323245 , june 2014 .
|
motivated by damage due to heating in sensor operation , we consider the throughput optimal offline data scheduling problem in an energy harvesting transmitter such that the resulting temperature increase remains below a critical level .
we model the temperature dynamics of the transmitter as a linear system and determine the optimal transmit power policy under such temperature constraints as well as energy harvesting constraints over an awgn channel .
we first derive the structural properties of the solution for the general case with multiple energy arrivals .
we show that the optimal power policy is piecewise monotone decreasing with possible jumps at the energy harvesting instants .
we derive analytical expressions for the optimal solution in the single energy arrival case .
we show that , in the single energy arrival case , the optimal power is monotone decreasing , the resulting temperature is monotone increasing , and both remain constant after the temperature hits the critical level .
we then generalize the solution for the multiple energy arrival case .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the introduction of effective chiral lagrangians to account for the basic symmetries of qcd and its application through @xmath6 to the study of meson meson interaction @xcite or meson baryon interaction @xcite has brought new light into these problems and allowed a systematic approach . yet , @xmath6 is constrained to the low energy region , where it has had a remarkable success , but makes unaffordable the study of the intermediate energy region where resonances appear . in recent years , however , the combination of the information of the chiral lagrangians , together with the use of nonperturbative schemes , have allowed one to make prediction beyond those of the chiral perturbation expansion . the main idea that has allowed the extension of @xmath6 to higher energies is the inclusion of unitarity in coupled channels . within the framework of chiral dynamics , the combination of unitarity in coupled channels together with a reordering of the chiral expansion , provides a faster convergence and a larger convergence radius of a new chiral expansion , such that the lowest energy resonances are generated within those schemes . a pioneering work along this direction was made in @xcite where the lippmann schwinger equation in coupled channels was used to deal with the meson baryon interaction in the region of the @xmath4 and @xmath7 resonances . similar lines , using the bethe salpeter equation in the meson meson interaction , were followed in @xcite and a more elaborated framework was subsequently developed using the inverse amplitude method ( iam ) @xcite and the n / d method @xcite . the iam method was also extended to the case of the meson baryon interaction in @xcite and in @xcite where a good reproduction of the @xmath8 resonance was obtained using second order parameters of natural size . the n / d method has also been used for @xmath9 scattering @xcite and in the @xmath10 and coupled channels system in @xcite . a review of these unitary methods can be seen in @xcite . the consideration of coupled channels to study meson baryon interactions at intermediate energies has also been exploited in @xcite using the k - matrix approach , although not within a chiral context . the bethe salpeter equation was also used in the study of the the meson baryon interaction in the strangeness @xmath11 sector in @xcite and in the @xmath0 sector around the @xmath4 region in @xcite . in this latter work , aimed at determining the @xmath12 coupling , only the vicinity of the resonance was studied and no particular attention was given to the region of lower energies . subsequently a work along the same lines using the bethe salpeter equation , but considering all the freedom of the chiral constraints , was done in @xcite and a good reproduction of the experimental observables was obtained for the @xmath13 sector . those works considered only states of meson baryon in the coupled channels and both in @xcite and @xcite the @xmath2 channel was omitted . this channel plays a moderate role in the @xmath13 sector @xcite but , as we shall see , it plays a crucial role in the @xmath3 sector . our aim in the present work is to extend the chiral unitary approach to account for the @xmath2 channel , including simultaneously some other corrections inspired by vector meson dominance ( vmd ) which finally allow one to have a reasonable description of the meson baryon interaction up to meson baryon energies of around 1600 mev . the @xmath4 resonance is generated dynamically in this approach and the mass , width and branching ratios are obtained in fair agreement with the experiment . the phase shifts and inelasticities for @xmath9 scattering in that region are also evaluated and good agreement with experiment is also found both in the @xmath13 and @xmath3 sectors . in addition some trace of the @xmath5 is found , linked to the introduction of the @xmath2 channel , with a pole in the second riemann sheet with the right energy , albeit a large width . a side effect of the calculations is that we determine the s - wave part of the @xmath14 transition amplitude , revising previous determinations in @xcite and @xcite , and together with the p - wave amplitudes previously determined , we obtain a good reproduction of the cross sections for these reactions . in this section we study the meson baryon scattering in s - wave in the strangeness @xmath0 sector . we shall make use of the bethe salpeter equation in coupled channels considering states of a meson of the @xmath15 octet and a baryon of the @xmath16 octet , as required by the su(3 ) chiral formalism . for total zero charge we have six channels , @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21 , and @xmath22 . the bethe salpeter equation is given by @xmath23 where @xmath24 is the product of the meson and baryon propagators . the diagrammatic expression of the @xmath25 matrix is shown in fig.[fig : bseq ] . following @xcite we take the kernel ( potential ) of the bethe salpeter equation from the lowest order chiral lagrangian involving mesons and baryons @xmath26 \label{eqn : lagrangian}\ ] ] where @xmath27 and @xmath28 are the su(3 ) matrices for the octet baryon field and the octet meson field respectively , and @xmath29 is the weak decay constant of the meson . from this lagrangian , the transition potentials between our six channels are given by @xmath30 with the initial(final ) baryon spinor @xmath31 ( @xmath32 ) and the initial(final ) meson momentum @xmath33 ( @xmath34 ) . the coefficients @xmath35 , reflecting the su(3 ) symmetry of the problem , are obtained from eq . ( [ eqn : lagrangian ] ) and shown in table [ tbl : cij ] . .@xmath35 coefficient in the potential . @xmath36 [ cols="^,^,^,^,^,^,^",options="header " , ] the lower part of table [ tbl : coupling ] shows the same quantities obtained from a breit - wigner fit of the real energy scattering amplitudes @xmath37 . we fit them by the breit - wigner form together with a background @xmath38 at @xmath39 mev . the unknown parameters @xmath40 , @xmath41 and @xmath42 are determined by the method of least squares . we obtain the values of @xmath43 from the @xmath44 corresponding to the @xmath19 final state amplitudes . their absolute value agree fairly well with the values obtained from the pole residues and gives us confidence about our numerical evaluation of the couplings . for instance , the @xmath9 , @xmath19 and @xmath2 branching ratios are now 24 % , 67 % and 9 % respectively . this latter analysis allows us to determine the sign of the couplings . the signs given are relative to that of @xmath45 . it is instructive to decompose the resonance in the su(3 ) representations . in fact , our result , ignoring the coupling to the @xmath2 channel , leads to @xmath46 and tells us that the @xmath4 resonance is almost an equal weight mixture of the r - parity even su(3 ) octet @xmath47 and the r - parity odd su(3 ) octet @xmath48 . it would be interesting to compare these results with the results from other models or lattice qcd simulations . we have studied the s - wave @xmath9 scattering , together with that of coupled channels , in a chiral unitary model in the region of center of mass energies from threshold to 1600 mev . we calculated the t matrix using the bethe - salpeter equation in the eight coupled channels including six meson - baryon channels and two @xmath2 channels . we took the transition potentials between the meson - baryon systems from the lowest order chiral lagrangian and improved them taking into account the vector meson dominance hypothesis . then we introduced the appropriate @xmath49 transition potentials which influence both the elastic scattering and the pion production processes . in the present model the renormalization due to higher order contributions is included by means of subtraction constants in the real part of the propagators of the two or three - body systems , which are taken as free parameters and determined through comparison with the t matrix of the data analysis . the imaginary part of the meson - baryon or @xmath2 propagators is fixed and ensures unitarity in the s matrix . a realistic t matrix is obtained with a few free parameters for energies up to 1600 mev . the phase - shifts and the inelasticities are well reproduced in both isospin 1/2 and 3/2 . we find that the correction of the chiral coefficient and the @xmath2 channels are important to obtain an accurate t matrix , especially in isospin 3/2 . our analysis allowed us to determine the s - wave amplitudes for @xmath50 and we found that the isospin 3/2 @xmath49 amplitude is different from the two previous empirical ones . the resonance @xmath4 is generated dynamically and qualifies as a quasi bound state of meson and baryon . the corresponding pole is seen in the t matrix on the complex plane . we calculate the total and partial decay width of the resonance . the total width obtained , about 80 mev , is smaller than the pdg estimation , but agrees with the new data from bepc . also the large @xmath19 branching ratio observed in the data is reproduced . the present study has served to show the potential of the chiral unitary approach extending the predictions to higher energies than it would be possible with the use of @xmath6 . yet , we also saw that improvements in the basic information of the lowest order chiral lagrangians to account for phenomenology of vmd are welcome . on the other hand we found mandatory the inclusion of the @xmath51 channels in order to find an accurate reproduction of the data , particularly those in the isospin 3/2 sector . we also found that the introduction of these channels , forcing them to reproduce the inelasticities and other data , has as an indirect consequence that the @xmath5 resonance appears then as a pole in the complex plane indicating a large mixing of this resonance with @xmath51 states . this interpretation would be consistent with the large experimental coupling of this resonance to the @xmath51 channel . we would like to acknowledge discussions with j.c . ncher . one of us , t. i. , would like to thank j.a . oller and a. hosaka for useful discussions . this work has been partly supported by the spanish ministry of education in the program `` estancias de doctores y tecnlogos extranjeros en espaa '' , by the dgicyt contract number bfm2000 - 1326 and by the eu tmr network eurodaphne , contact no . erbfmrx - ct98 - 0169 . 99 j. gasser , h. leutwyler , nucl . b250 ( 1985 ) 465 u.g . meissner , rep . 56 ( 1993 ) 903 v. bernard , n. kaiser and u.g . meissner , int . e4 ( 1995 ) 193 a. pich , rep . 58 ( 1995 ) 563 g. ecker , prog . 35 ( 1995 ) 1 n. kaiser , p.b . siegel and w. weise , phys . lett . b362 ( 1995 ) 23 n. kaiser , p.b . siegel and w. weise , a594 ( 1995 ) 325 n. kaiser , t. waas and w. weise , nucl . a 612 ( 1997 ) 297 j.a . oller and e. oset , nucl . a 620 ( 1997 ) 438 ; erratum , nucl . phys . a 652 ( 1999 ) 407 j.a . oller , e. oset and j.r . pelez , phys . d59 ( 1999 ) 074001 ; erratum phys . d60 ( 1999 ) 099906 j.a . oller and e. oset , phys . d60 ( 1999 ) 074023 a. gmez nicola and j.r . pelez , phys . d62 ( 2000 ) 017502 a. gmez nicola , j. nieves , j.r . pelez and e. ruiz arriola , phys . b486 ( 2000 ) 77 u.g . meissner and j.a . oller , nucl . a673 ( 2000 ) 311 j.a oller and u.g . meissner , phys . b500 ( 2000 ) 263 j.a . oller , e. oset and a. ramos , prog . 45 ( 2000 ) 157 t. p. vrana , s.a . dytman and t.s.h . lee , phys . 328 ( 2000 ) 181 e. oset and a. ramos , nucl . a635 ( 1998 ) 99 j.c . ncher , a. parreo , e. oset , a. ramos , a. hosaka and m. oka , nucl . a678 ( 2000 ) 187 j. nieves and e. ruiz arriola , hep - ph/0104307 , phys . d in print a.m. green and s. wycech , phys . c60 ( 1999 ) 035208 f. mandl and g. show , quantum field theory , john wyley and sons , 1984 d.m . manley , phys . d30 ( 1984 ) 536 h. burkhardt and j.lowe , phys . 67 ( 1991 ) 2622 center of nuclear study , http://gwdac.phys.gwu.edu/ g. ecker , j. gasser , a. pich and e. de rafael , nucl . phys . b321 ( 1989 ) 311 j. nieves and e. ruiz arriola , phys . b455 ( 1999 ) 30 j. nieves and e. ruiz arriola , nucl . a679 ( 2000 ) 57 m. urban , m. buballa and j. wambach , nucl . a641 ( 1998 ) 433 f. klingl , n. kaiser and w. weise , z. phys . a356 ( 1996 ) 193 r.aaron , phys . rev d16 ( 1977 ) 50 e. oset and m.j . vicente vacas , nucl . phys . a446 ( 1985 ) 584 v. sossi , n. fazel , r.r . johnson and m.j . vicente vacas , phys . b298 ( 1993 ) 287 o. jaekel , m. dillig and c.a.z . vasconcellos , nucl . phys . a 541 ( 1992 ) 675 v. bernard , n. kaiser and u.g . meissner , nucl . b457 ( 1995 ) 147 v. bernard , n. kaiser and u.g . meissner , nucl . a619 ( 1997 ) 261 t.s . jensen and a.f . miranda , phys . c55 ( 1997 ) 1039 j. caro ramon , n. kaiser , s. wetzes and w. weise , nucl . a672 ( 2000 ) 249 m.batinic , i.slaus and a.svarc , phys . c52 ( 1995 ) 2188 h.c . chiang , e. oset and l.c . liu , phys . c44 ( 1991 ) 738 r.s.hyano , s.hirenzaki and a.grillitzer , eur . phys . j. a6 ( 1999 ) 99 h .- ch . schrder , , phys . b469 ( 1999 ) 25 particle data group , eur . j. c 15 ( 2000 ) 1 j.z . bai , phys . b510 ( 2001 ) 75
|
we study the s - wave interaction of mesons with baryons in the strangeness @xmath0 sector in a coupled channel unitary approach .
the basic dynamics is drawn from the lowest order meson baryon chiral lagrangians .
small modifications inspired by models with explicit vector meson exchange in the @xmath1channel are also considered .
in addition the @xmath2 channel is included and shown to have an important repercussion in the results , particularly in the @xmath3 sector .
the @xmath4 resonance is dynamically generated and appears as a pole in the second riemann sheet with its mass , width and branching ratios in fair agreement with experiment .
a @xmath5 resonance also appears as a pole at the right position although with a very large width , coming essentially from the coupling to the @xmath2 channel , in qualitative agreement with experiment .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
at certain stages of stellar evolution , some stars show peculiarity of spectral lines in their spectra that argues in favour of an enhanced or depleted abundance of several chemical species in their stellar atmospheres with respect to the solar abundance . abnormally strong lines of silicon in @xmath1 cvn were first reported by antonia maury @xcite , while the strong lines of ionised silicon and ionised strontium were found by annie cannon @xcite in some bright southern stars in the process of determining their spectral class . the first systematic study and classification of stars of spectral classes b - f with abnormally strong lines of various chemical species was performed by morgan @xcite . to explain the mystery of the observed abnormally strong lines , shapley @xcite suggested the possibility of abundance abnormalities in the atmospheres of these stars . several decades later , this idea was confirmed by the results of a differential coarse analysis using the curves of growth @xcite . it should be noted , that the observed peculiar abundance of chemical species is only related to the stellar atmosphere and does not reflect the chemical composition of the entire star . we turn our attention to two types of stars with abundance anomalies : blue horizontal - branch ( bhb ) stars that burn helium in their core and hydrogen in a shell @xcite , and main sequence stars that burn hydrogen in their core . bhb stars are found mostly in globular clusters . comprehensive surveys show that the hot bhb stars have abundance anomalies as compared to the cool bhb stars in the same cluster @xcite . hot bhb stars show enhanced abundances of iron , magnesium , titanium and phosphor , and depleted helium @xcite . surveys of rotational velocities of bhb stars indicate that stars with @xmath2 11500 k possess modest rotation with v@xmath3 10 km s@xmath4 , while the cooler stars rotate more rapidly on average @xcite . slow rotation is an indicator of a hydrodynamically stable atmosphere and the observed abundance peculiarities as well as the brighter @xmath5-magnitudes ( as compared to the theoretical prediction for @xmath5-magnitudes based on stellar evolution models ) of hot bhb stars @xcite can be explained in the frame of atomic diffusion mechanism @xcite at play in their atmospheres . competition between the gravitational and radiative forces in a hydrodynamically stable atmosphere can cause accumulation or depletion of chemical elements at certain optical depths and lead to a vertical stratification of element abundances . hui - bon - hoa et al . @xcite have constructed stellar atmosphere models of hot bhb stars with vertical stratification of elements , that successfully reproduced some of the anomalies mentioned above . analysing the available high resolution spectra of some hot bhb stars , khalack et al . @xcite have detected vertical stratification of the abundance of several chemical elements , including iron . the stellar atmosphere models of hot bhb stars calculated self - consistently with abundance stratifications predict that the iron abundance stratification decreases as a function of @xmath6 and becomes negligible for bhb stars hotter than @xmath7 14000 k @xcite . this theoretical result is consistent with the detected iron abundance slopes found by khalack et al . @xcite from spectral line analysis of hot bhb stars with different effective temperatures . a significant portion of upper main sequence stars show abundance peculiarities of various chemical species and are commonly named chemically peculiar ( cp ) stars @xcite . george preston @xcite introduced this collective term and divided the cp stars in four groups : cp1 metallic - line stars ( type am - fm ) , cp2 magnetic peculiar b and a - type stars ( type bp - ap ) , cp3 mercury - manganese stars ( type hg - mn ) and cp4 helium weak stars . following a large number of extensive studies of cp stars , new types of abundance peculiarities have been discovered and an extension of preston s classification was proposed @xcite . in this new scheme , the cp4 group includes only magnetic he - weak stars ( sr - ti - si branch ) , while the non - magnetic he - weak stars ( p - ga branch ) are placed in the group cp5 . the he - rich magnetic stars ( usually having spectral class b2 ) are placed in the group cp6 . the cp7 group is reserved for the non - magnetic he - rich stars . in this classification the odd numbers correspond to non - magnetic stars , while groups with the even numbers contain stars that possess a detectable magnetic field . more details about the history of discovery and study of cp stars can be found in @xcite . an extensive list of known and suspected ap , hgmn and am stars was recently published by renson & manfroid @xcite , while a list of known magnetic cp stars has been compiled by bychkov et al . some upper main sequence cp stars show variability of absorption line profiles in their spectra with the period of stellar rotation due to horizontal inhomogeneous distributions of elements abundance in their stellar atmosphere @xcite . one can see the periodic ( days , months , years ) variation of spectral line profiles with the axial rotation of a cp star due to the variation of contribution from different parts of the stellar atmosphere to these lines . ap stars also show signatures of a strong magnetic field and its structure correlates with the patches of overabundances ( or underabundances ) of certain elements @xcite . observationally , it was shown that the properties of the line profile variability due to abundance patches as well as the magnetic field structure of ap stars do not change during several decades @xcite and , therefore , it is usually assumed that stellar atmospheres of ap stars are hydrodynamically stable . in a hydrodynamically stable atmosphere the presence of a magnetic field can intensify accumulation or depletion of chemical elements at certain optical depths @xcite . from the analysis of line profiles of ionised iron and ionised chromium in the spectra of @xmath8 crb , ryabchikova et al . @xcite have shown that iron and chromium abundances increase towards the deeper atmospheric layers . this can be explained in terms of the mechanism of atomic diffusion @xcite . it appears that the light and the iron - peak elements are concentrated in the lower atmospheric layers of some ap stars ( hd 133792 and hd 204411 ) @xcite . meanwhile the rare - earth elements ( for example , pr and nd ) are usually pushed into the upper atmosphere of ap stars @xcite . detection of vertical stratification of element abundances in stellar atmospheres of stars is an indicator of the effectiveness of the atomic diffusion mechanism that may be responsible for the observed abundance peculiarities . the vertical stratification of element abundances in a hydrodynamically stable atmosphere can be estimated through the analysis of multiple line profiles that belong to the same ion of the studied element @xcite using the zeeman2 code @xcite , for instance . this approach has been successfully employed to study the vertical abundance stratification in the atmospheres of several bhb @xcite and hgmn stars @xcite . to search for the signatures and to study the abundance stratification of chemical species with optical depth in the atmospheres of cp stars , we have initiated project veselka , which means rainbow in ukrainian and stands for vertical stratification of element abundances . slowly rotating ( v@xmath9 < 40 km / s ) cp stars of the upper main sequence were selected for our study using the catalogue of ap , hgmn and am stars of @xcite . this limitation for the rotational velocity is imposed with the aim to increase the probability for a star to have a hydrodynamically stable atmosphere , where the atomic diffusion mechanism can produce vertical stratification of element abundances . a low value of v@xmath9 also leads to narrow and mostly unblended line profiles in the observed spectra , which is beneficial for our abundance analysis . .cp stars observed with espadons in the frame of project veselka . [ cols= " < , > , > , < " , ] + @xmath10known spectroscopic binary stars [ tab1 ] the selected slowly rotating cp stars and two normal main sequence stars ( used as reference stars ) have been observed during 2013 - 14 with espadons ( echelle spectropolarimetric device for observations of stars ) at the cfht ( canada - france - hawaii telescope ) employing the deep - depletion e2v device olapa . the instrument performances as well as the optical characteristics of the spectropolarimetre are described in @xcite . espadons acquires high resolution ( r= 65000 ) stokes iv spectra throughout the spectral range from 3700 to 10500 in a single exposure @xcite . in order to be able to find convincing signatures of vertical stratification of element abundances from the spectral analysis , we require spectra with high signal - to - noise ratio ( close to one thousand per bin in the spectral region around 5150 ) . table [ tab1 ] presents a list of the observed slowly rotating chemically peculiar stars and some normal upper main sequence stars . the first , second and third columns provide respectively the name of a star , its apparent visual magnitude and v@xmath9 value , while the fourth column contains information about its spectral type . the spectral classes of stars selected for the project veselka are given in table [ tab1 ] taking into account the data from @xcite , simbad astronomical database and the estimates of effective temperature and surface gravity obtained by khalack & leblanc @xcite . some of the observed program stars are spectroscopic binaries . our sample also includes several hgmn stars and magnetic bp - ap stars . for each star , we have obtained at least two spectra to verify if it is a binary star , or if the observed line profiles are variable with the phase of stellar rotation due to horizontal stratification of element abundances . the obtained spectra have been reduced using the dedicated software package libre - esprit @xcite which yields both the stokes i spectrum and the stokes v circular polarisation spectrum . the stokes v spectra are required to estimate and study the stellar magnetic field and its potential contribution to the atomic diffusion mechanism @xcite . the level of continuum ( free from absorbtion and emission lines ) in each echelle order of the stokes i spectra was approximated with a polynomial function and its coefficients were derived . using this polynomial function , the stokes iv spectra were normalized resulting in a continuum with variations no larger than two to three percent . to estimate the effective temperature and gravity of the observed stars ( see next section ) , we have used their non - normalized spectra @xcite . the abundance analysis of normalized polarimetric spectra has been carried out employing a modified version @xcite of the zeeman2 radiative transfer code @xcite , which requires atomic data and a synthetic stellar atmosphere model ( local temperature , pressure , electronic density , etc . relative to atmospheric optical depth ) characterized by the effective temperature , surface gravity and metallicity to simulate the synthetic line profiles . the atomic data provided by the vald-2 @xcite and nist @xcite databases have been used for our simulations . to determine the parameters of stellar atmospheres , one can fit the observed balmer line profiles with synthetic spectra from a grid of stellar atmosphere models with different values of @xmath6 , @xmath11 and metallicity . the stellar atmosphere models used here were calculated with the phoenix atmospheric code @xcite . with the aim to estimate the effective temperature , surface gravity and metallicity of stars observed in the frame of the project veselka , we have generated a new library of high resolution ( 0.05 in the visible range from 3700 to 7700 ) synthetic spectra @xcite using version fifteen of the phoenix code @xcite . a grid of stellar atmosphere models and corresponding fluxes has been calculated for 5000 k @xmath12 9000 k with a step of 250 k , for 9000 k @xmath13 15000 k with a step of 500 k and for 3.0 @xmath14 4.5 with a step of 0.5 . the grids of models have been produced for the solar metallicity @xcite as well as for the metallicities [ m / h]= -1.0 , -0.5 , + 0.5 , + 1.0 , + 1.5 assuming a nil microturbulent velocity . nine balmer line profiles observed in the non - normalized spectra of several cp stars selected for project veselka were fitted using the code fitsb2 @xcite to determine the fundamental parameters of their stellar atmosphere . an example of the best fit of balmer lines in hd 110380 obtained for the solar metallicity is presented in fig . this is a f2 m star ( see table . [ tab1 ] ) with @xmath6 = 6980 k , @xmath11 = 4.19 @xcite , for which our best fit approximates quite well also the caii 3934 and 3968 line profiles in the left side of the balmer @xmath15 line and in its left wing respectively ( see fig . [ fig1 ] ) . we have found that our grids provide almost the same values of fundamental parameters of stellar atmospheres for the studied cp stars as do the atlas9 grids calculated by castelli & kurucz @xcite . however , while studying the sensitivity of the determined values of @xmath6 and @xmath11 to the set of balmer lines used , we have found that the use of the atlas9 grids may produce some ambiguity in the determination of fundamental stellar parameters if the effective temperature is close to 10000 k depending on which set of balmer lines is used . meanwhile , the phoenix-15 grids are not sensitive to the choice of balmer lines in the range of effective temperatures from 9700 k to 12000 k @xcite . from the analysis of observed line profiles while using the modified zeeman2 code @xcite , we can determine the radial velocity of the studied star , the projection of its rotational velocity to the line of sight v@xmath9 and the abundance of the chemical element responsible for this absorption line . fig [ fig2 ] shows an example of the feii 4635 line profile observed in hd 95608 that is well fitted by the theoretical profile @xcite . preliminary analysis has shown that iron lines are not variable in the hd 95608 spectra obtained in a time span of few weeks . therefore , in the case of hd 95608 , we can use the line profile composed from the data of several spectral observations for our analysis ( see fig [ fig2 ] ) . for the abundance analysis , we usually select unblended ( not contaminated by a contribution from other chemical species ) absorbtion line profiles that belong to a particular ion . in our study , we assume that the core of the line profile is formed mainly at line optical depth @xmath16=1 , which corresponds to a particular layer of the stellar atmosphere.therefore , from the simulation of each line profile that belongs to a particular ion , we can obtain its abundance at a particular layer of the stellar atmosphere ( that corresponds to a particular continuum optical depth @xmath17 ) . taking into account that the analysed lines usually have different lower energetic levels and oscillator strengths , their cores are generally formed at different optical depths @xmath17 . in this way , we can study the vertical distribution of an element s abundance from the analysis of a large number ( at least ten or more ) line profiles that belong to one or two ions of this element that are detected in the analyzed spectrum @xcite . a statistically significant vertical stratification of the element s abundance is considered when the abundance change with optical depth exceeds 0.5 dex . more details on the fitting procedure are given in @xcite . this method was successfully used by khalack et al . @xcite to study the vertical abundance stratification of chemical species in bhb stars and by thiam et al . @xcite in hgmn stars . using our grids of stellar atmosphere models and synthetic spectra calculated with the phoenix-15 code @xcite we have estimated the values of effective temperature , surface gravity and metallicity for sixteen stars observed in the frame of veselka project @xcite . our results on @xmath6 and @xmath11 obtained for twelve of these stars are consistent with the previously published data . meanwhile , for the four other stars ( hd 23878 , hd 83373 , hd 95608 and hd 164584 ) we have for the first time reported the estimates of their effective temperature , gravity and metallicity in @xcite . estimation of average abundances and detection of vertical abundance stratification of some chemical species were carried out for several cp stars selected for project veselka @xcite . clear evidences of vertical stratification of iron and chromium abundances were found in the stellar atmospheres of hd 95608 and hd 116235 ( see fig . [ fig3 ] for an example ) . for the analysed chemical species , no evidence of vertical stratification was found in the atmospheres of hd 71030 and hd 186568 @xcite . the radial and rotational velocities determined for the four stars under consideration are consistent with the values found in previous studies . according to leblanc et al . @xcite , hd 71030 does not show large abundance anomalies and it might be a normal main sequence star . two normal b - type stars hd 186568 and hd 209459 @xcite have been selected for project veselka as the reference stars ( see table [ tab1 ] ) . we plan to use them to test the applied method for the detection of vertical abundance stratification and to verify our estimates of average abundance of chemical species well represented by a large number of spectral lines in the analysed spectra . as was mentioned above , we have not found signatures of vertical abundance stratification from the analysis of hd 186568 spectra . the solar abundances of titanium and iron were also obtained for hd 209459 , which is known to have no abundance peculiarities @xcite . the average abundances of oxygen , silicon , iron and chromium were obtained for different rotational phases of hd 22920 that shows clear signatures of the presence of a moderate magnetic field . all the analyzed elements show variability of their line profiles with the rotational phase . this argues in favour of a non - uniform horizontal distribution of their abundances . among the studied elements , the siii line profiles show the strongest variability with rotational phase @xcite . to study the vertical abundance stratification of chemical species , the observed line profiles were fitted with the synthetic ones , simulated assuming a homogeneous horizontal distribution of element abundances and no magnetic field . this simplification is not physically correct , but for a given rotational phase it provides an estimate for the average abundances of chemical species . meanwhile , the contribution of the magnetic field is accounted for by the slightly larger value of v@xmath9 obtained as the final result of fitting . among the chemical species represented by a number of unblended spectral lines , only silicon and chromium appear to show strong signatures of vertical stratification of their abundance in the stellar atmosphere of hd 22920 @xcite . [ fig4 ] shows that chromium has a tendency to increase its abundance towards the deeper atmospheric layers in hd 22920 . similar behaviour with atmospheric depth was also found for the silicon abundance @xcite . accumulation or depletion of chemical elements at certain optical depths brought about by atomic diffusion @xcite can modify the structure of stellar atmospheres of cp stars @xcite and lead to the observed horizontal @xcite and vertical @xcite stratification of element abundances . for the bp - ap stars , it is believed that the substantial magnetic field suppresses convection and contribute significantly to the mechanism of atomic diffusion @xcite . in this way , the magnetic field can stabilise the structure of abundance peculiarities in atmospheres of magnetic cp stars , so that they remain stable over more than fifty years of spectral observations @xcite . therefore , for a comprehensive study of the structure of stellar atmospheres in cp stars , it is important to detect and observationally estimate the intensity of vertical abundance stratification of different chemical species @xcite . such detections are the main goal of project veselka . these results may lead to improved calculations of the self - consistent stellar atmosphere models with vertical stratification of elements using the phoenix code @xcite , or other atmospheric models such as those of stift & alecian @xcite . for several program cp stars , we have estimated their effective temperature , surface gravity and metallicity through the fitting of nine balmer line profiles with synthetic spectra calculated for a grid of stellar atmosphere models . for this aim , a new library of grids of stellar atmosphere models and corresponding fluxes has been generated and tested @xcite . the obtained stellar atmosphere parameters have been used to calculate homogeneous stellar atmosphere models , that were employed to perform an abundance analysis of hd 22920 , hd 23878 , hd 95608 , hd 116235 and hd 186568 @xcite with the help of the modified zeeman2 code @xcite . among the studied cp stars , signatures of vertical stratification of silicon and chromium abundances were found in hd 22920 @xcite , and of iron and chromium abundances in hd 95608 and hd 116235 @xcite . in order to improve the precision of measurement of the abundance stratification with atmospheric depths for different chemical elements we need to analyse more unblended spectral line profiles that belong to each element . therefore , we plan to expand our study towards the near infrared spectral region , where absorbtion lines of many metals can be found for the cool upper main sequence stars ( spectral classes a0-f5 ) . for this aim , we look forward to analyse the spectra from the near ir spectropolarimetre / velocimetre spirou @xcite proposed to be installed in 2017 at the cfht . the canada foundation for innovation ( cfi ) has approved the production of a copy of spirou that is planned to be installed at a telescope with high aperture located in the southern hemisphere . the spectropolarimetre spirou will allow the acquisition of high resolution ( r>70000 ) stokes ivqu spectra throughout the spectral range from 0.98 to 2.5 @xmath18 m ( yjhk bands ) in a single exposure @xcite . such high spectral resolution in the near ir will provide the measurement of radial velocities with a precision higher than one meter per second @xcite . taking into account the aforementioned characteristics of spirou , we consider it to be very useful to search for signatures of vertical abundance stratification in stars . the high resolution near ir spectra of cp stars , even with the relatively low signal - to - noise ratio , in combination with their optical spectra could significantly improve the precision of the statistical evaluation of vertical stratification of element abundances . the abundance analysis of other cp stars selected for project veselka is underway . the use of available high - resolution and high signal - to - noise spectra of cp stars obtained by other astronomers will be analysed to accumulate an extensive database for vertical stratification of element abundances in these stars . the database will be used to search for a potential dependence of the vertical abundance stratification of different chemical species relative to the effective temperature similar to the one that we have found for bhb stars @xcite . we are also going to study the spectra of some magnetic bp - ap stars with the aim to detect vertical stratification of element abundances and to search for possible correlation of the abundance peculiarities with the magnetic field structure . special attention will be paid to the cp stars in binary systems , because due to the tidal interaction its members usually have a slow axial rotation @xcite , which can lead to a hydrodynamically stable atmosphere . [ fig5 ] shows an example of two overlapping spectra corrected for radial velocity of the main component in the spectroscopic binary hd 159082 . one can see there , that the fei and feii line profiles are relatively stable and can be used for the analysis of vertical abundance stratification . meanwhile , the cri , moi , nii and hei line profiles change significantly during a period of time shorter than one day . the spectra of the binary stars found in table [ tab1 ] have been incorporated in the database of the binamics project @xcite that provides high quality spectra of magnetic cp stars in binary systems . we are thankful to the facult des tudes suprieures et de la recherche de luniversit de moncton and nserc for research grants . the grids of stellar atmosphere models and corresponding fluxes have been generated using the supercomputer _ briarree _ of luniversit de montral , that operates under the guidance of calcul qubec and calcul canada . the use of this supercomputer is funded by the canadian foundation for innovation ( cfi ) , nanoqubec , rmga and research fund of qubec - nature and technology ( frqnt ) . this paper has been typeset from a tex / latex file prepared by the authors . abt h.a . , levato h. , grosso m. 2002 , apj , 573 , 359 alecian g. , & stift m.j . 2010 , a&a , 516 , 53 alecian e. , neiner c. , wade g.a . 2015 , in _new windows on massive stars : asteroseismology , interferometry and spectropolarimetry_ , proc . of iaus 307 , g. meynet , c. georgy , j.h . groh & ph . stee , 330 barrick g.a . , vermeulen t. , baratchart s. , et al . 2012 , in _software and cyberinfrastructure for astronomy ii._ , proc . of spie , 8451 , 15 , [ article i d . 84513j ] behr b.b . 2003a , apjs , 149 , 67 behr b.b . 2003b , apjs , 149 , 101 burbidge g.r . & burbidge e.m . 1955 , astrophys . , 1 , 431 bychkov v.d . , bychkova l.v . , & madej j. 2003 , a&a 407 , 631 cannon a.j . & pickering e.c . 1901 , annals of the astronomical observatory of harvard college , 28 , 129 castelli f. , & kurucz r.l . , 2003 , in _modelling of stellar atmospheres_ , proc . of iaus 210 , eds . : n. piskunov , w.w . weiss , & d.f . gray , a20 delfosse x. , donati j .- f . , kouach d. et al . 2013 , in proc . of the annual meeting of the french soc . of astronomy and astrophysics , sf2a-2013 , l. cambresy , f. martins , e. nuss , & a. palacios , 497 donati j .- f . , semel m. , carter b.d . , et al . , 1997 , mnras , 291 , 658 donati j .- f . , catala c. , wade g.a . , et al . , 1999 , a&ass , 134 , 149 donati j .- f . , catala c. , landstreet j.d . , petit p. , & espadons team 2006 , in _solar polarization 4_ , asp conf . series , 358 , eds . : r. casini , & b.w . lites , 362 dworetsky m.m . , & budaj j. , 2000 , mnras , 318 , 1264 glaspey j.w . , michaud g. , moffat a.f.j . , & demers s. 1989 , apj , 339 , 926 grevesse n. , asplund m. , suaval a.j . , & scott p. 2010 , ap&ss , 328 , 179 grundahl f. , vandenberg d.a , andersen m.i . 1998 , apj , 500 , l179 grundahl f. , catelan m. , landsman w.b . , stetson p.b . , & andersen m. 1999 , apj , 524 , 242 hauschildt p.h . , shore s.n . , schwarz g.j . , et al . , 1997 , apj , 490 , 803 hearnshaw j.b . 2014 , _the analysis of starlight : two centuries of astronomical spectroscopy_ , cambrige university press , new york hubrig s. , & castelli f. , 2001 , a&a , 375 , 963 hui - bon - hoa a. , leblanc f. , hauschildt p.h . 2000 , apj , 535 , l43 khalack v. , & wade g. 2006 , a&a , 450 , 1157 khalack v. , leblanc f. , bohlender d. , wade g.a . , behr b.b . 2007 , a&a , 466 , 667 khalack v.r . , leblanc f. , behr b.b . , wade g.a . , bohlender d. 2008 , a&a , 477 , 641 khalack v.r . , leblanc f. , behr b.b . 2010 , mnras , 407 , 1767 khalack v. , yameogo b. , thibeault c. , leblanc f. 2014 , in _magnetic fields throughout stellar evolution_ , proc . of iaus 302 , eds . : p. petit , m. jardine , & h.c . spruit , 272 khalack v. , & poitras p. 2015 , in _new windows on massive stars : asteroseismology , interferometry and spectropolarimetry_ , proc . of iaus 307 , eds . : g. meynet , c. georgy , j.h . groh & ph . stee , 383 khalack v.r . , leblanc f. 2015 , aj , accepted , [ arxiv:1505.08158 ] khochukhov o. , piskunov n. , ilyin i. , ilyina s. , tuominen i. 2002 , a&a , 389 , 420 khokhlova v.l . 1975 , astron . , 52 , 950 kramida a. , ralchenko yu . , reader j. , & nist asd team 2013 , nist atomic spectra database ( ver . available : http://physics.nist.gov/asd . national institute of standards and technology , gaithersburg , md kupka f. , piskunov n. , ryabchikova t.a . , stempels h.c . , weiss w.w . , 1999 , a&as , 138 , 119 landstreet j.d . 1988 , apj , 326 , 967 leblanc f. , monin d. , hui - bon - hoa a. , hauschildt p.h . , 2009 , a&a , 495 , 937 leblanc f. , hui - bon - hoa a. , khalack v. 2010 , mnras , 409 , 1606 leblanc f. , khalack v. , yameogo b. , thibeault c. , gallant i. 2015 , mnras , submitted maitzen h.m . 1984 , a&a , 138 , 493 mashonkina l. , ryabchikova t. , ryabtsev a. 2005 , a&a , 441 , 309 mathys g. , hubrig s. 1997 , a&as , 124 , 475 maury ac . & pickering e.c . 1897 , annals of the astronomical observatory of harvard college , 28 , 1 michaud g. 1970 , apj , 160 , 641 moehler s. , 2004 , in _the a - star puzzle_ , proc . of iaus 224 , : zverko j. , weiss w.w . , iovsk j. , & adelman s.j . , 119 morgan w.w . 1931 , astrophys . j. , 73 , 104 morgan w.w . 1933 , astrophys . j. , 77 , 77 napiwotzki r. , yungelson l. , nelemans g. , et al . 2004 , in _spectroscopically and spatially resolving the components of the close binary stars_ , asp conf . series , 318 , eds . hilditch , h. hensberge & k. pavlovski , 402 peterson r.c . , rood r.t . , & crocker d.a . 1995 , apj , 453 , 214 preston g.w . 1974 , astron . 12 , 257 romanyuk i.i . , semenko e.a . , kudryavtsev d.o . 2014 , astrophysical bulletin , 69 , 4 , 427 ryabchikova t. , wade g.a . ; leblanc f. 2003 , in _modelling of stellar atmospheres_ , proc . of iaus 210 , n. piskunov , w.w . weiss , & d.f . gray , 301 ryabchikova t. , leone f. , kochukhov o. , bagnulo s. 2004 , in _the a - star puzzle_ , proc . of iaus 224 , : zverko j. , weiss w.w . , iovsk j. , & adelman s.j . , 580 ryabchikova t. , kochukhov o. , bagnulo s. 2008 , a&a , 480 , 811 recio - blanco a. , piotto g. , aparicio a. , & renzini a. 2004 , a&a , 417 , 597 renson p. , manfroid j. 2009 , a&a 498 , 961 shapley h. 1924 , harvard coll . observ . bull . 798 , 2 shavrina a.v . , glagolevskij yu.v . , silvester j. , et al . 2010 , mnras , 401 , 1882 smith k.c . 1996 , asptrophys . & space sci . , 237 , 77 song h.f . , maeder a. , meynet g. , et al . 2013 , a&a , 556 , 100 stift m.j . , alecian g. 2012 , mnras , 425 , 2715 thiam m. , leblanc f. , khalack v. , wade g.a . 2010 , mnras , 405 , 1384
|
a new research project on spectral analysis that aims to characterize the vertical stratification of element abundances in stellar atmospheres of chemically peculiar ( cp ) stars is discussed in detail .
some results on detection of vertical abundance stratification in several slowly rotating main sequence cp stars are presented and considered as an indicator of the effectiveness of the atomic diffusion mechanism responsible for the observed peculiarities of chemical abundances .
this study is carried out in the frame of project veselka ( vertical stratification of elements abundance ) for which 34 slowly rotating cp stars have been observed with the espadons spectropolarimeter at cfht . + * key words : * diffusion , line : profiles , stars : abundances , atmospheres , chemically peculiar , magnetic field , rotation @xmath0dpartement de physique et dastronomie , universit de moncton , moncton , n .- b . , canada e1a 3e9 + [email protected] + [email protected]
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
inasmuch as studies of the interaction between a particle and a quantum field are basic to particle physics and field theory , understanding the interaction between an atom and a quantum field is essential to current atomic and optical physics research @xcite . the interaction of an accelerated charge or detector ( an object with some internal degrees of freedom such as an atom or harmonic oscillator ) in a quantum field is a simple yet fundamental problem with many implications in quantum field theory @xcite , thermodynamics @xcite and applications in radiation theory and atomic - optical physics . it is common knowledge that accelerating charges give rise to radiation @xcite . but it is not entirely straightforward to derive the radiation formula from quantum field theory . how are vacuum fluctuations related to the emitted radiation ? when an atom or detector moves at constant acceleration , according to unruh @xcite , it would experience a thermal bath at temperature @xmath0 , where @xmath1 is the proper acceleration . is there emitted radiation with an energy flux in the unruh effect ? unruh effect , and the related effect for moving mirrors studied by davies and fulling @xcite , were intended originally to mimic hawking radiation from black holes . because of this connection , for some time now there has been a speculation that there is real radiation emitted from a uniformly accelerated detector ( uad ) under steady state conditions ( i.e. , for atoms which have been uniformly accelerated for a time sufficiently long that transient effects have died out ) , not unlike that of an accelerating charge @xcite . in light of pending experiments both for electrons in accelerators @xcite and for accelerated atoms in optical cavities @xcite this speculation has acquired some realistic significance . there is a need for more detailed analysis for both the uniform acceleration of charges or detectors and for transient motions because the latter can produce radiation and as explained below , sudden changes in the dynamics can also produce emitted radiation with thermal characteristics . after unruh and wald s @xcite earlier explication of what a minkowski observer sees , grove @xcite questioned whether an accelerated atom actually emits radiated energy . raine , sciama and grove @xcite ( rsg ) analyzed what an inertial observer placed in the forward light cone of the accelerating detector would measure and concluded that the oscillator does not radiate . unruh @xcite , in an independent calculation , basically concurred with the findings of rsg but he also showed the existence of extra terms in the two - point function of the field which would contribute to the excitation of a detector placed in the forward light cone . massar , parantani and brout @xcite ( mpb ) pointed out that the missing terms in rsg contribute to a polarization cloud " around the accelerating detector . for a review of earlier work on accelerated detectors , see e.g. , @xcite . for work after that , see , e.g. , hinterleitner @xcite , audretsch , mller and holzmann @xcite , massar and parantani @xcite . our present work follows the vein of raval , hu , anglin ( rha ) and koks @xcite on the minimal coupling model and uses some results of lin @xcite on the unruh - dewitt model @xcite . with regard to the question is there a radiation flux emitted from an unruh detector ? " the findings of rsg , unruh , mpb , rha and others show that , at least in ( 1 + 1 ) dimension model calculations , _ there is no emitted radiation from a linear uniformly accelerated oscillator under equilibrium conditions _ , even though , as found before , that there exists a polarization cloud around it . hu and johnson @xcite emphasized the difference between an equilibrium condition ( steady state uniform acceleration ) where there is no emitted radiation , and nonequilibrium conditions where there could be radiation emitted . nonequilibrium conditions arise for non - uniformly accelerated atoms ( for an example of finite time acceleration , see raval , hu and koks ( rhk ) @xcite ) , or during the initial transient time for an atom approaching uniform acceleration , when its internal states have not yet reached equilibrium through interaction with the field . hu and raval ( hr ) @xcite presented a more complete analysis of the two - point function , calculated for two points lying in arbitrary regions of minkowski space . this generalizes the results of mpb in that there is no restriction for the two points to lie to the left of the accelerated oscillator trajectory . they show where the extra terms in the two - point function are which were ignored in the rsg analysis . more important to answering the theme question , they show that at least in ( 1 + 1 ) dimension the stress - energy tensor vanishes everywhere except on the horizon . this means that there is no net flux of radiation emitted from the uniformly accelerated oscillator in steady state in ( 1 + 1)d case . most prior theoretical work on this topic was done in ( 1 + 1 ) dimensional spacetimes . however since most experimental proposals on the detection of unruh effect are designed for the physical four dimensional spacetime , it is necessary to do a detailed analysis for ( 3 + 1 ) dimensions . although tempting , one can not assume that all ( 3 + 1 ) results are equivalent to those from ( 1 + 1 ) calculations . first , there are new divergences in the ( 3 + 1 ) case to deal with . second , the structure of the retarded field in ( 3 + 1 ) dimensional spacetime is much richer : it consists of a bound field ( generalized coulomb field ) and a radiation field with a variety of multipole structure , while the ( 1 + 1 ) case has only the radiation field in a different form . third , an earlier work of one of us @xcite showed that there is some constant negative monopole radiation emitted from a detector initially in the ground state and uniformly accelerated in ( 3 + 1)d minkowski space , and claimed that this signal could be an evidence of the unruh effect . this contradicts the results reported by hr @xcite and others from the ( 1 + 1)d calculations . we need to clarify this discrepancy and determine the cause of it , by studying the complete process from transient to steady state . in particular , since radiation only exists under nonequilibrium conditions in the ( 1 + 1 ) case , it is crucial to understand the transient effects in the ( 3 + 1 ) case to gauge our expectation of what could be , against what would be , observed in laboratories . in conceptual terms , one is tempted to invoke stationarity and thermality conditions for the description of an uad . this is indeed a simple and powerful way to understand its physics if the detector undergoes uniform acceleration and interacts with the field all throughout ( e.g. , because of the stationarity of the problem in the rindler proper time it is guaranteed that the total boost energy - operator is conserved ) . however , this argument is inapplicable for transient epochs during which the physics is quite different ( see , e.g. , the inertial to uniform acceleration motion treated in @xcite ) . likewise , one can invoke the thermality condition ( i.e. , the thermal radiance experienced by a uad is equivalent to that of an inertial detector in a thermal bath ) to obtain results based on simple reasonings . but then we note that the thermality condition does not uniquely arise from uniform acceleration conditions . for example if the motion is rapidly altered @xcite the radiation produced can be approximately thermal . this thermality in emitted radiation ( e.g. , from sudden injection of atoms into a cavity ) is similar to those encountered in cosmological particle creation @xcite , but has a different physical origin from unruh effect which is similar to particle creation from black holes ( hawking effect ) @xcite ( see , e.g. , @xcite ) . in terms of methodology , instead of using the more sophisticated influence functional method as in the earlier series of papers on accelerated detectors @xcite and moving charges @xcite , our work here follows more closely the work of hr who used the heisenberg operator method to calculate the two - point function and the stress - energy tensor of a massless quantum scalar field . in our analysis based on the ( 3 + 1)d unruh - dewitt detector theory we found the full and exact dynamics of the detector and the field in terms of their heisenberg operator evolution , thus making available the complete quantum and statistical information for this detector - field system , enabling us to address the interplay of thermal radiance in the detector , vacuum polarization cloud around the detector , quantum fluctuations and radiation , and emitted flux of classical radiation . the paper is organized as follows . in sec.ii we introduce the unruh - dewitt detector theory . then in sec.iii we describe the quantum dynamics of the detector - field system in the heisenberg picture , yielding the expectation values of the detector two - point function with respect to the minkowski vacuum and a detector coherent state in sec.iv . with these results we derive the two - point function of the quantum field and describe what constitutes the vacuum polarization " around the detector in sec.v . then in sec.vi we calculate the quantum expectation values of the stress - energy tensor induced by the uniformly accelerated detector . this allows us to explore the conservation law and derive the quantum radiation formula . a comparison with the results in ref.@xcite follows in sec.vii . finally , we summarize our findings in sec.viii . the total action of the detector - field system is given by @xmath2 where @xmath3 is the internal degree of freedom of the detector , assumed to be a harmonic oscillator with mass @xmath4 and a ( bare ) natural frequency @xmath5 : @xmath6.\ ] ] here @xmath7 is the detector s proper time . henceforth we will use an overdot on @xmath3 to denote @xmath8 . the scalar field @xmath9 is assumed to be massless , @xmath10 the interaction action @xmath11 for the unruh - dewitt ( ud ) detector theory has the form @xcite , @xmath12 where @xmath13 is the coupling constant . this can be regarded as a simplified version of an atom . below we will consider the ud detector moving in a prescribed trajectory @xmath14 in a four - dimensional minkowski spacetime with metric @xmath15 diag@xmath16 and line element @xmath17 . by prescribed " we mean the trajectory of the detector is not considered as a dynamical variable , thus we ignore the backreaction effect of the field on the trajectory . ( see ref.@xcite for an example where the trajectory and the field are determined self - consistently by each other . ) the detector is made ( by the act of an external agent ) to go along the worldline @xmath18 parametrized by its proper time @xmath7 . this is the trajectory of a uniformly accelerated detector situated in rindler wedge r ( the portion @xmath19 and @xmath20 of minkowski space ; see chapter 4 of ref.@xcite ) . the conjugate momenta ( @xmath21 ) of dynamical variables ( @xmath22 ) are defined by @xmath23 by treating the above dynamical variables as operators and introducing the equal time commutation relations , @xmath24 & = & i\hbar , \label{qpcm } \\ \left . [ \hat{\phi}(t,{\bf x}),\hat{\pi } ( t,{\bf x ' } ) ] \right . & = & i\hbar\delta^3 ( { \bf x}-{\bf x'}),\label{phipicm}\end{aligned}\ ] ] one can write down the heisenberg equations of motion for the operators and obtains @xmath25 which have the same form as the classical euler - lagrange equations . suppose the system is prepared before @xmath26 , and the coupling @xmath11 is turned on precisely at the moment @xmath26 when we allow all the dynamical variables to begin to interact and evolve under the influence of each other . ( the consequences of this sudden switch - on and the assumption of a factorizable initial state for the combined system a quantum brownian oscillator plus oscillator bath is described in some details in @xcite ) . by virtue of the linear coupling @xmath27 , the time evolution of @xmath28 is simply a linear transformation in the phase space spanned by the orthonormal basis @xmath29 , that is , @xmath30 can be expressed in the form @xmath31+f^q(x)\hat{q}(0)+ f^p(x)\hat{p}(0 ) . \label{defphi}\ ] ] here @xmath32 and @xmath33 are c - number functions of spacetime . similarly , the operator @xmath34 can be written as @xmath35 + q^q(\tau ) \hat{q}(0)+ q^p(\tau ) \hat{p}(0 ) , \label{defq}\ ] ] with c - number functions @xmath36 and @xmath37 . for the case with initial operaters being the free field operators , namely , @xmath38 , @xmath39 , @xmath40 and @xmath41 , one can go further by introducing the complex operators @xmath42 and @xmath43 : @xmath44 , \\ \hat{\pi}_0({\bf x } ) & = & \int { d^3k\over ( 2\pi)^3 } \sqrt{\hbar\over 2\omega}(-i\omega ) \left [ e^{i{\bf k\cdot x } } \hat{b}_{\bf k}-e^{-i{\bf k\cdot x}}\hat{b}^\dagger_{\bf k}\right]\end{aligned}\ ] ] with @xmath45 , and @xmath46 note that , instead of @xmath5 , we use the renormalizd natural frequency @xmath47 ( to be defined in @xmath48 ) in the definition of @xmath43 . then the commutation relations @xmath49 and @xmath50 give @xmath51=1 , \;\;\;\;\ ; [ \hat{b}_{\bf k } , \hat{b}_{\bf k'}^\dagger ] = ( 2\pi)^3\delta^3({\bf k}-{\bf k'}),\ ] ] and the expressions @xmath52 and @xmath53 can be re - written as @xmath54 where @xmath55,\label{phiv}\\ \hat{\phi}_a(x ) & = & \sqrt{\hbar \over 2\omega_r m_0}\left[f^a(t,{\bf x } ) \hat{a}+ f^{a*}(t,{\bf x})\hat{a}^\dagger \right ] , \label{phib}\\ \hat{q}_b(\tau ) & = & \int { d^3 k\over ( 2\pi)^3}\sqrt{\hbar\over 2\omega } \left[q^{(+)}(\tau,{\bf k})\hat{b}_{\bf k } + q^{(-)}(\tau,{\bf k})\hat{b}_{\bf k}^\dagger\right],\label{qv } \\ \hat{q}_a(\tau ) & = & \sqrt{\hbar\over 2\omega_r m_0}\left [ q^a(\tau)\hat{a}+q^{a*}(\tau)\hat{a}^\dagger \right ] .\label{qb}\end{aligned}\ ] ] the whole problem therefore can be transformed to solving c - number functions @xmath56 and @xmath57 from @xmath58 and @xmath59 with suitable initial conditions . since @xmath60 and @xmath61 are hermitian , one has @xmath62 and @xmath63 . hence it is sufficient to solve the c - number functions @xmath64 , @xmath65 , @xmath66 and @xmath67 . to place this in a more general setting , let us perform a lorentz transformation shifting @xmath68 to @xmath69 , and define @xmath70 this does not add any complication to our calculation . now the coupling between the detector and the field would be turned on at @xmath69 . we are looking for solutions with the initial conditions @xmath71 the method to solve for @xmath74 and @xmath75 are analogous to what we did in classical field theory . first , we find an expression relating the harmonic oscillator and the field amplitude right at the detector . then substituting this relation into the equation of motion for the oscillator , we obtain a complete equation of motion for @xmath75 with full information of the field . last , we solve this complete equation of motion for @xmath75 , and from its solution determine the field @xmath74 consistently . eq.@xmath59 implies that @xmath76 the general solution for @xmath72 reads @xmath77 where @xmath78 is the free field solution , and @xmath79 is the retarded solution , which looks like the retarded field in classical field theory . here @xmath80 and the retarded green s function @xmath81 in minkowski space is given by @xmath82 with @xmath83 . applying the explicit form of the retarded green s function , one can go further to write @xmath84 where @xmath85 with @xmath86 , @xmath87 and @xmath88 . the formal retarded solution @xmath89 is singular on the trajectory of the detector . to deal with the singularity , note that the ud detector here is a quantum mechanical object , and the detector number would always be one . this means that at the energy threshold of detector creations , there is a natural cutoff on frequency , which sets an upper bound on the resolution to be explored in our theory . thus it is justified to assume here that the detector has a finite extent @xmath90 , which will introduce the back reaction on the detector . let us regularize the retarded green s function by invoking the essence of effective field theory : @xmath91 ( for more details on this regularization scheme , see refs.@xcite . ) with this , right on the trajectory , the retarded solution for large @xmath92 is @xmath93,\ ] ] where @xmath94 . substituting the above expansion into @xmath58 and neglecting @xmath90 terms , one obtains the equation of motion for @xmath73 with back reaction , @xmath95 fortunately , there is no higher derivatives of @xmath75 present in the above equation of motion . now @xmath73 behaves like a damped harmonic oscillator driven by the vacuum fluctuations of the scalar field , with the damping constant @xmath96 and the renormalized natural frequency @xmath97 in @xmath98 , the solution for @xmath73 compatible with the initial conditions @xmath99 is @xmath100 where @xmath101 has been given in @xmath102 , @xmath103 and @xmath104 are defined as @xmath105 with @xmath106 throughout this paper we consider only the under - damped case with @xmath107 , so @xmath108 is always real . similarly , from @xmath58 , @xmath59 , @xmath111 and @xmath112 , the equations of motion for @xmath109 and @xmath110 read @xmath113 the general solutions for @xmath109 , similar to @xmath114 , is @xmath115 however , according to the initial condition @xmath116 , one has @xmath117 , hence @xmath118 again , the value of @xmath109 is singular right at the position of the detector . performing the same regularization as those for @xmath73 , @xmath119 becomes ( cf . @xmath98 ) @xmath120 which describes a damped harmonic oscillator free of driving force . the solution consistent with the initial condition @xmath121 and @xmath122 reads @xmath123 . \label{qb}\ ] ] as shown in the previous section , as @xmath60 evolves , some non - zero terms proportional to @xmath61 and @xmath124 will be generated . suppose the detector is initially prepared in a state that can be factorized into the quantum state @xmath125 for @xmath3 and the minkowski vacuum @xmath126 for the scalar field @xmath9 , that is , @xmath127 then the two - point function of @xmath3 will split into two parts , @xmath128 \left[\hat{q}_b(\tau ) + \hat{q}_a(\tau)\right]\left|\right.q\left . \right > |0_m\left.\right > \nonumber\\ & = & \left<\right.q \,|\ , q\left.\right > \left < \right . q(\tau)q(\tau')\left.\right>_{\rm v}+ \left < \right . q(\tau)q(\tau')\left.\right>_{\rm a}\left < 0_m| 0_m\right>. \label{splitqq}\end{aligned}\ ] ] where , from @xmath112 , @xmath129 similar splitting happens for every two - point function of @xmath30 as well as for the stress - energy tensor . observe that @xmath130 depends on the initial state of the field , or the minkowski vacuum , while @xmath131 depends on the initial state of the detector only . one can thus interpret @xmath132 as accounting for the response to the vacuum fluctuations , while @xmath133 corresponds to the intrinsic quantum fluctuations in the detector . in the following , we will demonstrate the explicit forms of some two - point functions we have obtained and analyze their behavior . to distinguish the quantum or classical natures of these quantities , the initial quantum state @xmath134 will be taken to be the coherent state @xcite , @xmath135 where @xmath136 is the @xmath137-th excited state for the free detector , and @xmath138 with a constant @xmath139 . the representation of @xmath134 in @xmath3-space reads @xmath140 which is a wave - packet centered at @xmath139 with the spread identical to the one for the ground state . along the trajectory @xmath14 in @xmath141 , performing a fourier transformation with respect to @xmath7 on @xmath102 , one has @xmath142 where the frequency spectrum of the minkowski mode from the viewpoint of the uad , @xmath143 is not trivial any more . given the result of the integration , @xmath144 a planck factor with the unruh temperature @xmath145 emerges . then from @xmath146 , @xmath147 and @xmath148 , one has @xmath149 \left [ e^{w^*_{j'}(\tau'-\tau'_0)}-e^{i\kappa(\tau'-\tau'_0)}\right],\end{aligned}\ ] ] where the integrand has poles at @xmath150 and @xmath151 , @xmath152 . let @xmath153 and taking the coincidence limit , one obtains @xmath154\nonumber\\ & - & { 1\over 4}\left [ \left({i\omega\over\gamma}+ e^{-2\gamma\eta } \left({i\omega\over\gamma}+1 -e^{-2i\omega\eta}\right)\right ) \left ( \psi_{\gamma+i\omega}+ \psi_{-\gamma - i\omega}\right)\right . \nonumber\\ & & \left.\left . - \left(-{i\omega\over\gamma}+ e^{-2\gamma\eta}\left({i\omega\over\gamma}+1 -e^{-2i\omega\eta } \right)\right)i\pi\coth{\pi\over a}(\omega - i\gamma)\right]\right\}. \label{q^2v}\end{aligned}\ ] ] here @xmath155 is defined by the hyper - geometric function as @xmath156 and @xmath157 is the poly - gamma function . the divergent @xmath158-term is produced by the coincidence limit : as @xmath159 , @xmath160 with the euler s constant @xmath161 . since @xmath162 characterizes the time scale that the interaction is turned on , @xmath158 could be finite in real processes . in any case , for every finite value of @xmath158 , the first line of the result in @xmath163 vanishes as @xmath164 . ( dotted line , eq.(@xmath165 ) with @xmath158-term excluded ) , @xmath166 ( dashed line , eq.(@xmath167 ) ) , and the sum of these two ( @xmath168 , solid line ) . here we have taken @xmath169 , @xmath170 , @xmath171 and @xmath172 . , width=302 ] in fig . [ qqvseta ] , we show the @xmath173 without @xmath158-term in dotted line . roughly speaking the curve saturates exponentially in the detector s proper time . as @xmath164 , @xmath174 saturates to the value @xmath175.\label{qqsat}\ ] ] for @xmath176 , the time scale of the rise is about @xmath177 , which can be read off from the @xmath178 in @xmath163 . from there one can also see that the small oscillation around the rising curve has a frequency of @xmath179 . for @xmath180 , it will be clear that what is interesting for the calculation of the flux is the combined quantities like @xmath181 . notice that @xmath182 with the result of @xmath183 , this calculation is straightforward . let us turn to the two - point functions of @xmath184 . similar to @xmath183 , one has @xmath185 \left [ w^*_{j'}e^{w^*_{j'}(\tau'-\tau'_0)}- i\kappa e^{i\kappa(\tau'-\tau'_0)}\right ] \label{qtqt'}\end{aligned}\ ] ] from @xmath147 , @xmath148 and @xmath186 . the coincidence limit of the above two - point function reads @xmath187 \right . \nonumber\\ & + & { a\over 2}(\gamma+i\omega)^2 e^{-(\gamma + a)\eta } \left[{f_{\gamma+i\omega } ( e^{-a\eta } ) \over \gamma+i\omega+a}\left ( { i\omega\over\gamma}\right ) e^{-i\omega\eta } + { f_{-\gamma - i\omega}(e^{-a\eta})\over \gamma + i\omega - a}\left ( \left(1-{i\omega\over\gamma}\right)e^{i\omega\eta } -e^{-i\omega\eta}\right ) \right]\nonumber\\ & + & { 1\over 4 } ( \gamma + i\omega)^2\left[\left ( { i\omega\over\gamma}+ e^{-2\gamma\eta}\left({i\omega\over\gamma}-1+e^{-2i\omega\eta}\right)\right ) \left(\psi_{\gamma+i\omega}+ \psi_{-\gamma - i\omega}\right)\right.\nonumber\\ & & \left.\left . -\left(-{i\omega\over\gamma}+e^{-2\gamma\eta } \left({i\omega\over\gamma}-1+e^{-2i\omega\eta}\right)\right)i\pi\coth { \pi\over a}(\omega - i\gamma)\right]\right\ } , \label{dotq^2v}\end{aligned}\ ] ] where @xmath188 can be subtracted safely . this will be justified later . ( dotted line , eq.(@xmath189 ) ) , @xmath190 ( dashed line , eq.(@xmath191 ) ) and their sum ( @xmath192 , solid line).,width=302 ] the subtracted @xmath193 is illustrated in fig . [ ppvseta ] ( in which @xmath158-term has also been excluded ) . one can immediately recognize that @xmath194 when @xmath195 approaches zero ; a new divergence occurs at @xmath196 . mathematically , this logarithmic divergence comes about because the divergences in the hyper - geometric functions in @xmath197 do not cancel each other , unlike in @xmath198 . physically , this divergence at the initial time @xmath199 could be another consequence of the sudden switch - on at @xmath69 or @xmath200 . we expect that these ill - behaviors at the start could be tamed if we turn on the coupling adiabatically . ( see @xcite for a discussion on this issue . ) for large @xmath195 , the behavior of the dotted curve in fig . [ ppvseta ] is quite similar to the one in fig . [ qqvseta ] for @xmath201 . it saturates to @xmath202 - 4\gamma \omega \ln a \right\}. \label{ppsat}\ ] ] comparing @xmath197 and @xmath163 , their time scales of saturation ( @xmath177 for @xmath203 ) and the frequency of the small ripples on the rising curve ( @xmath179 ) are also the same . note that when @xmath204 and @xmath1 is finite , @xmath205 and @xmath206 implies @xmath207 which justifies the subtraction of @xmath208-term in @xmath197 . we now derive the expectation values of the detector two - point functions with respect to the coherent state @xmath209 . subsituting @xmath210 into @xmath211 and using @xmath212 and @xmath209 , one finds that @xmath213 where @xmath214{\rm re}\left[q^a(\tau')\right ] = \bar{q}(\tau)\bar{q}(\tau'),\label{barq}\end{aligned}\ ] ] with the mean value @xmath215 while the qm " term is of purely quantum nature , the cl " term is of classical nature : @xmath216 is real and @xmath217 does not involve @xmath218 . thus the cl " term is identified as the semiclassical part of the two - point functions . the coincidence limits of the above two - point functions are @xmath219,\label{q2bqm}\ ] ] and @xmath220 . similarly , it is easy to find @xmath221 : @xmath222 % { \rm re}\left[\dot{q}^{\rm a}(\tau')\right]= \dot{\bar{q}}(\tau)\dot{\bar{q}}(\tau'),\end{aligned}\ ] ] and their coincidence limit , @xmath223 , \label{dotq2aqm}\ ] ] and @xmath224 . also one has @xmath225 . ( see eq.@xmath226 and below ) . its behavior is quite different from the quantum part shown in the previous figures . here we take @xmath227 , with other parameters unchanged . , width=302 ] note that the above two - point functions with respect to the coherent state are independent of the proper acceleration @xmath1 . @xmath166 and the variance ( squared uncertainty ) of @xmath3 , @xmath228 ^ 2\left.\right > = \left<\right . q ( \eta)^2\left.\right>_{\rm v}+ \left<\right.q(\eta)^2\left.\right > _ { \rm a}^{\rm qm}\ ] ] have been shown in fig . [ qqvseta ] . @xmath229 decays exponentially due to the dissipation of the zero - point energy to the field . as @xmath230 decays , @xmath231 grows and compensates the decrease , then saturates asymptotically . similar behavior can be found in fig . [ ppvseta ] , in which @xmath232 ^ 2\left.\right > = \left<\right . ( \eta)^2\left.\right>_{\rm v}+ \left<\right.\dot{q}(\eta)^2\left.\right > _ { \rm a}^{\rm qm}\ ] ] is illustrated . in fig . [ qqclvseta ] we show the semiclassical two - point funciton @xmath233 . its behavior is quite different from the quantum part shown in the previous figures . the saturated value @xmath234 in eq.@xmath206 is the late - time variance of @xmath3 , namely , @xmath235 . its dependence on the proper acceleration @xmath1 is shown in fig . [ qqvsa ] . against the proper acceleration @xmath1 ( solid line , eq.(@xmath236 ) ) with other parameters the same as fig . [ qqvseta ] . for small @xmath1 , the value of @xmath237 is less than @xmath238 ( dotted line , see eq.(@xmath239 ) ) . for large @xmath1 , @xmath234 is nearly proportional to @xmath1.,width=302 ] one can see that , when @xmath1 is large , @xmath240 is nearly proportional to @xmath1 , while in the zero - acceleration limit @xmath241 with @xmath242 , the saturated value goes to a positive number . from @xmath206 and @xmath243 , one finds that @xmath244 thus @xmath245 is smaller than @xmath246 for every @xmath247 when @xmath248 . in other words , for a non - accelerated detector , whose unruh temperature is zero , the variance of @xmath3 in the detector - field coupled system is still finite and smaller than the one for the ground state in the free theory . actually , @xmath249 will become smaller than @xmath250 whenever @xmath1 is small enough . observing fig . [ qqvsa ] , there is a critical value of @xmath1 that gives the late - time variance identical to the initial one ( @xmath251 in fig . [ qqvsa ] ) . does this mean that the @xmath3-component of the final wave - packet with @xmath252 is in the original ground state of the free theory ? the answer is no . what happens is that the quantum state of @xmath3 has been highly entangled with the quantum state of @xmath9 at late times , and the value of @xmath253 simply represents the width of the projection of the whole wave - packet ( in the @xmath3-@xmath9 representation of the state ) onto the @xmath3-axis . there is actually no factorizable @xmath3-component of the wave - packet , and the final configuration of the wave - packet in @xmath3-@xmath9 space looks totally different from the initial one . indeed , with the same critical value of @xmath1 , @xmath254 is not equal to @xmath255 for every @xmath256 . but one can still imagine that , at @xmath200 , the coherent state for the free detector is an ensemble of particles with a distribution function like @xmath257 in @xmath3-space . @xmath258 is the width of this distribution function . when @xmath259 , due to the dissipation which comes with the coupling , all particles in the ensemble are going to fall into the bottom of the potential of @xmath3 , so @xmath260 shrinks to zero . on the other hand , the vacuum fluctuations of the field act like a pressure which can push the ensemble of particles outwards , so that the width of the projection of the wave - packet in @xmath3-space , @xmath261 , remains finite . a larger @xmath1 gives a higher unruh temperature , and a higher outward pressure , so eventually the wave - packet reaches equilibrium with a wider projection in the potential well of @xmath3 . a natural definition of the energy of the dressed detector ( a similar concept is that of a dressed atom " , see e.g. , ref . @xcite ) is @xmath262 , \label{grounde}\ ] ] with @xmath263 and @xmath264 according to @xmath265 . in figs . [ qqvseta]-[qqclvseta ] , one can see that @xmath216 , @xmath266 and @xmath267 eventually die out . so the late - time energy of the dressed detector is @xmath268\nonumber\\ & = & { \hbar\over 2\pi}\left\ { a-2{\rm re}\left [ ( \gamma+ i\omega ) \psi_{\gamma+i\omega}\right]-2\gamma \ln a\right\}\end{aligned}\ ] ] from @xmath206 and @xmath205 . this is actually the true ground - state energy of the dressed detector , with the vacuum fluctuations of the field incorporated . the first term in @xmath269 could be interpreted as the total energy of a harmonic oscillator in thermal bath , @xmath270 , with the unruh temperature @xmath271 . the ground - state energy of the dressed detector is not identical to the one for the free detector , @xmath272 . in particular , if @xmath1 is small enough , the subtracted @xmath269 is lower than @xmath273 , though there is an ambiguity of a constant in determining the value of the energy . this is analogous to the lamb shift in atomic physics @xcite . similar to the two - point functions of the detector , for the initial quantum state @xmath274 , the two - point function of @xmath9 could be split into two parts , @xmath275\left[\phi_a(x')+\phi_b(x')\right ] \left|\right.q\left . \right > |\left . 0_m\right > \nonumber\\ & = & g_{\rm v}(x , x ' ) + g_{\rm a}(x , x ' ) , \label{twopt}\end{aligned}\ ] ] where , from @xmath52 and @xmath111 , @xmath276 eqs.@xmath114-@xmath277 and @xmath278 suggest that @xmath279 accounts for the back reaction of the vacuum fluctuations of the scalar field on the field itself , while @xmath280 corresponds to the dissipation of the zero - point energy of the internal degree of freedom of the detector . substituting @xmath114 into @xmath281 , @xmath279 can be decomposed into four pieces , @xmath282 in which @xmath283 are defined by @xmath284 with @xmath285 . @xmath286 is actually the green s function for free fields , which should be subtracted to obtain the renormalized green s function for the interacting theory , namely , @xmath287 since @xmath288^*$ ] by definition , it is sufficient to calculate @xmath289 and @xmath290 in the following . the structure of @xmath291 is quite simple . comparing @xmath89 , @xmath292 and the definition @xmath293 , one concludes that @xmath294 the result @xmath295 can be substituted directly to get the coincidence limit of @xmath291 . by definition , @xmath296 accounts for the interference between the retarded solution @xmath297 and the free solution @xmath101 . since we are interested in the coincidence limit of @xmath296 , and @xmath298 vanishes in the l - wedge ( @xmath299 , @xmath300 ) and p - wedge ( @xmath301 ) of minkowski space , below only the @xmath302 with @xmath303 and @xmath304 in the f - wedge ( @xmath305 ) and r - wedge would be calculated . it has been given in ref.@xcite that @xmath306,\end{aligned}\ ] ] where @xmath307 , @xmath308 and @xmath309 for @xmath303 in r and f - wedges , should have been on the right hand side of @xmath310 . ] respectively , @xmath311 and @xmath312 were defined in @xmath313 and @xmath314 , and @xmath315 hence , from @xmath89 and @xmath102 , one has @xmath316 \left [ e^{i\kappa\eta_-'}-z(\kappa)e^{i\kappa\eta_+'}\right ] \label{formalg10}\ ] ] with @xmath317 . the coincidence limit of @xmath296 reads @xmath318\right\ } , \label{g10v}\end{aligned}\ ] ] with @xmath319 and @xmath320 for @xmath303 in r and f - wedges , respectively . near the event horizon @xmath321 , @xmath322 diverges , and the last two terms in @xmath323 vanish . as for @xmath324 , since @xmath325 , one has @xmath326 , and only @xmath327 contributes to @xmath280 . inserting @xmath278 into @xmath328 and comparing with @xmath329 , one finds that @xmath330 it can also be divided into a quantum part @xmath331 and a semiclassical part @xmath332 according to @xmath329 and below . in our ( 3 + 1 ) dimensional ud detector theory , the coincidence limit of the quantum part of @xmath333 reads @xmath334 owing to @xmath335 , @xmath336 and @xmath337 . collecting the results in @xmath338 , @xmath323 and @xmath339 , it is found that @xmath340 is singular at @xmath341 , and one has to be more cautious . as can be seen from @xmath338 and @xmath339 , @xmath342 and @xmath343 look like the squares of the retarded field with effective squared scalar charge @xmath344 and @xmath345 , respectively . since the detector is accelerating , these two terms do carry radiated energy ( this will be shown explicitly later ) . the interfering term @xmath346 , is more intriguing : at first glance , it acts like a polarization in the medium , which screens the radiation field carried by @xmath342 and @xmath343 . however , the interfering term @xmath347 does not respond to @xmath348 at all it is independent of @xmath109 and impervious to any information about the quantum state of @xmath3 . hence the interfering term can not be interpreted as the polarization in the medium . the total effect is simply a destructive interference between the field induced by the vacuum fluctuations , and the vacuum fluctuations themselves . for physical interpretations one should group @xmath347 and @xmath291 together and leave @xmath348 alone . ( dashed line ) , @xmath349 ( dotted line ) and their sum ( solid line ) against @xmath350 near the event horizon @xmath351 . other parameters are the same as those in fig . [ qqvseta ] . one can see the feature that positive @xmath352 is screened by negative @xmath353.,width=302 ] ( dashed line ) , @xmath354 ( dotted line ) and their sum ( solid line ) against @xmath355 at @xmath356 and @xmath357 . the cusp in the right ( @xmath358 in this plot ) locates at the position of the detector with @xmath359 . it is due to the weaker divergence than @xmath360 for @xmath347.,width=302 ] these quantities , together with their sum , are illustrated in figs . [ gtot2 ] and [ gtot3 ] . in fig . [ gtot2 ] , one can see that , soon after the coupling is turned on at @xmath361 , @xmath362 build up and pull the solid curve down . observing @xmath323 , the time scale ( in proper time of the detector ) of this pull - down is about @xmath363 , which is shorter than the time scale @xmath177 for @xmath344 and @xmath364 , since we take @xmath365 here . in fig . [ gtot3 ] , one can also see that @xmath366 diverge as @xmath367 around the trajectory , while the divergence of @xmath353 as @xmath368 is a bit weaker than @xmath367 , such that @xmath369 goes to zero on the trajectory of the detector . in prior work for ( 1 + 1)d spacetime @xcite the counterpart of @xmath370 has been considered as evidence for the existence of a vacuum polarization cloud " around the detector @xcite . this is because @xmath370 around the detector does not vanish even after the system reaches equilibrium , it exchanges particles with the detector , and the mean energy it carries is zero . nevertheless , vacuum polarization is a concept pertinent to field - field quantum interacting systems . in quantum electrodynamics , electrons are described in terms of a field , which distributes in the whole spacetime , so vacuum polarization is pictured as the creation and annihilation of virtual electron - positron pairs everywhere in spacetime . these virtual electron - positron pairs do modify the field strength around the location of a point charge , yielding a non - vanishing variance of the electromagnetic ( em ) field . but in the ud detector theory , at the level of precision explored here , the detector - field interaction ( hence the virtual processes ) only occurs on the trajectory of the detector . there is no virtual detector or scalar charge at any spatial point off the location of the ud detector . hence in ud detector theory vacuum polarization cloud " is not a precise description of @xmath370 in steady state . at late times @xmath370 simply shows the characteristics of the field in the true vacuum state , in contrast to @xmath371 for the true ground state of the detector . in classical theory , the modified stress - energy tensor for a massless scalar field @xmath9 in minkowski space is @xcite @xmath372 = \left ( 1 - 2\xi\right)\phi_{,\mu}\phi_{,\nu}-2\xi\phi \phi_{;\mu\nu } + \left ( 2\xi -{1\over 2}\right)g_{\mu\nu}\phi^{,\rho}\phi_{,\rho } + { \xi\over 2}g_{\mu\nu}\phi\box\phi , \label{tmnxi}\ ] ] where @xmath373 is a field coupling parameter , set to zero here . denote @xmath374 as the four velocity of the detector , and define the null distance @xmath375 and the spacelike unit vector @xmath376 by @xmath377 @xcite with normalization @xmath378 and @xmath379 ( see fig . [ defr ] ) . then the stress - energy tensor for the classical retarded field @xmath380 induced by the ud detector moving along the trajectory @xmath381 can be written as @xcite @xmath382_{\xi=0 } = { \lambda_0 ^ 2\over ( 4\pi)^2 } \theta(\eta_-)\left\{{1\over r^4 } q^2(\tau_-)\left ( -{1\over 2 } g_{\mu\nu}+u_\mu u_\nu\right)\right.\nonumber\\&&+{1\over r^3 } q(\tau_-)\left [ \dot{q}(\tau_- ) + q(\tau_- ) a_\rho u^\rho\right ] \left ( -g_{\mu\nu}+2 u_\mu u_\nu + u_\mu v_\nu + v_\mu u_\nu\right ) \nonumber\\ & & + \left . { 1\over r^2}\left [ \dot{q}(\tau_-)+ q(\tau_- ) a_\rho u^\rho\right]^2(u_\mu + v_\mu)(u_\nu + v_\nu ) \right\ } . \label{tmnret}\end{aligned}\ ] ] the @xmath383 term in the above expression corresponds to the radiation field , which carries radiated power given by @xcite @xmath384 ^ 2 \label{dwdtoii}\ ] ] to the null infinity of minkowski space . here the @xmath3-term corresponds to dipole radiation ( @xmath385 , @xmath386 in multipole expansion of the radiation field ) with the angular distribution @xmath387 , in the modified stress - energy tensor . for @xmath388 , the angular distribution is @xmath389 , which is the same as the one for the em radiation emitted by a electric charge in electrodynamics . ] while the @xmath184-term corresponds to monopole radiation ( @xmath390 ) isotropic in the rest frame instantaneously for the ud detector at @xmath312 . the solid angle @xmath391 could be further integrated out , then one obtains the classical radiation formula @xmath392 , \label{eloss}\ ] ] which is the counterpart of the larmor formula for em radiation . the second term is the usual radiation formula for the massless scalar field emitted by a constant , point - like scalar charge in acceleration @xcite . naively , one may expect that the quantum version of the radiation formula could look like @xmath393 $ ] . in the following , we shall calculate the quantum expectation value of the flux @xmath394 , from which we will see that the quantum radiation formula is more complicated than expected . the expectation value of the renormalized stress - energy tensor @xmath395 is obtained by calculating @xmath396\right>_{\rm ren } = \lim_{x'\to x } \left [ { \partial\over \partial x^\mu}{\partial\over\partial x'^\nu } -{1\over 2}g_{\mu\nu } g^{\rho\sigma}{\partial\over \partial x^\rho } { \partial\over\partial x'^\sigma}\right ] g_{\rm ren}(x , x ' ) , \label{exptmn}\ ] ] according to @xmath397 with @xmath398 . with the results in the previous section , it is straightforward to obtain @xmath399 induced by the uad : @xmath400\right>_{\rm ren } & = & { \lambda_0 ^ 2\theta(\eta_-)\over(2\pi)^2 a^2 x^2}\left [ g_\mu{}^\rho g_\nu{}^\sigma -{1\over 2}g_{\mu\nu}g^{\rho\sigma}\right ] \times\nonumber\\ & & \left [ \eta_{-,\rho}\eta_{-,\sigma } \left<\right . \dot{q}(\tau_-)^2\left.\right>_{\rm tot } + { x_{,\rho } x_{,\sigma}\over x^2}\left<\right.q(\tau_-)^2\left.\right>_{\rm tot } \right.\nonumber\\ & & - { x_{,\rho}\over x}\eta_{-,\sigma}\left<\right . q(\tau_-)\dot{q}(\tau_-)\left.\right>_{\rm tot}- \eta_{-,\rho}{x_{,\sigma}\over x}\left < \right.\dot{q}(\tau_- ) q(\tau_-)\left.\right>_{\rm tot}\nonumber\\ & & \left . + \left(\eta_{-,\rho}\eta_{+,\sigma}+ \eta_{+,\rho}\eta_{-,\sigma}\right){\hbar\theta_{+-}\over 2\pi m_0 } - \left({x_{,\rho}\over x}\eta_{+,\sigma } + \eta_{+,\rho}{x_{,\sigma}\over x}\right ) { \hbar\theta_{+x}\over 2\pi m_0}\right].\label{tmnren}\end{aligned}\ ] ] upon collecting @xmath401 and @xmath402 as well as those from @xmath403 . here @xmath404 , @xmath405 and @xmath406 with @xmath407 defined in @xmath408-@xmath409 . to see the properties of quantum nature , we define the total variances by subtracting the semiclassical part from @xmath410 as @xmath411 their evolution against @xmath412 are illustrated in figs . [ qqtot1 ] and [ pptot1 ] . , @xmath413 and @xmath375.,width=302 ] in our case , the minkowski coordinate @xmath414 of a spacetime point in f and r - wedge can be transformed to the coordinate @xmath415 by @xmath416 , \label{vtor}\\ u & = & r e^{-a\tau_-}\left [ 1 - \cos\theta - ( ar)^{-1}\right ] , \label{utor}\end{aligned}\ ] ] so that @xmath417 . also one has @xmath418 with @xmath419 . now eq.@xmath420 can be directly compared with @xmath421 and @xmath422 . one can see clearly that the @xmath423 term has the same angular distribution as the one for the @xmath424 term in @xmath421 , hence would be recognized as a monopole radiation by the minkowski observer . the angular distributions of the remaining terms in @xmath420 are , however , much more complicated because of their dependence on @xmath425 . we have mentioned in the previous section that @xmath348 and @xmath291 in @xmath339 and @xmath338 carry radiated energy , now this becomes clear . observing that what correspond to @xmath426 $ ] are those proportional to @xmath427 in @xmath428 terms of @xmath420 . these terms contribute a positive flux . nevertheless , due to the presence of the interfering terms @xmath407 , most of this positive flux of quantum nature will be screened when the system reaches steady state as @xmath429 . ( eq.@xmath430 ) near the event horizon for the detector ( @xmath431 ) . this plot is virtually the same as fig . [ gtot2 ] except the time " variable is @xmath412 here . the values of parameters are still the same as before . the total variance finally saturates to the value @xmath432 . one can compare this plot with fig . [ qqvseta ] directly and see the suppression.,width=302 ] as shown in fig . [ qqtot1 ] , the total variance @xmath433 near the event horizon @xmath351 drops exponentially in proper time ( power - law in the minkowski time ) after the coupling is turned on . note that @xmath434 is proportional to @xmath340 defined in @xmath435 , and @xmath311 is independent of @xmath350 on the event horizon , so fig . [ qqtot1 ] is virtually the same plot as fig . [ gtot2 ] except that the time variable here is @xmath412 . thus , similar to the behavior of @xmath436 near the event horizon , @xmath437 ( @xmath438 ) builds up and the total variance @xmath439 is pulled down during the time scale @xmath363 ( for @xmath440 ) according to @xmath163 , @xmath243 and @xmath441 . then @xmath442 turns into a tail ( @xmath443 in fig . [ qqtot1 ] ) which exponentially approaches the saturated value @xmath444 with the time scale @xmath177 . ( eq.@xmath445 ) with the same parameters . note that @xmath446 is independent of @xmath322 , and this plot is not restricted around the event horizon . one can compare with fig . [ ppvseta ] and see the suppression.,width=302 ] for @xmath447 and @xmath448 , their behaviors are similar ( see fig . [ pptot1 ] ) . in particular , @xmath449 goes to zero at late times from @xmath197 , @xmath450 and @xmath408 with @xmath451 , so the corresponding monopole radiation vanishes after the transient . from the calculations of ref.@xcite based on perturbation theory , it was suggested that the existence of a monopole radiation could be an experimentally distinguishable evidence of the unruh effect . here we find from a non - perturbative calculation that , in fact , only the transient of it could be observed . ( a comparison of both results will be given in sec.vii . ) this appears to agree with the claim that for a uad in ( 1 + 1)d , emitted radiation is only associated with nonequilibrium process @xcite . the negative tail of @xmath447 in fig . [ pptot1 ] and the corresponding quantum radiation could last for a long time with respect to the minkowski observer ( @xmath452 ) , but this is essentially a transient . the interference between the quantum radiation induced by the vacuum fluctuations and the vacuum fluctuations themselves totally screens the information about the unruh effect in this part of the radiation . what is the physics behind the interfering term in @xmath453 ? by inserting our results into @xmath454 and @xmath445 , one can show that , @xmath455,\label{econserv}\end{aligned}\ ] ] for all proper time interval after the interaction is turned on @xmath456 . the left hand side of this equality is the energy - loss of the dressed detector from @xmath457 to @xmath458 , while the right hand side is the radiated energy via the monopole radiation corresponding to @xmath453 during the same period . therefore @xmath459 is simply a statement of energy conservation between the detector and the field _ in this channel _ , and the interfering terms @xmath460 must be included so that @xmath461 is present on the right hand side instead of the naively expected @xmath462 . a simpler but more general derivation of this relation is given in appendix @xmath463 . eq.@xmath459 also justifies that @xmath454 is indeed the correct form of the internal energy of the dressed detector . with the relation @xmath459 we can make two observations pertaining to results and procedures given before . first , while the @xmath158-terms in @xmath464 ( eq.(@xmath165 ) ) and @xmath465 ( eq.(@xmath189 ) ) are not included in any figure of this paper , they are consistent with the conservation law @xmath459 . actually the @xmath158-term in @xmath163 satisfy the driving - force - free equation of motion @xmath466 , just like the semiclassical @xmath216 does . second , eq.@xmath459 implies that all the internal energy of the dressed detector dissipates via a monopole radiation , and the external agent which drives the detector along the trajectory @xmath141 has no additional influence on this channel . transforming @xmath420 to the form of @xmath421 by applying @xmath467-@xmath468 , one can calculate the radiation power @xmath469 following a similar argument in classical theory . before calculating , let us observe the behavior of the steady - state @xmath470 in the forward light cone . as @xmath375 increses , the developments of two terms in late - time @xmath471 are illustrated in figs . [ thxx1 ] and [ thpm1 ] . it turns out that both are regular and non - vanishing at the null infinity of minkowski space ( @xmath472 ) even in steady state ( @xmath473 ) [ thxx1 ] and [ thpm1 ] also indicate that , near the the null infinity of minkowski space , almost all the equal-@xmath375 surface lies in the f - wedge , except the region around @xmath474 is still in the r - wedge . the contribution to the integral around @xmath474 can be totally neglected because the value of @xmath470 is regular there while the measure for this portion in the angular integral is zero when @xmath472 . so the radiation power can be written as @xmath475 \right\ } , \label{qradformula1}\ ] ] by inserting @xmath420 , @xmath476 and @xmath477 into @xmath478 . this is the quantum radiation formula for the massless scalar field emitted by the uad in ( 3 + 1)d spacetime . at late times ( eq.([qtot ] ) with @xmath479 ) as the null distance @xmath375 increases . the values of parameters are the same as before except that here we choose @xmath480 . the horizontal line indicates the saturated value of @xmath481 shown in fig . [ qqtot1 ] . when @xmath375 starts with @xmath482 , @xmath483 decreases from @xmath484 ( top - left ) . the light cone hits the event horizon at @xmath485 when @xmath486 , and @xmath487 always has the value @xmath432 at the event horizon ( top - right , the curve and the horizontal line intersect right at the event horizon ) . as @xmath375 further increases , @xmath488 sinks more and more ( bottom - left ) , and some oscillations begin to develop near @xmath485 . finally @xmath489 is non - vanishing at the null infinity @xmath490 , with the value smaller than @xmath444 whenever @xmath491 ( bottom - right).,title="fig:",width=226 ] at late times ( eq.([qtot ] ) with @xmath479 ) as the null distance @xmath375 increases . the values of parameters are the same as before except that here we choose @xmath480 . the horizontal line indicates the saturated value of @xmath481 shown in fig . [ qqtot1 ] . when @xmath375 starts with @xmath482 , @xmath483 decreases from @xmath484 ( top - left ) . the light cone hits the event horizon at @xmath485 when @xmath486 , and @xmath487 always has the value @xmath432 at the event horizon ( top - right , the curve and the horizontal line intersect right at the event horizon ) . as @xmath375 further increases , @xmath488 sinks more and more ( bottom - left ) , and some oscillations begin to develop near @xmath485 . finally @xmath489 is non - vanishing at the null infinity @xmath490 , with the value smaller than @xmath444 whenever @xmath491 ( bottom - right).,title="fig:",width=226 ] at late times ( eq.([qtot ] ) with @xmath479 ) as the null distance @xmath375 increases . the values of parameters are the same as before except that here we choose @xmath480 . the horizontal line indicates the saturated value of @xmath481 shown in fig . [ qqtot1 ] . when @xmath375 starts with @xmath482 , @xmath483 decreases from @xmath484 ( top - left ) . the light cone hits the event horizon at @xmath485 when @xmath486 , and @xmath487 always has the value @xmath432 at the event horizon ( top - right , the curve and the horizontal line intersect right at the event horizon ) . as @xmath375 further increases , @xmath488 sinks more and more ( bottom - left ) , and some oscillations begin to develop near @xmath485 . finally @xmath489 is non - vanishing at the null infinity @xmath490 , with the value smaller than @xmath444 whenever @xmath491 ( bottom - right).,title="fig:",width=226 ] at late times ( eq.([qtot ] ) with @xmath479 ) as the null distance @xmath375 increases . the values of parameters are the same as before except that here we choose @xmath480 . the horizontal line indicates the saturated value of @xmath481 shown in fig . [ qqtot1 ] . when @xmath375 starts with @xmath482 , @xmath483 decreases from @xmath484 ( top - left ) . the light cone hits the event horizon at @xmath485 when @xmath486 , and @xmath487 always has the value @xmath432 at the event horizon ( top - right , the curve and the horizontal line intersect right at the event horizon ) . as @xmath375 further increases , @xmath488 sinks more and more ( bottom - left ) , and some oscillations begin to develop near @xmath485 . finally @xmath489 is non - vanishing at the null infinity @xmath490 , with the value smaller than @xmath444 whenever @xmath491 ( bottom - right).,title="fig:",width=226 ] ( eq.(@xmath492 ) ) at late times as @xmath375 increases . parameters are the same as those in fig . [ thxx1 ] . when @xmath375 starts with @xmath482 , @xmath493 grows from @xmath482 ( top - left ) . after the light cone hits the event horizon , @xmath493 keeps regular on the event horizon ( top - right , the vertical line indicates the position of the event horizon ) . as @xmath375 further increases , a nodal point enters from the right ( bottom - left ) . then more and more nodal points can be seen . finally @xmath493 is non - vanishing at the null infinity @xmath472 , with infinitely many nodal points close to @xmath485 ( bottom - right).,title="fig:",width=226 ] ( eq.(@xmath492 ) ) at late times as @xmath375 increases . parameters are the same as those in fig . [ thxx1 ] . when @xmath375 starts with @xmath482 , @xmath493 grows from @xmath482 ( top - left ) . after the light cone hits the event horizon , @xmath493 keeps regular on the event horizon ( top - right , the vertical line indicates the position of the event horizon ) . as @xmath375 further increases , a nodal point enters from the right ( bottom - left ) . then more and more nodal points can be seen . finally @xmath493 is non - vanishing at the null infinity @xmath472 , with infinitely many nodal points close to @xmath485 ( bottom - right).,title="fig:",width=226 ] ( eq.(@xmath492 ) ) at late times as @xmath375 increases . parameters are the same as those in fig . [ thxx1 ] . when @xmath375 starts with @xmath482 , @xmath493 grows from @xmath482 ( top - left ) . after the light cone hits the event horizon , @xmath493 keeps regular on the event horizon ( top - right , the vertical line indicates the position of the event horizon ) . as @xmath375 further increases , a nodal point enters from the right ( bottom - left ) . then more and more nodal points can be seen . finally @xmath493 is non - vanishing at the null infinity @xmath472 , with infinitely many nodal points close to @xmath485 ( bottom - right).,title="fig:",width=226 ] ( eq.(@xmath492 ) ) at late times as @xmath375 increases . parameters are the same as those in fig . [ thxx1 ] . when @xmath375 starts with @xmath482 , @xmath493 grows from @xmath482 ( top - left ) . after the light cone hits the event horizon , @xmath493 keeps regular on the event horizon ( top - right , the vertical line indicates the position of the event horizon ) . as @xmath375 further increases , a nodal point enters from the right ( bottom - left ) . then more and more nodal points can be seen . finally @xmath493 is non - vanishing at the null infinity @xmath472 , with infinitely many nodal points close to @xmath485 ( bottom - right).,title="fig:",width=226 ] at late times , while @xmath494 ceases , it still remains a positive radiated power flow @xmath495 \right\}\nonumber\\ & & = { \hbar\lambda_0 ^ 2 \over 8\pi^2 m_0 } \left\ { { a^3\over 3\omega_r^2 } -a % \right.\nonumber\\ & & \left . -{2\over 3}\left [ { a^3\over \omega_r^2}-a + 2\gamma + { \rm re}\left[{i(\gamma+i\omega)\over a\omega } \left [ ( \gamma+i\omega)^2-a^2\right ] \psi^{(1 ) } \left({\gamma+i\omega \over a}\right)\right]\right ] \right\ } \label{qmrad}\end{aligned}\ ] ] to the null infinity of minkowski space . thus we conclude that there exists a steady , positive radiated power of quantum nature emitted by the detector even when the detector is in steady state . for large @xmath1 , the first term in @xmath496 dominates , and the radiated power is approximately @xmath497 where @xmath498 is the unruh temperature . this could be interpreted as a hint of the unruh effect . note that it does not originate from the energy flux that the detector experiences in unruh effect , since the internal energy of the dressed detector is conserved only in relation to the radiated energy of a monopole radiation corresponding to @xmath494 . learning from the em radiation emitted by a uniformly accelerated charge @xcite , we expect that the above non - vanishing radiated energy of quantum origin could be supplied by the external agent we introduced in the beginning to drive the motion of the detector . further analysis on the quantum radiations of the detector involving the dynamics of the trajectory is still on - going . we can recover the results in ref.@xcite , which is obtained by perturbation theory , as follows . in ref.@xcite , the first order approximation of the flux @xmath499 through the event horizon @xmath351 has been calculated . here , the expectation value of @xmath500 near the event horizon reads , from @xmath420 , @xmath501 g_{\rm ren}(x , x')|_{u\to 0}\nonumber\\ & = & { 2\lambda_0 ^ 2\over ( 2\pi)^2 a^4 \left(\rho^2+a^{-2}\right)^2 } \theta\left[-{1\over a}\ln { a\over v}\left(\rho^2+a^{-2}\right)-\tau_0\right ] \left\ { { 1\over v^2}\left<\right.\dot{q}(\tau_-)^2\left.\right>_{\rm tot}+ \right.\nonumber\\ & & \left.\left . { \rho^2\over ( \rho^2+a^{-2})^2 } \left [ a^2 \left<\right.q(\tau_-)^2\left.\right>_{\rm tot } + a \left < \right . \dot{q}(\tau_-)q(\tau_-)+q(\tau_-)\dot{q}(\tau_-)\left.\right>_{\rm tot } + \left<\right.\dot{q}(\tau_-)^2\left.\right>_{\rm tot}\right ] \right\ } \right|_{u\to 0},\label{powerflux}\end{aligned}\ ] ] in which @xmath502 and @xmath503 terms vanish . by letting @xmath504 with @xmath412 finite , then taking @xmath505 , the total variance @xmath445 becomes @xmath506\right\ } \nonumber\\ & \stackrel{\eta_-\to \infty}{\longrightarrow } & { \hbar\omega_r\over m_0(1-e^{2\pi\omega_r / a } ) } , \label{wrong}\end{aligned}\ ] ] owing to @xmath197 , @xmath450 and @xmath408 . this is identical to the corresponding part of eq.(66 ) in ref.@xcite , @xmath507 by noting that there , @xmath172 , @xmath508 , and @xmath509 there is equal to @xmath510 here . the monopole radiation corresponding to @xmath511 looks like a constant negative flux since @xmath512 . accordingly it was concluded in ref.@xcite that such a quantum monopole radiation could be experimentally distinguishable from the bremsstrahlung of the detector . at first glance this constant monopole radiation seems to contradict the knowledge gained from ( 1 + 1)d results . but actually similar results for ( 1 + 1)d cases were also obtained by massar and parentani ( mp ) @xcite , who found that a detector initially prepared in the ground state and coupled to a field under a smooth switching function does emit radiation during thermalization . they pointed out that the radiated flux in what they refered to as the golden rule limit " ( @xmath164 with @xmath513 small , while the switching function becomes nearly constant ) is approximately a constant negative flux for all @xmath514 . in spite of the long interaction time and the nearly constant radiated flux , the detector will remain in dis - equilibrium . note that the initial conditions in @xcite are similar to those in sec.ii of mp , and the limiting condition for obtaining @xmath515 is exactly what mp assumed there . hence , the constant negative flux in @xcite is essentially a transient effect , which exists only in the period that the above stated condition holds . when the interaction time @xmath195 exceeds @xmath516 , this approximation breaks down . to obtain the correct late - time behavior , one should take the limit @xmath505 before @xmath504 . then @xmath517 goes to zero . in this paper , we consider the unruh - dewitt detector theory in ( 3 + 1 ) dimensional spacetime . a uniformly accelerated detector is modeled by a harmonic oscillator @xmath3 linearly coupled with a massless scalar field @xmath9 . the cases with the coupling constant @xmath518 less than the renormalized natural frequency @xmath519 of the detector are considered . we solved exactly the evolution equations for the combined system of a moving detector coupled to a quantum field in the heisenberg picture , and from the evolution of the operators we can obtain complete information on the combined system . for the case that the initial state is a direct product of a coherent state for the detector and the minkowski vacuum for the field , we worked out the exact two - point functions of the detector and similar functions of the field . by applying the coherent state for the detector , we can distinguish the classical behaviors from others . the quantum part of the coincidence limit of two - point functions , namely , the variances of @xmath3 and @xmath9 , are determined by the detector and the field together . from the exact solutions , we were able to study the complete process from the initial transient to the final steady state . in particular , we can identify the time scales of transient behaviors analytically . when the coupling is turned on , the zero - point fluctuations of the free detector dissipates exponentially , then the vacuum fluctuations take over . the time scales for both processes are the same . eventually the variance of @xmath3 saturates at a finite value , where the dissipation of the detector is balanced by the input from the vacuum fluctuations of the field . even in the zero - acceleration limit , the variances of @xmath3 and @xmath184 , thus the ground state energy of the interacting detector , shifted from the ones for the free detector . this fluctuations - induced effect share a similar origin with that of the lamb shift . the variance of @xmath3 yields an effective squared scalar charge , which induce a positive variance in the scalar field . this variance of the field contributes a positive radiated energy at the quantum level . however , the interference between the vacuum fluctuations and the retarded solution induced by the vacuum fluctuations screens part of the emission of quantum radiation . the time scale of the screening is proportional to @xmath363 for @xmath520 , where @xmath1 is the proper acceleration and @xmath521 is the damping constant proportional to @xmath522 . after the screening the renormalized green s functions of the field are still non - zero in steady state . a quantum radiation formula determined at the null infinity of minkowski space has been derived . we found that even in steady state there exists a positive radiated power of quantum nature emitted by the uniformly accelerated ud detector . for large @xmath1 the radiated power is proportional to @xmath523 , where @xmath498 is the unruh temperature . this could be interpreted as a hint of the unruh effect . however , the nearly constant negative flux obtained in ref.@xcite for ( 3 + 1)d case is essentially a transient effect . only part of the radiation is connected to the internal energy of the detector . the total energy of the dressed detector and the radiated energy of a monopole radiation from the detector is conserved for every proper time interval after the coupling is turned on . the external agent which drives the detector s motion has no additional influence on this channel . since the corresponding monopole radiation of quantum nature ceases in steady state , the hint of the unruh effect in the late - time radiated power is not directly from the energy flux experienced by the detector in unruh effect . this extends the result in ref.@xcite that there is no emitted radiation of quantum origin in unruh effect in ( 1 + 1 ) dimensional spacetime . since all the relevant quantum and statistical information about the detector ( atom ) and the field can be obtained from the results presented here , when appropriately generalized , they are expected to be useful for addressing issues in atomic and optical schemes of quantum information processing , such as quantum decoherence , entanglement and teleportation . these investigations are in progress . @xmath524 * acknowledgments * blh thanks alpan raval and mei - ling tseng for discussions in an earlier preliminary attempt on the 3 + 1 problem . we appreciate the referees queries and suggestions for improvements on the presentation of this paper . this work is supported in part by the nsc taiwan under grant nsc93 - 2112-m-001 - 014 and by the us nsf under grant phy03 - 00710 and phy-0426696 . from @xmath291 in @xmath338 , it is easy to see that @xmath525 . \label{phi1phi1}\end{aligned}\ ] ] note that the @xmath526-functions at @xmath527 , coming from the derivative of the step functions , have been neglected here . with @xmath347 , one can write down in closed form of the interfering terms in the r - wedge of the rindler space . under the coincidence limit , it looks like , @xmath528 + ( x\leftrightarrow x')\right\}\nonumber\\&= & { \hbar\lambda_0 ^ 2\theta(\eta_-)\over(2\pi)^3 m_0 a^2 x^2 } \left[\eta_{-,\mu}\eta_{-,\nu}\theta_{--}+ { x_{,\mu}x_{,\nu}\over x^2}\theta_{xx } -{x_{,\mu}\over x}\eta_{-,\nu}\theta_{x- } - \eta_{-,\mu}{x_{,\nu}\over x } \theta_{-x}\right.\nonumber\\ & & \left . + \eta_{-,\mu}\eta_{+,\nu}\theta_{-+}+ \eta_{+,\mu}\eta_{-,\nu}\theta_{+- } - { x_{,\mu}\over x}\eta_{+,\nu } \theta_{x+ } - \eta_{+,\mu}{x_{,\nu}\over x}\theta_{+x}\right].\label{xterm}\end{aligned}\ ] ] where @xmath529,\label{th11}\\ \theta_{xx } & = & % -{a\over \omega_r^2 } ? ? + { 2\over \omega } { \rm re } \left [ i\psi_{\gamma+i\omega}+ { ia\over\gamma+i\omega+a } e^{-(\gamma+i\omega+a)\eta_- } f_{\gamma+i\omega}(e^{-a\eta_- } ) \right]+2 { \cal f}_0(x),\label{th00}\\ \theta_{-x}&= & \theta_{x- } = -{a\over\omega}{e^{-\gamma\eta_-}\over e^{a\eta_-}-1 } \sin\omega\eta_- + { \cal f}_1(x),\label{thmx}\\ \theta_{+x } & = & \theta_{x+ } = { a\over\omega}{e^{-\gamma\eta_-}\over \pm e^{a\eta_+}-1 } \sin\omega\eta_- - { \cal f}_1(x),\label{thpx}\\ \theta_{+- } & = & \theta_{-+}= -{a\over\omega}{e^{-\gamma\eta_-}\over \pm e^{a\eta_+}-1 } \left(\gamma\sin\omega\eta_- -\omega\cos\omega\eta_-\right)- { a \over \pm e^{a(\eta_+-\eta_-)}-1 } + { \cal f}_2(x),\label{thpm}\end{aligned}\ ] ] with @xmath530 \right\},\ ] ] with @xmath531 and @xmath532 for @xmath303 in r and f - wedge , respectively . note that @xmath460 is independent of @xmath322 . @xmath437 is actually proportional to @xmath296 in @xmath323 . another observation is that , combining @xmath401 and @xmath402 , one finds that the divergent @xmath208 term in @xmath460 is canceled by the one in @xmath533 . as for @xmath403 , the result is similar to @xmath401 with @xmath534 being replaced by @xmath535 . again it splits into the quantum and semiclassical parts as in sec.iv b. the conservation law @xmath459 is directly obtained by arranging our somewhat complicated results . it can also be derived in a simpler way as follows . from the definition of the ground - state energy @xmath454 of the dressed detector , its first derivative of @xmath7 is @xmath536\ ] ] introducing the equations of motion @xmath98 and @xmath119 to eliminate @xmath537 , one has @xmath538\right>_{\rm v } + \omega_r^2 \left<\right.q(\tau)\dot{q}(\tau)\left.\right>_{\rm v}\right . \nonumber\\ & & \left.+ \left<\dot{q}(\tau)\left[-2\gamma\dot{q}(\tau)- \omega_r^2 q(\tau)\right]\right>_{\rm a}+ \omega_r^2 \left<\right.q(\tau)\dot{q}(\tau)\left.\right>_{\rm a}\right\}\nonumber\\ & = & -2\gamma m_0 \left<\right.\dot{q}^2(\tau)\left.\right > + \lambda_0 { \rm re}\left<\right.\dot{q}(\tau)\phi_0(y(\tau))\left.\right>_{\rm v } , \label{dotenergy}\end{aligned}\ ] ] where the last term is a short - hand for @xmath539 ( there is an additional term @xmath540 if a background field @xmath541 is present . ) substituting @xmath147 and @xmath148 into @xmath542 , integrating out @xmath543 with the help of eq.@xmath186 , then comparing with @xmath544 in @xmath402 where @xmath545 has the formal expression @xmath546 , one finds that @xmath542 is actually identical to @xmath547 . hence @xmath548 integrating both side from @xmath549 to @xmath550 , one ends up with eq.@xmath459 . c. r. galley and b. l. hu , phys . rev . * d72 * , 084023 ( 2005 ) ; c. r. galley , b. l. hu and s .- y . lin , electromagnetic and gravitational self - force on a relativistic particle from quantum fields in curved space " [ gr - qc/0603099 ] .
|
in this paper we analyze the interaction of a uniformly accelerated detector with a quantum field in ( 3 + 1)d spacetime , aiming at the issue of how kinematics can render vacuum fluctuations the appearance of thermal radiance in the detector ( unruh effect ) and how they engender flux of radiation for observers afar .
two basic questions are addressed in this study : a ) how are vacuum fluctuations related to the emitted radiation ?
b ) is there emitted radiation with energy flux in the unruh effect ?
we adopt a method which places the detector and the field on an equal footing and derive the two - point correlation functions of the detector and of the field separately with full account of their interplay . from the exact solutions , we are able to study the complete process from the initial transient to the final steady state ,
keeping track of all activities they engage in and the physical effects manifested .
we derive a quantum radiation formula for a minkowski observer .
we find that there does exist a positive radiated power of quantum nature emitted by the detector , with a hint of certain features of the unruh effect .
we further verify that the total energy of the dressed detector and a part of the radiated energy from the detector is conserved .
however , this part of the radiation ceases in steady state .
so the hint of the unruh effect in radiated power is actually not directly from the energy flux that the detector experiences in unruh effect . since all the relevant quantum and statistical information about the detector ( atom ) and
the field can be obtained from the results presented here , they are expected to be useful , when appropriately generalized , for addressing issues of quantum information processing in atomic and optical systems , such as quantum decoherence , entanglement and teleportation .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in order to construct @xmath1-dimensional homotopy quantum field theories ( hqft s ) , the second author introduced for a group @xmath0 the notions of a ribbon @xmath0-category and a modular ( ribbon ) @xmath0-category . the aim of this paper is to analyze the categories of representations of quantum groups ( at roots of unity ) from this prospective . the role of @xmath0 will be played by the center of the corresponding lie algebra . we begin with a general theory of ribbon @xmath0-categories with abelian @xmath0 . according to @xcite , any modular ( ribbon ) @xmath0-category @xmath8 gives rise to a 3-dimensional hqft with target @xmath2 . such an hqft comprises several ingredients including a homotopy modular functor " assigning modules to pairs ( a surface @xmath9 , an element of @xmath10 ) and a numerical invariant of pairs ( a closed oriented 3-manifold @xmath6 , a cohomology class @xmath11 ) . we introduce here a larger class of weakly non - degenerate premodular @xmath0-categories . similarly to @xcite , each such category @xmath8 gives rise to a numerical invariant @xmath12 of pairs @xmath13 as above . if @xmath14 satisfies an additional assumption of regularity and @xmath0 is finite then the standard witten - reshetikhin - turaev invariant @xmath15 is also defined and splits as @xmath16 where @xmath17 is the first betti number of @xmath6 . note that the invariant @xmath12 extends to an hqft provided @xmath8 is a modular @xmath0-category . let @xmath3 be a simple complex lie algebra and @xmath5 be a complex root of unity . we show that under certain assumptions on the order of @xmath5 , the pair @xmath18 gives rise to a premodular ribbon @xmath0-category @xmath19 where @xmath20 is the center of @xmath3 . ( for the center groups of simple lie algebras , see table 1 below ) . the definition of @xmath8 is based on a study of the representations of the quantum group @xmath4 , cf . we specify conditions on @xmath5 which ensure that @xmath8 is regular so that we have numerical invariants of 1-cohomology classes and a splitting of the standard wrt - invariant as above . another set of conditions ensures that @xmath21 is a modular @xmath0-category . the resulting 3-dimensional hqft is however not very interesting since it splits as a product of a standard tqft and a homological hqft ( cf . remark [ quq ] ) . the paper consists of three sections . in sect . 1 we discuss the theory of premodular @xmath0-categories with abelian @xmath0 . in sect . 2 we recall the definition of @xmath12 and briefly discuss the homotopy modular functor . in sect . 3 we consider the category @xmath22 . the first author would like to thank a. beliakova , c. blanchet , a. bruguieres , g. masbaum for helpful discussions . throughout this section , @xmath0 denotes an abelian group and @xmath23 a field of characteristic zero . we shall use the standard notions of the theory of monoidal categories , see @xcite . recall that a left duality in a monoidal category @xmath8 associates to any object @xmath24 an object @xmath25 and two morphisms @xmath26 and @xmath27 satisfying the identities @xmath28 here @xmath29 denotes the unit object of @xmath30 and for simplicity we omit the associativity isomorphisms and the canonical isomorphisms @xmath31 . an object of @xmath30 isomorphic to @xmath29 is said to be _ trivial_. a monoidal category @xmath14 is _ @xmath23-additive _ if all the @xmath32 in @xmath14 are @xmath23-modules and both the composition and the tensor product of morphisms are bilinear over @xmath23 . we say that a @xmath23-additive monoidal category @xmath14 _ splits as a disjoint union of subcategories _ @xmath33 numerated by certain @xmath34 if : - each @xmath35 is a full subcategory of @xmath14 ; - each object of @xmath14 belongs to @xmath35 for a unique @xmath34 ; - if @xmath36 and @xmath37 with @xmath38 then @xmath39 . a _ monoidal @xmath0-category over @xmath23 _ is a @xmath23-additive monoidal category with left duality @xmath14 which splits as a disjoint union of subcategories @xmath40 numerated by @xmath41 such that \(i ) if @xmath36 and @xmath37 then @xmath42 ; \(ii ) if @xmath36 then @xmath43 . we shall write @xmath44 and call the subcategories @xmath45 of @xmath14 the _ components _ of @xmath14 . the category @xmath46 corresponding to the neutral element @xmath47 is called the _ neutral component _ of @xmath14 . conditions ( i ) and ( ii ) show that @xmath46 is closed under tensor multiplication and taking the dual object . condition ( i ) implies that @xmath48 . thus , @xmath46 is a monoidal category with left duality . the standard notions of braidings and twists in monoidal categories apply in this setting without any changes . a braiding ( resp . twist ) in a monoidal @xmath0-category @xmath14 is a system of invertible morphisms @xmath49 ( resp . @xmath50 ) satisfying the usual conditions , see @xcite . we say that a monoidal @xmath0-category is _ ribbon _ if it is ribbon in the sense of @xcite , i.e. , if it has braiding and twist compatible with each other and with duality . the standard theory of ribbon categories applies to any ribbon @xmath0-category @xmath8 . suppose that @xmath51 is a framed oriented @xmath52-component link in @xmath53 whose components are ordered , and @xmath54 are @xmath23-linear combinations of objects of @xmath55 . then there is defined the quantum reshetikhin - turaev invariant @xmath56 . in particular , for any object @xmath57 , we have a dimension @xmath58 where @xmath59 is an oriented unknot with framing 0 . for any endomorphism @xmath60 , we have a well - defined trace @xmath61 so that @xmath62 . let @xmath14 be a ribbon @xmath0-category . an object @xmath63 of @xmath14 is _ simple _ if @xmath64 . it is clear that an object isomorphic or dual to a simple object is itself simple . the assumption that @xmath23 is a field and ( * ? ? ? * lemma ii.4.2.3 ) imply that any non - zero morphism between simple objects is an isomorphism . we say that an object @xmath63 of @xmath14 is _ dominated by simple objects _ if there is a finite set of simple objects @xmath65 of @xmath14 ( possibly with repetitions ) and morphisms @xmath66 such that @xmath67 . clearly , if @xmath36 then without loss of generality we can assume that @xmath68 for all @xmath69 . we say that a ribbon @xmath0-category @xmath14 is _ premodular _ if it satisfies the following three axioms : ( 1.3.1 ) the unit object @xmath70 is simple ; ( 1.3.2 ) for each @xmath71 , the set @xmath72 of the isomorphism classes of simple objects of @xmath35 is finite ; ( 1.3.3 ) for each @xmath71 , any object of @xmath35 is dominated by simple objects of @xmath35 . according to ( * ? ? ? * remark 7.8 ) the set @xmath73 is a subgroup of @xmath0 . the theory certainly reduces to @xmath74 . for simplicity , we will assume from now on that @xmath75 . recall that @xmath59 is an unknot with framing @xmath76 . let @xmath77 be an unknot with framing @xmath78 . for each @xmath71 , consider the formal linear combination @xmath79 where @xmath80 is a simple object in @xmath35 representing @xmath81 . define @xmath82 @xmath83 where the twist @xmath84 acts on @xmath80 as the scalar operator @xmath85 . we recall here the following result ( * ? ? ? * lemma 6.6.1 ) . if @xmath86 , then @xmath87 . [ product ] the equality here is understood as an equality in the verlinde algebra of @xmath8 . applying to both sides the @xmath23-linear homomorphism @xmath88 from the verlinde algebra to @xmath23 , sending the class of any object to its dimension , we obtain that @xmath89 . taking as @xmath63 a simple object in @xmath90 and using the fact that @xmath91 ( see ( * ? ? ? * lemma 6.5 ) ) we obtain the following corollary . for every @xmath92 , @xmath93 [ equal ] one has that @xmath94 . [ product2 ] this follows from lemma [ product ] and corollary [ equal ] . an important property of any premodular category is the sliding property . here is the version for premodular g - category . ( graded sliding property ) let @xmath8 be a premodular @xmath0-category . let @xmath95 be framed oriented ordered links in @xmath53 such that @xmath96 is obtained from @xmath51 by sliding the second component over the first one ( see figure [ sliding ] ) . then for every @xmath97 , one has @xmath98 @xmath99 the proof is the same as in the case of a premodular category , see for example @xcite . one has to take into account the colors of the components of tensor product decomposition . this proposition extends to tangles in the obvious way . one has to be a bit careful about colorings of a tangle . any non - circle component must be colored with an object of @xmath8 , not a @xmath23-linear combination of objects as for circle components . suppose that @xmath100 is a set of objects of a ribbon @xmath0-category @xmath8 . an object @xmath101 is _ @xmath102-transparent _ if for every @xmath103 , @xmath104 this means that one can always move a string colored by @xmath63 past a string colored by @xmath105 , see figure [ transparent ] . @xmath106 let @xmath107 denote the set of all @xmath108-transparent simple objects of @xmath8 . let @xmath63 be a simple object of a premodular @xmath0-category @xmath8 with @xmath109 . the operator of the tangle in figure [ s]a is non - zero for some @xmath71 if and only if @xmath110 , i.e. , @xmath63 is @xmath108-transparent . [ trans3 ] @xmath111 since @xmath63 is simple , the operator in figure [ s ] is equal to a scalar operator @xmath112 with @xmath113 . suppose @xmath114 . let @xmath115 . figure [ transparent2 ] shows that @xmath110 ; the second equality uses the graded sliding property . ( this argument was first used by blanchet and beliakova in @xcite . ) @xmath116 now suppose @xmath117 . it is clear that @xmath118 . by corollary [ product2 ] , one has @xmath119 . the operator of the tangle in figure [ s]b is @xmath120 . it is also equal to ( by combining the two parallel components ) @xmath121 . thus @xmath122 , and @xmath123 . under the assumption of lemma [ trans3 ] , if @xmath63 is @xmath46-transparent , then @xmath63 is almost " transparent with respect to any object in @xmath8 . more precisely , the same argument as in the proof of lemma [ trans3 ] shows that for any @xmath124 , one has @xmath125 a premodular @xmath0-category @xmath14 is _ weakly non - degenerate _ if @xmath126 . it is known ( see @xcite ) that weak non - degeneracy implies @xmath109 . a premodular @xmath0-category @xmath14 is _ regular _ if it is weakly non - degenerate and @xmath127 . if @xmath8 is a regular premodular @xmath0-category and @xmath128 then @xmath129 [ vanish ] consider the equality in figure [ proof4 ] which is obtained by sliding the top component of the left hand side over the bottom one . @xmath130 by the regularity of @xmath14 , there are no @xmath108-transparent simple objects in @xmath35 . hence , by lemma [ trans3 ] , the right hand side on figure [ proof4 ] is @xmath76 . it follows that @xmath131 , since @xmath132 . similarly @xmath133 . let @xmath8 be a premodular @xmath0-category . for @xmath134 , choose simple objects @xmath135 representing @xmath136 , respectively , and set @xmath137 here @xmath138 is the standard hopf link in @xmath53 with framing @xmath76 . it follows from the properties of the trace that @xmath139 does not depend on the choice of @xmath140 in the isomorphism classes @xmath136 . a premodular @xmath0-category @xmath8 is _ modular _ if the following axiom is satisfied . ( _ non - degeneracy axiom _ ) the ( finite ) square matrix @xmath141_{i , j\in i_0 } $ ] is invertible over @xmath23 . it follows from this axiom that the neutral component @xmath46 of @xmath14 is a modular ribbon category in the usual , ungraded sense ( see @xcite ) . it is known that the non - degeneracy implies the weak non - degeneracy . a modular ribbon @xmath0-category @xmath8 may be non - modular in the ungraded sense because the set @xmath142 of the isomorphism classes of simple objects in @xmath8 may be infinite , or because the full @xmath143-matrix @xmath141_{i , j\in i } $ ] may be non - invertible . suppose that @xmath0 is finite and a modular g - category @xmath55 is regular . then @xmath8 is a modular category in the ungraded sense . [ ttt ] according to @xcite , @xcite , a premodular category @xmath8 is modular if and only if it has no non - trivial @xmath8-transparent simple objects and @xmath144 . if @xmath63 is a @xmath8-transparent simple object of @xmath8 then @xmath145 . but since @xmath46 is modular , @xmath63 is a trivial object . by corollary [ equal ] , @xmath146 , since @xmath46 is modular . * 2.1 . invariants of 3-dimensional @xmath0-manifolds . * fix an abelian group @xmath0 . let @xmath21 be a weakly non - degenerate premodular ribbon @xmath0-category over a field of zero characteristic @xmath23 . we explain here following @xcite that @xmath21 gives rise to a topological invariant of 1-dimensional cohomology classes of 3-manifolds with coefficients in @xmath0 . fix an element @xmath147 such that @xmath148 . let @xmath6 be a closed connected oriented 3-dimensional manifold and @xmath149 . present @xmath6 as the result of surgery on @xmath53 along a framed oriented link @xmath150 . recall that @xmath6 is obtained by gluing @xmath52 solid tori to the exterior of @xmath51 in @xmath53 this allows us to consider for @xmath151 , the value , @xmath152 , of @xmath153 on the meridian of @xmath154 and provide @xmath154 with the color @xmath155 . let @xmath156 ( resp . @xmath157 ) be the number of positive ( resp . negative ) squares in the diagonal decomposition of the intersection form @xmath158 where @xmath159 is the compact oriented 4-manifold bounded by @xmath6 and obtained from the 4-ball @xmath160 by attaching 2-handles along tubular neighborhoods of the components of @xmath51 in @xmath161 . set @xmath162 where @xmath163 is the first betti number of @xmath6 . it follows from ( * ? ? ? * theorem 7.3 ) that @xmath164 is a homeomorphism invariant of the pair @xmath165 ( in @xcite this is stated for modular @xmath0-categories , but the proof remains true for weakly non - degenerate premodular categories ) . the factor @xmath166 appears here for normalization purposes . as in the standard theory , the invariant @xmath167 generalizes to an invariant of triples @xmath168 where @xmath6 is as above , @xmath169 is a colored ribbon graph in @xmath6 and @xmath170 . here a coloring is understood as the usual coloring of @xmath169 over @xmath8 such that the color of every 1-stratum @xmath171 of @xmath169 is an object in @xmath172 where @xmath173 is the value of @xmath153 on the meridian of @xmath171 . this notion of a coloring applies in particular to framed oriented links in @xmath6 so that we obtain an isotopy invariant of triples @xmath174 where @xmath6 is as above , @xmath51 is a colored framed oriented link in @xmath6 and @xmath175 . suppose now that @xmath0 is finite and @xmath8 is regular . set @xmath176 . by lemma [ vanish ] , @xmath177 we can therefore consider the standard witten - reshetikhin - turaev invariant @xmath178 of @xmath6 where @xmath179 . if @xmath0 is finite and @xmath8 is a regular premodular @xmath0-category , then for any closed connected oriented 3-manifold @xmath6 , @xmath180 the proof in @xcite can be applied to get the result . a modular @xmath0-category @xmath8 ( not necessarily regular ) gives rise to a @xmath1-dimensional hqft , see @xcite . in particular , the homotopy modular functor @xmath181 associated with @xmath55 assigns @xmath23-modules to so - called extended @xmath0-surfaces and assigns @xmath23-linear isomorphisms of these modules to weak homeomorphisms of such surfaces . we recall here the definition of an extended @xmath0-surface and the corresponding @xmath23-module in our abelian case . let @xmath182 be a closed oriented surface . a point @xmath183 is _ marked _ if is equipped with a sign @xmath184 and a tangent direction , i.e. , a ray @xmath185 where @xmath186 is a non - zero tangent vector at @xmath187 . a _ marking _ of @xmath182 is a finite ( possibly void ) set of distinct marked points @xmath188 . @xmath0-marking _ of @xmath182 is a marking @xmath188 endowed with a cohomology class @xmath189 . a @xmath0-marking @xmath188 is _ colored _ if it is equipped with a function @xmath190 which assigns to every point @xmath191 an object @xmath192 where @xmath193 is a small loop in @xmath194 encircling @xmath187 in the direction induced by the orientation of @xmath182 if @xmath195 and in the opposite direction otherwise . an _ extended @xmath0-surface _ comprises a closed oriented surface @xmath182 , a colored @xmath0-marking @xmath188 , and a lagrangian space @xmath196 . the corresponding @xmath23-module is defined as follows . assume that @xmath182 is a connected surface of genus @xmath197 and @xmath198 . for @xmath199 , set @xmath200 and @xmath201 . the group @xmath202 is generated by the homotopy classes of loops @xmath203 homotopic to @xmath204 , respectively , and by @xmath205 elements @xmath206 subject to the only relation @xmath207\ , ... \,[a_n , b_n]= 1\ ] ] where @xmath208= ab a^{-1}b^{-1}$ ] . then @xmath209 @xmath210 where for an object @xmath211 we set @xmath212 and @xmath213 . observe that the dimension of @xmath214 over @xmath23 does not depend on the values of @xmath153 on @xmath215 . this implies that this dimension does not depend on the choice of @xmath153 . for @xmath216 , this dimension is computed by the verlinde formula . in particular , if @xmath217 we obtain @xmath218 the same formula can be deduced from the generalised verlinde formula for hqft s given in ( * ? ? ? 4.11 ) . since we restrict ourselves here to abelian @xmath0 , the homotopy classes of maps from a manifold @xmath6 to the eilenberg - maclane space @xmath219 are classified by elements of @xmath220 this allows one to formulate hqft s with target @xmath219 in terms of 1-dimensional cohomology classes . we call the hqft s arising from modular @xmath0-categories with abelian @xmath0 and trivial crossed structure ( as everywhere in this paper ) _ abelian_. the abelian hqft s are simpler than the general hqft s . one of simplifications is that in the abelian case we can forget about base points ( in the general case one has to work in the pointed category ) . also , in the abelian case the canonical action of @xmath0 on the spaces of conformal blocks @xmath221 defined in ( * ? ? ? 10.3 ) is trivial . let @xmath222 be a finite - dimensional simple lie algebra over @xmath223 . fix a cartan subalgebra and a system of basis roots @xmath224 . choose the inner product on @xmath225-span@xmath226 such that the square length of any _ short _ root is 2 . let @xmath227 be the quotient of the weight lattice @xmath228 by the root lattice @xmath229 . it is known that @xmath0 is isomorphic to the center of the simply - connected lie group associated with @xmath222 , and @xmath230 is equal to the determinant of the cartan matrix . let @xmath231 be the maximal absolute value of the non - diagonal entries of the cartan matrix . let @xmath232 denote the coxeter number of @xmath3 and @xmath233 the dual coxeter number of @xmath3 . by @xmath234 we denote the smallest positive integer such that @xmath235 for all @xmath236 . the data associated with simple lie algebras is given in table 1 . the quantum group @xmath237 , as defined in @xcite , is a hopf algebra over @xmath238 , with @xmath186 a formal variable . there is an integral version of @xmath239 , defined over the ring @xmath240 $ ] , introduced by lusztig . for @xmath241 , let @xmath242 be the hopf algebra over @xmath223 obtained by tensoring the integral version of @xmath237 with @xmath223 , where @xmath243 is considered as a @xmath240$]-module by @xmath244 ( see @xcite ) . let @xmath5 be a root of unity . the fusion category @xmath245 is the quotient of the category of all tilting @xmath242-modules by negligible modules and negligible morphisms ( see @xcite , there @xmath245 is denoted by @xmath246 ) . the category @xmath245 is a semisimple monoidal @xmath223-abelian category with duality . the isomorphism classes of simple objects in @xmath245 are parametrized by the dominant weights @xmath247 such that @xmath248 where @xmath249 is the half - sum of all positive roots and @xmath250 is defined as follows . there are two cases depending on the order @xmath69 of the root of unity @xmath251 . case 1 : the numbers @xmath69 and @xmath231 are co - prime and @xmath252 . then @xmath253 where @xmath254 is the weyl chamber and @xmath255 is the short highest root , i.e. , the only root in @xmath254 with square length 2 . for a dominant weight @xmath247 such that @xmath248 there is a simple @xmath242-module @xmath263 , known as the weyl module , which is a deformation of the corresponding classical @xmath3-module . it represents the isomorphism class numerated by @xmath247 . the decomposition of a tensor product @xmath264 is described in @xcite ( the tensor product is denoted there by , but we will use the usual notation ) . for our purposes , we notice that if @xmath265 appears as a summand in @xmath266 , then @xmath267 for some @xmath268 where @xmath269 is the group of affine transformations of @xmath270 generated by the reflections in the walls of the simplex @xmath271 , the topological closure of @xmath250 , and the dot action is the one shifted by @xmath249 so that @xmath272 . note that @xmath273 is a fundamental domain of @xmath269 . for further use , we state here a simple lemma . it is known that @xmath277 , where @xmath105 is the usual weyl group , and @xmath278 is the @xmath279-lattice spanned by the long roots . ( one considers @xmath278 as a group of translations ) . obviously , @xmath280 . if @xmath281 , the statement is trivial . if @xmath282 , the statement is well - known in the theory of simple lie algebras . there are a braiding and a twist in @xmath245 which make @xmath245 a ribbon category . they depend on a choice of a complex root of unity @xmath283 such that @xmath284 , where @xmath234 is as in table 1 . let us denote the resulting ribbon category by @xmath285 . the tensor structure in @xmath285 and the corresponding verlinde algebra do not depend on the choice of @xmath283 . the ribbon category @xmath285 is premodular , since it has only a finite number of isomorphism classes of simple objects . it is also hermitian , see @xcite , so that @xmath286 for every @xmath287 . ( this can also be deduced from the explicit formulas for the quantum dimensions of the weyl modules ) . thus for every simple object @xmath190 , one has @xmath288 is a positive real number ( since @xmath289 ) . in case 2 if @xmath290 , then this hermitian structure is known to be unitary , see @xcite . then @xmath291 for every @xmath287 . again this can also be deduced from the explicit formulas for the quantum dimensions of the weyl modules . it is clear that @xmath292 where @xmath69 is the order of @xmath251 . when @xmath69 is divisible by @xmath231 , @xmath293 , and @xmath283 has order @xmath294 , the category @xmath285 is modular . these are the well - known , and so far the main , examples of modular category . [ [ tildecal - cfrak - gvarepsilonzeta - as - a - premodular - g - category ] ] @xmath285 as a premodular @xmath0-category ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ let @xmath295 be the projection . for @xmath41 , let @xmath72 be the set of dominant weights @xmath247 such that @xmath296 and @xmath297 . let @xmath298 be the set of all objects in @xmath285 which are direct sums of @xmath299 with @xmath300 . set @xmath301 . we consider @xmath8 as a full subcategory of @xmath285 . if all the colors of a framed link @xmath51 are simple @xmath3-modules with highest weights in the root lattice , then the quantum invariant of @xmath51 is in @xmath311 $ ] , according to the integrality , see @xcite . hence the fact that @xmath8 is a modular , or weakly non - degenerate , @xmath0-category does not depend on the choice of the @xmath234-th root @xmath283 of @xmath5 ; it totally depends on the order @xmath69 of @xmath251 . note that the fact @xmath285 is a ( non - graded ) modular category does depend on the choice of @xmath283 . the following proposition shows that in case 1 of sect . [ cases ] , the category @xmath312 constructed above is a modular @xmath0-category , at least under the assumption that @xmath69 is co - prime with @xmath230 . [ ref22 ] suppose @xmath3 is a lie algebra of series @xmath316 with odd @xmath317 . assume that the order @xmath69 of @xmath251 is even but not divisible by 4 and @xmath318 . then @xmath8 is a modular @xmath0-category . ( note that for @xmath316 , one has @xmath319 , and @xmath320 . ) in the cases of propositions [ ref11 ] and [ ref22 ] , it can be shown that the category @xmath21 is the product of its neutral component and a modular category associated with the center group @xmath0 ( see the definition in @xcite , see also @xcite ) . the corresponding invariant of a 1-cohomology class on a 3-manifold @xmath6 is then the product of the invariant of the cohomology class @xmath76 with an invariant depending only on @xmath321 and the linking form . the theory in this case is rather trivial . [ quq ] one may ask when the @xmath0-category @xmath8 is weakly - nondegenerate . the answer is only when the order @xmath69 of @xmath251 is as one described in propositions [ ref11 ] , [ ref22 ] , [ pre ] , or one in the following additional cases : for the lie algebra @xmath338 with odd @xmath317 the number @xmath69 must satisfies @xmath339 ; for the lie algebra @xmath340 , one must have @xmath341 , where @xmath342 . our aim here is to study the set @xmath107 of @xmath46-transparent objects of @xmath343 . to cover both cases 1 and 2 of sect . [ cases ] we introduce a group @xmath344 equal to @xmath0 in case 1 and equal to @xmath345 in case 2 . more precisely , set @xmath346 where @xmath347 , @xmath348 in case 1 and @xmath349 are the @xmath279-dual lattices in case 2 . in both cases one has @xmath350 . the group @xmath344 acts on @xmath271 as follows . let @xmath351 with a lift @xmath352 and let @xmath353 . the element @xmath354 may not lie in the simplex @xmath355 anymore , but there is @xmath275 which maps @xmath356 into @xmath271 . set @xmath357 . let the dot version of this action be the one shifted by @xmath249 so that @xmath358 . \(ii ) if @xmath69 is divisible by @xmath231 , @xmath360 , and @xmath283 is a root of unity of order @xmath294 then every @xmath108-transparent simple object in @xmath361 is isomorphic to @xmath362 with @xmath363 . [ sub ] to prove proposition [ sub ] we first recall the so - called second symmetry principle for quantum link invariants ( see @xcite ) . it describes how these invariants change under the action of @xmath344 on the colors of link components . we record two corollaries of this principle . first , @xmath366 this means for the hopf link colored by @xmath63 and any element in @xmath46 , we can unlink the hopf link not altering the value of the quantum invariant . thus the operator of figure [ s]a with @xmath378 is equal to @xmath379 . applying lemma [ trans3 ] ( for strings piercing the unknot with color @xmath384 ) , we obtain that @xmath385 where @xmath386 is the set of isomorphism classes of objects in @xmath387 . on the other hand , @xmath8 is known to be modular in the ungraded sense . hence a string colored by a simple object piercing through an unknot with color @xmath388 is non - zero only when the color of the string is a trivial object . hence @xmath389 which , by corollary [ equal ] , is @xmath390 . thus @xmath391 from part ( i ) we know that @xmath392 for every @xmath351 , and @xmath393 by ( [ dim ] ) . since @xmath394 , we can have @xmath395 only when @xmath396 . hence we have at least @xmath397 pairwise non - isomorphic simple objects in @xmath107 , each has the square of the quantum dimension equal 1 . since @xmath398 is a positive real number for any @xmath399 , we obtain that @xmath386 consists only of the isomorphism classes of @xmath400 with @xmath351 . now we prove that @xmath411 . by the second symmetry principle in @xcite , the twist @xmath84 acts on each @xmath400 as multiplication by @xmath412 where @xmath352 is a lift of @xmath351 . now ( b ) implies that the twist acts as the identity operator on all @xmath46-transparent objects . from ( [ dim ] ) we see that for @xmath413 one has @xmath414 . if @xmath415 ( in that case @xmath416 ) then @xmath417 is positive , hence @xmath418 . when @xmath283 is an arbitrary root of unity of order @xmath294 , by considering a galois action , we also have @xmath418 . we have @xmath420 . let @xmath368 be the non - trivial element of @xmath344 . explicit calculation shows that @xmath400 is not in @xmath108 . it follows from proposition [ sub ] part ( ii ) that @xmath108 does not have any non - trivial transparent objects . hence by a criterion of bruguieres ( simplified in ( * ? ? ? * lemma 4.3 ) ) , @xmath46 is modular . \1 . in the @xmath421-case , there is a decomposition of quantum invariants considered by blanchet and beliakova , using idempotents in the birman - wenzl - murakami category . their construction is probably different from ours because they do not consider spin representations . see also @xcite . \2 . for a quotient group @xmath422 of @xmath0 , one can consider any @xmath0-category as a @xmath422-category . by choosing @xmath422 appropriately , we can make the proof of proposition [ pre ] valid even when the order @xmath69 of @xmath251 does not satisfy the conditions there . in this way one can get a cohomology decomposition for @xmath337 in the case @xmath423 , as in @xcite . similar result holds true for the lie series @xmath234 . suppose @xmath424 ( otherwise the theory is reduced to the ungraded case ) . if the order @xmath69 of @xmath251 does not satisfy the conditions of proposition [ ref22 ] or [ ref11 ] , then it can be shown that @xmath46 contains a non - trivial @xmath108-transparent element of the form @xmath425 , hence @xmath8 can not be modular ( in the @xmath0-category sense ) . kirby , r. , melvin , p. , on the 3-manifold invariants of witten and reshetikhin - turaev for @xmath426 . ( 1991 ) , 473545 . kirillov , a. _ on an inner product in modular categories _ , jour . ams , * 9 * ( 1996 ) , 11351169 . murakami , h. , _ quantum invariants for 3-manifolds _ in the 3rd korea - japan school of knots and links , proc . of applied math . workshop , * 4 * ( 1994 ) , 129143 . sawin , s. _ jones - witten invariants for non - simply connnected lie groups and the geometry of the weyl alcove _ , preprint math.qa/9905010 .
|
for a group @xmath0 , the notion of a ribbon @xmath0-category was introduced in @xcite with a view towards constructing @xmath1-dimensional homotopy quantum field theories ( hqft s ) with target @xmath2 .
we discuss here how to derive ribbon @xmath0-categories from a simple complex lie algebra @xmath3 where @xmath0 is the center of @xmath3 .
our construction is based on a study of representations of the quantum group @xmath4 at a root of unity @xmath5 . under certain assumptions on @xmath5 ,
the resulting @xmath0-categories give rise to numerical invariants of pairs ( a closed oriented 3-manifold @xmath6 , an element of @xmath7 ) and to 3-dimensional hqft s .
-0.2 cm -0.2 cm [ section ] [ thm]lemma [ thm]proposition [ thm]corollary [ thm]remark [ section ] [ section ] c i d zz rr = 12.5 cm = 19.0 cm = 0.5 cm = 0pt = 12pt = 12pt hom i d sgn
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the capacitated vehicle routing problem ( @xmath3 ) is an extensively studied combinatorial optimization problem ( see e.g. , the book @xcite and references therein ) . @xmath3is defined on a metric space @xmath4 , where @xmath5 is a finite set of locations / vertices and @xmath6 a distance function that is symmetric and satisfies triangle inequality . there is a depot vertex @xmath7 that contains an infinite supply of an identical item , and each vertex @xmath8 demands some units @xmath9 of this item . a single vehicle of capacity @xmath10 is used to distribute the items . the objective is to find a minimum length tour of the vehicle that satisfies all demands subject to the constraint that the vehicle carries at most @xmath11 units at any time . @xmath3is closely related to the traveling salesman problem ( @xmath12 ) . it is clear that @xmath3reduces to @xmath12 in the absence of capacity constraint . more interestingly , a reverse relation is also known essentially the best known approximation algorithm for @xmath3 @xcite achieves a guarantee of @xmath13 , where @xmath14 is the best approximation ratio for @xmath12 . in practice , it is natural to have a fleet of _ multiple _ vehicles that can run in parallel . the objective can then be to either minimize the sum of completion times of all the vehicles or to minimize the maximum completion time over all vehicles ( or the makespan of the routing ) . furthermore the vehicles can all be identical ( same speed ) or heterogeneous ( have different speeds ) . in either case , it is not hard to see that the total completion time objective reduces to the usual @xmath3on a single maximum - speed vehicle , and constant - factor approximation algorithms readily follow . when the objective is to minimize the makespan with identical vehicles , ideas for approximating the regular @xmath3problem using a tour - splitting heuristic introduced by frederickson et al . @xcite can be easily adapted to derive a constant - factor approximation algorithm ( see below ) . this motivates the _ heterogenous capacitated vehicle routing problem _ ( @xmath15 ) that we consider . in this problem , a fleet of @xmath1 vehicles with non - uniform speeds _ and uniform capacities is initially located at the depot vertex @xmath0 . the objective is to satisfy the demands subject to the capacity constraints while minimizing the makespan . our main result is a constant - factor approximation algorithm for @xmath15 . most of our algorithmic ideas lie in solving the special case of @xmath15when there is no capacity constraint . this problem , which we call @xmath16 , is a generalization of @xmath12that might be of independent interest . for most of this paper , we will focus on obtaining a constant - factor approximation for @xmath16 . * tour - splitting solutions : * to illustrate the use of known techniques , we outline how to obtain a constant - factor approximation algorithm for @xmath16with uniform speeds @xcite . first , notice that the union of the tours of @xmath17 connects all vertices , and hence a minimum spanning tree has length at most @xmath18 . then consider an @xmath19 , duplicate its edge and take an euler tour @xmath20 , which is of length @xmath21 . now split @xmath20 into @xmath1 segments of lengths at most @xmath22 by removing edges . finally , the tour for the @xmath23 vehicle is obtained by connecting both endpoints of the @xmath23 segment of @xmath20 to the depot . since twice the distance from the depot to any vertex is a lower bound on @xmath17 , the length of each tour is at most @xmath24 and hence this solution is a 3-approximation . we remark that this can be extended to obtain an @xmath25-approximation for @xmath15with uniform speeds ( e.g. , using theorem [ th : vrptotsp ] in section [ sec : hvrp ] ) . at a very high level , this strategy has two main components : ( 1 ) partitioning an @xmath19into manageable - sized connected parts ; ( 2 ) assigning these parts to vehicles . this simple idea which was already present in the 70s is the central piece of many heuristics and approximations for vehicle routing problems ( e.g. , @xcite ) . however , it is not clear how to employ this technique in the presence of vehicles with multiple speeds . this is because the two main components now need some correlation : a small part of the @xmath19 , which should be assigned to a slower vehicle , must also be relatively closer to the depot in order to be reachable by this vehicle . * set - cover based solutions : * for @xmath16with non - uniform speeds , previous approaches seem to give only a logarithmic approximation , as follows . guess the optimal makespan @xmath26(within a constant factor ) . if each vehicle of speed @xmath27 is given a length budget of @xmath28 , then the vehicles can collectively cover all vertices . using an approximation algorithm for @xmath1-@xmath19 @xcite ( or the related orienteering problem @xcite ) within a maximum - coverage framework ( see e.g .. @xcite ) , we can obtain tours of length @xmath26that cover a constant fraction of all vertices . repeating this coverage step until all vertices are covered gives a solution to @xmath16of makespan @xmath29 . the intrinsic problem of this approach is that it is too general in fact , the above algorithm also yields a logarithmic approximation even in the setting where the metric faced by each vehicle is arbitrary ( instead of just scaling by its speed ) , and this generalization of @xmath16can be shown to be set - cover hard . it is unclear whether the approximation of this set - covering based approach can be improved for @xmath16 . we extend the first tour - splitting approach described above to obtain the following result . there are constant - factor approximation algorithms for @xmath16and @xmath15 . in order to obtain the approximation for @xmath16 , we abstract the requirements of the two components in the tour - splitting strategy . as a preprocessing step , we round the speeds of vehicles to powers of two and guess the optimum makespan @xmath30 . first , we specify conditions which guarantee that a collection of @xmath0-rooted trees is `` assignable '' , that is , each vehicle can visit the nodes of the trees assigned to it within time @xmath31 ( definition [ def : assignable ] ) . the conditions in definition [ def : assignable ] are based on the lp to obtain a 2-approximation for _ scheduling in unrelated parallel machines _ by lenstra et al . @xcite . secondly , instead of partitioning an @xmath19as in the previous section , we consider more structured spanning trees which we call level - primtrees . consider grouping the vertices ordered according to their distance from @xmath0 into levels , where the @xmath32th level includes all vertices within distance @xmath33 . can only be served by vehicles of speed @xmath34 or higher given the makespan bound @xmath30 . ] the level - primtree is simply the tree resulting from running prim s algorithm with the restriction that all nodes in a level are spanned before starting to pull in nodes from the next . a level - primtree has two important properties : ( i ) the vertices along every root - leaf path are monotonically nondecreasing in level and ( ii ) for every suffix of levels , the subgraph induced on it costs at most @xmath25 times its induced @xmath19 . the first condition , which is the departing point from @xmath19s , greatly simplifies the decomposition procedure carried in the next step . the second property is related to the assignability conditions in definition [ def : assignable ] and guarantees that the we can decompose a level - primtree into an assignable collection . these properties are formalized in theorem [ thm : lprim ] . the level - primconstruction combine both @xmath19and shortest - path distances from a root , so it is not surprising that this structure is related to _ light approximate shortest - path trees _ ( last ) introduced by khuller et al . indeed , we use the existence of a suitably defined lastin proving theorem [ thm : lprim ] . we remark , however , that the properties guaranteed by lasts are not enough for our purposes ( see section [ sec : levelprim ] ) . the third main component of our approximation for @xmath16is decomposing level - priminto an assignable collection of @xmath0-rooted trees . roughly , we partition the edges of level - priminto subtrees while ensuring that each subtree consisting of vertices in levels up to @xmath32 ( and hence is at a distance of about @xmath33 from the root ) also has length approximately @xmath33 , and thus can be assigned to a vehicle of speed about @xmath34 . this partition , which relies on the two properties of level - prim , gives a collection of _ unrooted _ trees which is assignable . due to the length of these trees , the extra distance to connect them to the root @xmath0 can be charged to their edges , hence this collection can be turned into a @xmath0-rooted assignable collection . in order to obtain an approximation to @xmath15 , we reduce this problem to approximating @xmath16 in a suitably modified metric space . this new distance function encodes any additional trips to and from the root that a vehicle has to make if it runs out of capacity . the exact transformation is presented in section [ sec : hvrp ] . for the @xmath3 , the best known approximation ratio @xcite is essentially @xmath13 where @xmath14 is the best guarantee for @xmath12 . the current best values for @xmath14 are @xmath35 for general metrics @xcite , and @xmath36 ( for any constant @xmath37 ) for constant dimensional euclidean metrics @xcite . this has been improved slightly to @xmath38 when @xmath39 @xcite . recently , das and mathieu @xcite gave a quasi - polynomial time approximation scheme for @xmath3on the euclidean plane . several variants of @xmath12have been studied , most of which have a min - sum objective . one related problem with min - max objective is _ nurse station location _ @xcite , where the goal is to obtain a collection of trees ( each rooted at a distinct depot ) such that all vertices are covered and the maximum tree length is minimized . even et al . @xcite gave a 4-approximation algorithm for this problem . this is based on partitioning the @xmath19and assigning to trees along the lines of section [ sec : previoustech ] ; their second step , however , involves a non - trivial bipartite matching subproblem . in proving the properties of level - prim , we use _ light approximate shortest - path trees _ introduced by khuller , raghavachari and young @xcite , building on the work on shallow - light trees of awerbuch , baratz and peleg @xcite . an @xmath40-lastis a rooted tree that has ( a ) length at most @xmath41 times the @xmath19and ( b ) the distance from any vertex to the root ( along the tree ) is at most @xmath42 times the distance in the original metric . khuller et al . @xcite showed that every metric has an @xmath43-last(for any @xmath44 ) and this is best possible . one phase of our algorithm uses some ideas from scheduling on parallel machines @xcite , which also has a min - max objective . in this problem , job @xmath45 has processing time @xmath46 on machine @xmath32 and the goal is to assign jobs to machines while minimizing the maximum completion time . lenstra et al . @xcite gave an lp - based 2-approximation algorithm for this problem . the input to the _ heterogenous tsp _ ( @xmath16 ) consists of a metric @xmath4 denoting distances between vertices , a depot @xmath7 and @xmath1 vehicles with speeds @xmath2 greater than or equal to 1 . the goal is to find tours @xmath47 ( starting and ending at @xmath0 ) for each vehicle so that every vertex is covered in some tour and which minimize the _ maximum completion time _ @xmath48 . at the loss of a factor of two in the approximation , we assume that the @xmath49 s are all ( non - negative integral ) powers of @xmath50 . then , for each integer @xmath51 we use @xmath52 to denote the number of vehicles with speed @xmath34 . we let @xmath26denote the optimal value of this modified instance of @xmath16 . we let @xmath53 be the complete graph on vertices @xmath5 with edge - weights corresponding to the distance function @xmath54 . for any set @xmath55 of edges , we set @xmath56 . given any ( multi)graph @xmath57 and a subset @xmath58 of its vertices , @xmath59 $ ] denotes the subgraph induced on @xmath58 and @xmath60 denotes the graph obtained by contracting vertices @xmath58 to a single vertex ( we retain parallel edges ) . moreover , for any pair of vertices @xmath61 in @xmath57 , we use @xmath62 to denote the length of the shortest path in @xmath57 between @xmath63 and @xmath64 . assume that we have correctly guessed a value @xmath30 such that @xmath65 . ( this value can be found via binary search and we address this in the end of section [ sec : decomp ] . ) we partition the set of vertices @xmath5 according to their distance to @xmath0 : @xmath66 @xmath67\ } , \mbox { for all } i\ge 0.\ ] ] the vertices in @xmath68 are referred to as _ level @xmath32 _ vertices . for any @xmath51 , we use @xmath69 as a shorthand for @xmath70 and similarly @xmath71 . we also define the _ level of an edge _ @xmath72 as the larger of the levels of @xmath63 and @xmath64 . for each @xmath51 , @xmath73 denotes the edges in @xmath74 of level @xmath32 . note that @xmath75 for all @xmath76 , since both end - points of @xmath77 are in @xmath69 and the triangle inequality bounds its distance by the two - hop path via the root . we use the notation @xmath78 and @xmath79 . we start by studying collections of trees that can be assigned to vehicles in a way that each vehicle takes time @xmath31 to visit all of its assigned trees . [ def : assignable ] a collection of @xmath0-rooted trees @xmath80 covering all vertices @xmath5 is called @xmath40-_assignable _ if it satisfies the following properties . 1 . for each @xmath51 and every @xmath81 , @xmath82 . 2 . for each @xmath51 , @xmath83 . intuitively , the trees in @xmath84 can be assigned to vehicles with speed @xmath34 so as to complete in time @xmath85 . condition ( 2 ) guarantees that the trees @xmath86 targeted by vehicles of speed @xmath34 and above stand a chance of being handled by them within makespan @xmath87 . interestingly , these minimal conditions are enough to eventually assign all trees in collection to vehicles while guaranteeing makespan @xmath88 . [ lem : assignable ] given an assignable collection @xmath80 of @xmath0-rooted trees , we can obtain in polynomial time an @xmath89-approximation for @xmath16 . to prove this lemma is also possible , but the route we take reveals more properties of the requirement at hand and could potentially be useful in tackling generalizations of @xmath16 . ] , we show that condition ( 2 ) guarantees the existence of a fractional assignment of trees where each vehicle incurs load at most @xmath90 . then using condition ( 1 ) and a result on scheduling on parallel machines @xcite , we round this assignment into an integral one while increasing the load on each vehicle by at most @xmath91 . we loose an extra factor of 2 to convert the trees into routes . [ [ fractional - assignment . ] ] fractional assignment . + + + + + + + + + + + + + + + + + + + + + + consider the bipartite graph @xmath57 whose left side contains one node for each tree in @xmath92 and whose right side contains one node for each vehicle . ( we identify the nodes with their respective trees / vehicles . ) there is an arc between the tree @xmath81 and a vehicle of speed @xmath93 if @xmath94 . consider the following @xmath95-matching problem in @xmath57 : for each tree @xmath81 , we set @xmath96 and for each vehicle @xmath63 of speed @xmath93 we set @xmath97 . a ( left - saturating ) @xmath95-matching is one which fractionally assigns all @xmath98 units of each tree @xmath99 such that no vehicle @xmath63 is assigned more than @xmath100 units . notice that a feasible @xmath95-matching gives a fractional assignment of trees where each vehicle incurs load at most @xmath90 . then our goal is to show the existence of a @xmath95-matching in @xmath57 . using a standard generalization of hall s theorem ( e.g. , see page 54 of @xcite ) , we see that @xmath57 has a feasible @xmath95-matching iff for every set @xmath5 of trees , @xmath101 is at most @xmath102 , where @xmath103 is the neighborhood of @xmath5 . however , the structure of @xmath57 allows us to focus only on sets @xmath5 which are equal to @xmath104 for some @xmath32 . contains a tree in @xmath84 then @xmath103 already contains all vehicles of speed @xmath93 for @xmath94 . then adding to @xmath5 extra trees in @xmath104 does not change its neighborhood and thus leads to a dominating inequality . ] using this revised condition , @xmath57 has a @xmath95-matching iff for all @xmath32 , @xmath105 . since this is exactly condition ( 2 ) in definition [ def : assignable ] , it follows that @xmath57 indeed has a @xmath95-matching ( which can be obtained in polynomial time using any maximum flow algorithm @xcite ) . [ [ scheduling - parallel - machines . ] ] scheduling parallel machines . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we show how to round the fractional assignment obtained in the previous section . we consider each tree as a `` job '' and each vehicle as a machine " , where the `` processing time '' @xmath106 of a tree @xmath99 in a vehicle @xmath63 of speed @xmath93 is @xmath107 ; then the `` makespan '' of a vehicle is exactly equal to the sum of the processing times of the trees assigned to it . let @xmath108 denote the fraction of tree @xmath99 assigned to vehicle @xmath63 given by scaling down a @xmath95-matching in @xmath57 ( i.e. , if the matching assigns @xmath54 units of @xmath99 to vehicle @xmath63 , we have @xmath109 ) . the feasibility of the matching gives @xmath110 for all @xmath63 . moreover , by construction of the edges of @xmath57 , @xmath111 for @xmath81 implies that @xmath63 has speed at least @xmath112 . then using property ( 1 ) of assignable trees we get that @xmath111 implies @xmath113 . these two properties guarantee that @xmath114 is a feasible solution for the natural lp formulation for the scheduling problem with a feasible makespan value of @xmath90 and the maximum processing time @xmath115 set to @xmath116 . theorem 1 of @xcite then asserts that @xmath114 can be rounded into an integral assignment of trees to vehicles such that the load on any vehicle is at most @xmath117 . as in section [ sec : previoustech ] , we can transform each tree in @xmath118 into a cycle while at most doubling its length , which then gives a @xmath119 approximation for @xmath16 . this concludes the proof of lemma [ lem : assignable ] . in order to obtain an assignable collection of @xmath0-rooted trees for our instance , we formally introduce level - primtrees . these are the trees obtained by the the following procedure . [ alg : stagei ] for each @xmath120 , let @xmath121 be an @xmath19for @xmath122 / v _ { < i}$ ] . @xmath123 . note that level - primtrees can alternately be defined by modifying prim s algorithm such that nodes in level @xmath32 are only considered to be added to the tree after all nodes in levels below @xmath32 have already been added . [ thm : lprim ] a level - primtree @xmath124 satisfies the following : the vertex - levels along every root - leaf path are non - decreasing . for each @xmath51 , @xmath125 . note that the second property in theorem [ thm : lprim ] mirrors the second property in definition [ def : assignable ] . a formal connection between the two is established via the following lemma that uses an optimal vehicle routing solution to derive a feasible spanning tree connecting a suffix of the level sets . [ lem : lb ] for each level @xmath126 , @xmath127 . consider an optimal solution for @xmath16and let @xmath128 be the set of edges traversed by vehicles in this solution ; label each edge in @xmath128 by the vehicle that traversed it . clearly @xmath128 connects all vertices to the root @xmath0 . observe that only vehicles having speed at least @xmath129 can even reach any vertex in @xmath130 ( since a vehicle of speed @xmath27 travels distance at most @xmath131 ) . thus every edge in @xmath132 must be labeled by some vehicle of speed at least @xmath129 . this implies that @xmath133 , since the right hand side is a bound on the total length traversed by vehicles having speed at least @xmath129 . on the other hand , since @xmath128 connects all vertices , @xmath132 contains a spanning tree of @xmath134 . thus we have @xmath135 . we then get the following corollary of theorem [ thm : lprim ] . [ cor : lprim ] a level - primtree @xmath124 satisfies the following : the vertex - levels along every root - leaf path are non - decreasing . for each @xmath51 , @xmath136 . in the rest of this section , we prove theorem [ thm : lprim ] . it is easy to see that for every @xmath137 , @xmath138 spans @xmath139 $ ] , hence the procedure produces a spanning tree for @xmath140 . moreover , by construction we obtain that every root - leaf path in @xmath141 traverses the levels in non - decreasing order as desired . thus , we focus on proving the second property in the theorem . instead of comparing the length of the edges in @xmath141 with an @xmath142 , it turns out to be much easier to use a specific lasttree as proxy for the latter . the following lastis implicit in the construction given in @xcite ; for completeness we outline a proof in appendix [ app : spider ] . recall that a _ spider _ is a tree with at most one vertex ( the center ) having degree greater than two . [ thm : last ] given any metric @xmath4 with root @xmath0 , there exists a spanning spider @xmath143 with center @xmath0 such that : for each @xmath8 , the distance from @xmath0 to @xmath63 in @xmath143 is at most @xmath144 . the length of @xmath143 is at most four times the @xmath19 in @xmath4 , i.e. @xmath145 . we remark that we can not use a lastdirectly instead of level - primsince the former does not need to have the properties asserted by theorem [ thm : lprim ] ; it is easy to find a lastwhich does not satisfy the first property , while figure [ fig : last ] also shows that the second can also be violated by an arbitrary amount . using these spider lasts we can obtain the main lemma needed to complete the proof of theorem [ thm : lprim ] . [ lem : lprim - length ] for any graph @xmath140 and any level - primtree @xmath141 on @xmath140 , we have @xmath146 . consider a spider last@xmath143 for @xmath140 and let @xmath147denote the set of all root - leaf paths in @xmath143 ; note that @xmath148 is edge - disjoint . consider any root - leaf path @xmath149 in @xmath147 . we claim that @xmath150 crosses levels almost in an increasing order . specifically , there does not exist a pair of nodes @xmath151 with @xmath152 , @xmath153 and @xmath154 . suppose ( for a contradiction ) that this were the case ; then we would have that @xmath155 where the last inequality uses @xmath156 . on the other hand , @xmath157 since @xmath154 ; so we obtain @xmath158 , which contradicts the definition of @xmath143 ( see theorem [ thm : last ] ) . now we transform @xmath143 into another spider @xmath159 which traverses levels in non - decreasing order as follows . for each root - leaf path @xmath160 , perform the following modification . let @xmath161 be the subsequence of @xmath150 consisting of the vertices in _ even numbered _ levels , i.e. each @xmath162 for some @xmath163 . similarly , let @xmath164 be the subsequence of @xmath150 consisting of the vertices in _ odd numbered _ levels . define two paths @xmath165 ( shortcutting @xmath150 over nodes @xmath166 s ) and @xmath167 ( shortcutting @xmath150 over @xmath168 s ) . observe that both @xmath169 and @xmath170 cross levels _ monotonically _ : if not then there must be some @xmath152 in @xmath150 with @xmath153 and @xmath171 , contrary to the previous claim . also , by employing the triangle inequality we have that @xmath172 . finally define the spider @xmath159 as the union of the paths @xmath173 over all root - leaf paths @xmath174 . by construction , the vertex levels along each root - leaf path of @xmath159 are non - decreasing . additionally @xmath175 , by theorem [ thm : last ] . now partition the edges of @xmath159 as : @xmath176 & \mbox { if } \ell=0,\\ l'[v_{\le \ell}]\setminus l'[v_{\le \ell-1 } ] & \mbox { if } \ell\ge 1 . \end{array } \right.\ ] ] by the monotone property of paths in @xmath159 , it follows that @xmath177 $ ] is connected for every @xmath126 . thus @xmath178 is a _ spanning tree _ in graph @xmath139/v_{<\ell}$ ] . since @xmath179 in the level - primconstruction , is chosen to be an @xmath19 in @xmath139/v_{<\ell}$ ] , we obtain @xmath180 . so , @xmath181 . this completes the proof of the lemma . [ [ completing - proof - of - theoremthmlprim . ] ] completing proof of theorem [ thm : lprim ] . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we now prove the second property in theorem [ thm : lprim ] . lemma [ lem : lprim - length ] directly implies this property for @xmath182 . for any level @xmath183 consider the graph @xmath184 ; observe that @xmath185 is a level - primtree for @xmath186 ( due to the iterative construction of @xmath187 ) . thus applying lemma [ lem : lprim - length ] to graph @xmath186 and its level - prim@xmath185 , we have @xmath188 . in this section we decompose a level - primtree into an assignable collection @xmath118 of @xmath0-rooted trees . motivated by corollary [ cor : lprim ] , the idea is to essentially break each subgraph @xmath121 into many pieces and connect them to @xmath0 in order to form the set of trees @xmath84 . more specifically , assume for now that each connected component in @xmath121 is large enough , i.e. has length at least @xmath33 . then for each @xmath120 , break the connected components of @xmath121 into trees of length approximately @xmath33 ; add to each tree the shortest edge connecting them to @xmath0 and set @xmath84 as the collection of @xmath0-rooted trees obtained . by construction we get that @xmath118 satisfies the first property of an assignable collection . moreover , notice that each edge added to connect a tree to the root has approximately the same length as the tree itself ; this guarantees that @xmath189 . it then follows that the collection @xmath118 is assignable . notice that it was crucial to break @xmath121 into trees of length at least approximately @xmath34 . but this is problematic when @xmath121 has a small connected component . in this case we show that such a small component is always attached to ( `` dangling '' from ) a large enough component in @xmath190 ( otherwise the dangling edge to a much earlier level will already make this component heavy enough not to be small ) ; then we simply treat the small component as an integral part of the latter . now we formally describe the proposed decomposition procedure . * step 1 . * let @xmath191 contain the subtree @xmath192 . for each level @xmath193 : partition edges @xmath194 into a collection @xmath195 of ( unrooted ) subtrees such that each subtree contains exactly one edge from @xmath196 to @xmath68 . for any @xmath197 call the unique edge from @xmath196 to @xmath68 its _ head - edge _ @xmath198 . note that such a partition is indeed possible since @xmath199/v_{<i}$ ] is connected . any subtree in @xmath195 ( for @xmath51 ) is referred to as a level @xmath32 subtree . note that head - edges are defined only for subtrees in level @xmath200 and above . * step 2 . * for each level @xmath51 : _ mark _ those @xmath197 that have @xmath201 . in addition , _ mark _ the tree @xmath202 in @xmath191 . let @xmath203 and @xmath204 denote the marked and unmarked subtrees in @xmath195 . * for each level @xmath193 and unmarked @xmath205 : define @xmath206 as the subtree in @xmath207 containing the other end - point of @xmath208 . [ cl : decomp1 ] for @xmath193 and unmarked @xmath205 , @xmath209 . moreover , @xmath206 is marked . since @xmath210 is unmarked in level @xmath193 , @xmath211 . so the end - point @xmath64 of @xmath208 in @xmath206 satisfies @xmath212 , otherwise @xmath213 . in particular @xmath214 and thus @xmath215 . for the second part of the claim , notice that if @xmath216 then @xmath217 , which is always marked . so suppose @xmath218 . from the above , @xmath206 is in level @xmath219 and hence contains a head - edge . this implies that @xmath206 contains some vertex @xmath220 , namely an end - point of @xmath221 . but then @xmath222 , where we used @xmath223 since @xmath220 and @xmath224 from above . thus @xmath206 must be marked . * for each level @xmath51 and marked @xmath225 : define @xmath226 as the set of all unmarked @xmath227 having @xmath228 . clearly @xmath229 for all @xmath230 . * for each level @xmath51 and marked @xmath225 : partition the tree @xmath231 into subtrees @xmath232 such that the first @xmath233 trees have length in the range @xmath234 $ ] and @xmath235 has length at most @xmath236 . notice that this is possible since all edges of @xmath231 belong to @xmath237 and hence have length at most @xmath238 . finally , add the shortest edge from @xmath0 to each of these new subtrees to obtain a collection @xmath239 of @xmath0-rooted trees . [ cl : decomp2 ] for any @xmath240 , we have @xmath241 . notice that every @xmath242 consists of a @xmath243 ( for some @xmath244 ) and an edge from @xmath0 to a node in @xmath245 . since the former has length at most @xmath246 and the latter has length at most @xmath247 , it follows that @xmath248 . @xmath249 $ ] . [ cl : decomp3 ] we break the analysis into two cases depending of @xmath250 . suppose @xmath251 , namely @xmath239 consists of a single tree @xmath99 . in this case @xmath252 , where @xmath77 is an edge to @xmath0 . if @xmath253 then @xmath254 and the result holds directly . if @xmath183 then @xmath255 has a node in @xmath256 and hence @xmath257 . because @xmath255 is marked and different than @xmath202 , the lower bound on its length implies that @xmath258 . the result follows by adding the length of @xmath231 to both sides . now suppose @xmath259 . since all trees in @xmath239 lie in @xmath260 , each edge from the root in @xmath239 has length at most @xmath261 . so the left hand side is at most @xmath262 . but for @xmath263 we have @xmath264 , so the last term of the previous expression can be upper bounded by @xmath265 . this bound is smallest when @xmath266 , which then gives @xmath267 . this concludes the proof of the claim . * for each level @xmath51 : define @xmath268 . the following lemma summarizes the main property of our decomposition procedure . [ lem : isassignable ] the collection @xmath269 obtained from the above procedure is @xmath270-assignable . by claim [ cl : decomp2 ] , each tree in @xmath84 has length at most @xmath271 . so the collection satisfies condition ( 1 ) of definition [ def : assignable ] . fix any @xmath51 for condition ( 2 ) in definition [ def : assignable ] . due to corollary [ cor : lprim ] , it suffices to prove that @xmath272 . using claim [ cl : decomp3 ] we obtain that @xmath273 = 5\cdot d(\subt^m_j ) + 5\cdot d(\subt^u_{j+1}).\ ] ] the last equality above uses the fact that that @xmath274 is a partition of @xmath275 . thus : @xmath276 this concludes the proof of lemma [ lem : isassignable ] . [ [ summary - of - the - algorithm . ] ] summary of the algorithm . + + + + + + + + + + + + + + + + + + + + + + + + + our algorithm starts with an initial low guess of @xmath30 and runs the level - primprocedure . if the second condition in corollary [ cor : lprim ] does not hold for this run , we double the guess for @xmath30 and repeat until it is satisfied ( this happens the first time that @xmath30 reaches the condition for the correct guess : @xmath65 ) . we use the decomposition in this section summarized in lemma [ lem : isassignable ] to obtain a ( 6,40)-assignable collection of trees . using lemma [ lem : assignable ] on this collection gives us the desired constant approximation ratio by observing that the guess @xmath30 in this step obeys @xmath277 . the input to the _ heterogenous cvrp _ ( @xmath15 ) consists of a metric @xmath4 denoting distances between vertices , depot @xmath7 ( containing an infinite supply of items ) , demands @xmath278 and @xmath1 vehicles with speeds @xmath2 , each having capacity @xmath11 . a solution to @xmath15consists of tours @xmath279 ( starting and ending at @xmath0 ) for each vehicle so that all demands are satisfied and each vehicle carries at most @xmath11 items at any point in time . the objective is to minimize the _ maximum completion time _ , we study the `` split - delivery '' version of @xmath3here , where demand at a vertex may be served by multiple visits to it ; however , our result easily extends to the `` unsplit - delivery '' @xmath15 . we show that the @xmath15problem can be reduced to @xmath16 in an approximation preserving way ; so we also obtain an @xmath25-approximation for @xmath15 . the idea in this reduction is to modify the input metric based on lower - bounds for @xmath3 @xcite . in order to avoid ambiguity , we use @xmath281 to denote the optimum for @xmath282 and @xmath283 to denote the optimum for @xmath284 . [ th : vrptotsp ] consider an instance @xmath285of @xmath15 . there is a poly - time constructible instance @xmath286of @xmath16such that @xmath287 . moreover , a solution to @xmath286of makespan @xmath30 can be converted in poly - time to a solution to @xmath285with makespan @xmath31 . let @xmath285be an instance of @xmath15as specified above . standard scaling arguments can be used to ensure that @xmath11 is polynomial in @xmath288 and @xmath289 for all @xmath290 ( details in the full version ) . let @xmath53 denote the complete graph on vertices @xmath5 with edge - weights equal to distances @xmath54 . augment @xmath140 to a new graph @xmath57 by adding vertices @xmath291\}$ ] , and edges @xmath292\}$ ] ; each edge @xmath293 has weight @xmath294 . for any vertex @xmath290 , the vertices @xmath295\}$ ] are referred to as copies of @xmath64 . let @xmath296 denote the metric induced on vertices @xmath297 where @xmath137 denotes the shortest - path distances in graph @xmath57 . we let @xmath286be the instance of @xmath16on metric @xmath296 with depot @xmath0 and @xmath1 vehicles having speeds @xmath2 . since @xmath298 this reduction runs in polynomial time . for any graph @xmath143 and subset @xmath299 of vertices , let @xmath300 denote the minimum length steiner tree connecting @xmath299 . for any subset @xmath301 and @xmath290 let @xmath302 denote the number of @xmath64-copies in @xmath99 ; also define @xmath303 . observe that for any @xmath301 we have @xmath304 by the definition of graph @xmath57 . we first show that @xmath305 . consider an optimal solution @xmath279 to @xmath285 . for each @xmath306 $ ] , let @xmath307 $ ] denote the units of demand at vertex @xmath290 served by vehicle @xmath32 , and let @xmath308 . note that @xmath309 for all @xmath290 ; hence we can choose @xmath310 for each @xmath311 $ ] such that @xmath312 and @xmath313 for all @xmath314 $ ] . since @xmath315 is a capacitated tour in @xmath140 serving demands @xmath316 , we have @xmath317 using the ( connectivity and capacitated routing ) lower - bounds for @xmath3 @xcite . thus @xmath318 . now consider the solution to @xmath286where the @xmath23 vehicle visits vertices @xmath319 along the minimum tsp tour on @xmath320 , for all @xmath311 $ ] ; the distance traversed by the @xmath23 vehicle is at most @xmath321 . so the @xmath16objective value of this solution is at most @xmath322 } \frac{4\cdot d(\sigma_i)}{\lambda_i}=4\cdot { \ensuremath{{\sf opt_{vrp}}}\xspace}({\ensuremath{\mathcal{i}}\xspace})$ ] . now consider a solution @xmath47 to @xmath286with makespan @xmath30 . let @xmath323 denote the vertices that are served by each vehicle @xmath306 $ ] . since @xmath324 is a tsp tour on @xmath325 , we have @xmath326 . now fix @xmath327 $ ] and consider the instance of @xmath3on vertices @xmath328 with demands @xmath329 . as mentioned in the previous paragraph , @xmath330 is a lower - bound for this instance , and the algorithm from @xcite returns a solution @xmath315 within a @xmath331 factor of this lower - bound . it readily follows that @xmath332 is a feasible solution to @xmath285with makespan at most @xmath322 } \frac{\rho\cdot d(\tau_i)}{\lambda_i}= o(m)$ ] . we note that this algorithm returns a _ non - preemptive _ @xmath15solution , i.e. , each item once picked up at the depot stays in its vehicle until delivered to its destination . moreover , the lower - bounds used by the @xmath15algorithm also hold for the ( less restrictive ) _ preemptive _ version , where items might be left temporarily at different vertices while being moved from the depot to their final destination . thus our algorithm also bounds the `` preemption gap '' ( ratio of optimal non - preemptive to preemptive solutions ) in @xmath15by a constant . one interesting open question regards the approximability of @xmath284 and @xmath282 when vehicles are located in multiple different depots across the space . the current definition of an assignable collection and the definition of level - primcrucially depend on the assumption of a unique depot , hence an extension to the multi - depot case is likely to require new ideas . another interesting direction is to consider @xmath15with non - uniform capacities , where the ideas presented in section [ sec : hvrp ] do not seem to generalize directly . we will show that the following algorithm produces an @xmath333-spider for @xmath140 . consider an @xmath19for @xmath140 and traverse it in preoder to obtain a path @xmath334 such that @xmath335 . [ alg : shortcutif ] @xmath336 to @xmath299 and * mark * @xmath337 . [ alg : shortcut ] marked node @xmath337 , remove @xmath338 from @xmath299 . [ alg : delete ] @xmath339 . since the algorithm keeps adding edges to the path @xmath341 , it is clear that before step [ alg : delete ] only the root @xmath342 and marked nodes have degree larger than 2 . moreover , each marked node has degree exactly 3 . thus , after step [ alg : delete ] we have that only the root has degree larger then 2 , and the lemma follows . first we prove that @xmath343 for all @xmath32 . to see this , consider @xmath299 right before step [ alg : delete ] . it follows from step [ alg : shortcut ] that @xmath344 for all @xmath32 . noticing that @xmath340 is a shortest path tree of @xmath299 from node @xmath342 implies the desired result . now we prove that @xmath340 satisfies the second property of @xmath333-spider . define @xmath345 and let @xmath346 be the @xmath32th node marked by the algorithm . it is clear that @xmath347 ; so our goal is to upper bound the last summation . fix a node @xmath346 . consider the beginning of the iteration where @xmath346 is marked . notice that at this point @xmath348 , since edge @xmath349 was already added to @xmath299 ; since @xmath341 is subgraph of @xmath299 , it is also clear that the right hand side is at most @xmath350 . however , since @xmath346 was marked , we have that @xmath351 , and then using the previous bounds we obtain that @xmath352 . adding the previous inequality over all @xmath346 s we get that @xmath353 noticing that @xmath354 and reorganizing leads to @xmath355 . finally , notice that @xmath356 : this follows from traversing the path @xmath341 and using the triangle inequality . this gives the bound @xmath357 .
|
the _ capacitated vehicle routing problem _ ( cvrp ) @xcite involves distributing ( identical ) items from a depot to a set of demand locations , using a single capacitated vehicle .
we study a generalization of this problem to the setting of multiple vehicles having non - uniform speeds ( that we call _ heterogenous cvrp _ ) , and present a constant - factor approximation algorithm .
the technical heart of our result lies in achieving a constant approximation to the following tsp variant ( called _ heterogenous tsp _ ) .
given a metric denoting distances between vertices , a depot @xmath0 containing @xmath1 vehicles having respective speeds @xmath2 , the goal is to find a tour for each vehicle ( starting and ending at @xmath0 ) , so that every vertex is covered in some tour and the maximum completion time is minimized .
this problem is precisely heterogenous cvrp when vehicles are uncapacitated .
the presence of non - uniform speeds introduces difficulties for employing standard tour - splitting techniques . in order to get a better understanding of this technique in our context
, we appeal to ideas from the 2-approximation for scheduling in parallel machine of lenstra et al .
@xcite .
this motivates the introduction of a new approximate mst construction called _ level - prim _ , which is related to _
light approximate shortest - path trees _ @xcite .
the last component of our algorithm involves partitioning the level - prim tree and matching the resulting parts to vehicles .
this decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the vista variables in the va lctea ( vvv ) eso public survey @xcite is a time - series , near - infrared ( ir ) survey of the galactic bulge and an adjacent portion of the inner disk , covering 562 square degrees of the sky ( fig . [ fig : area ] ) . the survey has provided multi - color photometry in 5 broadband filters ( @xmath0 , @xmath1 , @xmath2 , @xmath3 , and @xmath4 ) , but its main goal is to provide , for the first time , a homogeneous database for a variability study of the observed regions in the @xmath4-band . vvv has much improved photometric precision compared with , and extends much deeper than , 2mass @xcite . in addition , in contrast to single - epoch surveys , which only allow the construction of 2-dimensional ( 2d ) maps , with the addition of temporal information for well - established distance indicators such as rr lyrae stars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the vvv survey will enable us to resolve the 3d structure not only of the milky way , but also of the sagittarius dwarf spheroidal galaxy ( sgr dsph ; * ? ? ? * ) , parts of which are also included in the vvv fields , and possibly even detect previously unknown galactic structures and streams . the survey is carried out at the visible and infrared survey telescope for astronomy ( vista ) , a 4m - class telescope operated by eso and located at cerro paranal , chile . the heart of the vista / vircam instrument is a @xmath6 array of raytheon virgo ir detectors ( @xmath7 pixels ) , with a pixel size of @xmath8 @xcite . the size of a uniformly covered field ( also called a `` tile '' ) is 1.501 deg@xmath9 , hence the vvv survey requires a total of 348 such `` tiles '' to cover the survey area and include a small overlap between neighboring tiles ( fig . [ fig : area ] ) . the data reduction is carried out at the cambridge astronomy survey unit ( casu ) in collaboration with the uk wide - field astronomy unit ( wfau ) in edinburgh . details about the data pipeline and the various steps of the calibrations can be found in @xcite , @xcite , and @xcite . briefly , vvv images are pipeline - processed by the vista data flow system ( vdfs ; @xcite ) , including all steps of data reduction from image processing to photometry and its calibration . individual exposures are subjected to standard steps of pre - processing , such as flat - fielding , dark subtraction , and non - linearity correction . science frames are composed of two dithered images , i.e. subsequent exposures taken with an offset of typically @xmath10 pixels , in order to remove detector artifacts , cosmic rays , and other cosmetic defects . the resulting detector frame stacks , a.k.a . `` pawprints , '' have non - contiguous areal coverage due to the large gaps between the 16 vircam chips . at each observational epoch , a sequence of six pawprints is acquired , and these are further combined to form a contiguous mosaic image , a.k.a . `` tile . '' further steps in the reduction consist of source extraction and aperture photometry . sets of small circular apertures with increasing radii are used in order to maximize the signal - to - noise ratio in highly time - varying seeing conditions , and to suppress systematics due to source crowding . furthermore , the pipeline includes point - spread function ( psf ) estimation and psf fits for each object . flux loss in the wings of the psfs is remedied by aperture corrections @xcite . sources are classified based on the shape of the psf ( see , e.g. , * ? ? ? source positions are astrometrized using 2mass @xcite stars as reference , with a median accuracy of @xmath11 mas , depending on magnitude @xcite . magnitudes are corrected for detector distortion , and in the case of the @xmath12 filters are zero - point ( zp ) calibrated on a frame - by - frame basis using local 2mass secondary standards . for the zp calibration of the @xmath0 and @xmath1 filters , the procedure described in @xcite is followed . the zp accuracy of @xmath12 magnitudes depends on the number of available non - saturated local standards ( i.e. , the sky conditions ) , and is usually within @xmath13 , while the @xmath14 photometry is typically accurate to @xmath15 mag . we note that all vvv photometric data are on the vista magnitude system . the vvv survey started its data - gathering phase in february 2010 , with the first semester focusing on complete multi - band coverage of the whole survey area ( 562 square degrees ) in its 5 broadband filters ( see [ sec : over ] ) . this was completed in 2011 . the @xmath4-band observations that comprise the vvv variability study started in parallel to the multi - color observations in 2010 , and are still ongoing . once completed , the bulge region will contain between 60 and 100 epochs for each tile , whereas the disk tiles will have @xmath16 epochs each . the first two vvv public data releases have already become available from eso , and @xcite contains a detailed description of the first vvv data release . as of this writing , the various bulge tiles have been observed between 31 and 70 times , whereas the disk tiles have been observed 17 to 36 times ( see also fig . [ fig : epochs - delays ] ) . only observations which were carried out under the observing constraints as specified in the survey proposal are given the status `` completed . '' with respect to the submitted and scheduled observations , the vvv survey is @xmath17 complete . including observations scheduled for years 5 and 6 of the survey , this translates to an overall completeness of @xmath18 . since several vvv tiles have already been observed a few dozen times , a first look at stellar variability in the vvv data is already possible and this is precisely the main goal of this paper . the vvv survey has been monitoring the bulge and the southern disk in the @xmath4-band since 2010 . it will provide , for the first time , a homogeneous database with long - baseline time - series photometry with up to @xmath19 epochs for close to @xmath20 point sources . a brief overview of stellar variability in the vvv data was provided in @xcite . at its current status , after the extensive monitoring of the bulge fields has started , vvv has already provided a considerable number of epochs , suitable for analyses of stellar variability . vvv provides a sparse sampling of the time - domain , usually a single epoch for a few fields on a night ( with an occasional second visit ) , distributed close to randomly over the seasonal visibility period of the area . figure [ fig : epochs - delays ] shows the time sampling of bulge and disk fields . most of the currently available time - series data ( see [ sec : status ] ) were taken in the third year , and epochs for some tiles have a moderate clustering in the time sampling . the reason for the similarity in the sampling between different tiles is that observations are carried out in concatenations of nearby fields , in order to provide information on the near - ir atmospheric foreground emission , as is necessary in the data reduction process . in the following , we provide a brief overview of the basic properties of the @xmath4-band light curves based on the data of the bulge tile b293 ( @xmath21 , @xmath22 ) . this tile has 64 epochs with available vdfs photometry , and lies over a moderately crowded stellar field , allowing us to give the current best assessment of the quality of the time - series data . the limiting magnitude in the @xmath4-band varies between @xmath23 and @xmath24 mag , depending on the galactic latitude , due to differences in extinction and crowding @xcite , and also showing strong nightly variations , mainly due to the highly variable brightness of the near - ir sky , and partly because of the variable seeing or / and cloud coverage during observations . we note that these conditions also cause significant variations in the photometric properties of bright objects , due to the changing saturation level . figure [ fig : epochs ] shows the number of @xmath4 detections for vdfs 1.2 photometry as a function of the @xmath4-band brightness for objects fainter than @xmath25 mag in field b293 . the drop beyond @xmath26 mag is due to the non - detection of faint objects when observing conditions are sub - optimal . the overwhelming majority of stars brighter than @xmath27 mag are detected under almost all observing conditions . most of the bright objects with fewer than 60 detections are crowded and suffer from seeing - dependent merging with nearby sources , while some epochs for the faintest objects are missing mostly due to the high variation in the limiting magnitude caused by the variable near - ir atmospheric foreground . the secondary falling ridge starting at @xmath28 mag is due to the lower sensitivity of certain vircam chips . figure [ fig : sigma - ks ] shows the density distribution of vvv variable objects on the root - mean - square ( rms ) scatter average @xmath4 magnitude plane for sources on tile b293 with at least 20 epochs , after applying a general @xmath29 threshold - rejection procedure to the light curves . note that the @xmath4 limiting magnitude of this field is about 0.5 mag fainter than the visible limit of the distribution in figure [ fig : sigma - ks ] @xcite , but the majority of the faintest objects was detected only at a few epochs . figure [ fig : sigma - ks ] demonstrates well the high quality of the vdfs photometry note that the photometric zp is calibrated at each epoch , thus the scatter shown contains not only the formal photometric errors , but also the dispersion from the uncertainties of the nightly zps . the low noise at faint magnitudes ( @xmath30 mag at @xmath31 mag ) will allow us to detect rr lyrae stars even beyond the bulge . for bright stars ( @xmath32 mag ) , the photometric precision is around @xmath33 mag , which will allow us to study low - amplitude variables , including planetary transits around nearby k and m dwarfs @xcite , and to investigate the detailed near - ir light curve properties of pulsating and chromospherically active stars alike . there are various strategies for improving the photometric quality of the vvv light curves with respect to the vdfs catalogs ( see [ sec : var - over ] ) , by means of more sophisticated source extraction and photometric algorithms which may be better suited for dense stellar fields . the vvv science team is developing highly automated software pipelines which implement both of the two main approaches that currently deal with crowded - field photometry : psf - fitting photometry ( e.g. , daophot / allframe , @xcite ; dophot , @xcite ) and difference image analysis ( dia ; e.g. , @xcite ) . one of these pipelines , based on daophot , is described in @xcite . ultimately , some of these codes will be incorporated into the vdfs , providing a highly robust and versatile data analysis pipeline , equally well suited to handle all types of target fields . in addition to using the vdfs photometry catalogs from casu and daophot/ dophot pipelines ( see [ sec : techn ] ) , vvv is also developing its own dia pipeline . dia is particularly effective for obtaining photometry of faint objects in highly crowded fields . we employ a modified version of @xcite s optimal image subtraction package isis @xcite . the standard approach taken by isis is to convolve and photometrically match a reference image with good seeing to a target image of poorer seeing in order to obtain a difference image showing only flux changes between the two epochs . a light curve can then be obtained through photometry of a time series of difference images . the advantage of dia for variability studies is that the difference images contain only varying objects and are therefore unaffected by blended non - varying sources . the convolution kernel to transform the reference image psf to that of the target image is determined through a least - squares minimization using corresponding sub - regions of the two images . difference image sub - region . most of the image is gray , reflecting non - varying regions of the image . varying regions show up as white ( black ) for flux increases ( decreases ) . saturated stars typically show up as large blackened structures , sometimes with masked central regions as shown by the example within the rectangle . genuine variations have a more compact psf such as the two circled examples.,scaledwidth=80.0% ] whilst the standard approach is to convolve a good seeing reference image to a poorer seeing target image , this may not work well when the psf is undersampled on the reference image . ideally , good difference image quality requires typically at least 2.5 pixels / fwhm ( i.e. , @xmath34 seeing , at the average vircam pixel size ) , whereas the seeing at the vista site in cerro paranal is often below this the best - seeing images may have less than 2 pixels / fwhm . for this reason , we choose to use a relatively poor - seeing image with a well - sampled psf as the reference image , and then convolve a target image with superior seeing to it . for a reference image @xmath35 , a geometrically registered target image @xmath36 and convolution kernel @xmath37 , we minimize the sum @xmath38 ^ 2 \,\ , , \label{newdia}\ ] ] where the sum is over image pixels @xmath39 and @xmath40 represents a smooth differential background model . @xcite s isis package minimizes @xmath41 with respect to a set of linear coefficients to basis functions which represent both the convolution kernel @xmath37 and the differential background model @xmath40 . for @xmath37 the basis functions are a superposition of gaussian - like functions , and for @xmath40 the basis functions are 2d polynomials . figure [ fig : dia ] shows an example of a subtracted image that was obtained following this approach . whilst using a poor - seeing image for the reference may result in a difference image with reduced signal - to - noise ratio , this can be compensated by the fact that all difference images will have the same psf ( unlike @xcite s original method ) and one can iterate the procedure using the stack of difference images to produce a new , high signal - to - noise reference image . the vvv dia pipeline uses a modified version of isis which uses equation ( [ newdia ] ) in place of the standard approach . once a difference image sequence has been produced , variable objects are identified from a squared stack of difference images , and psf photometry is performed for all epochs at the locations of all variable objects ( defined above some signal - to - noise threshold ) . resulting raw light curves are written to files . we can search these files using specific criteria for e.g. periodic variability , or use some sutiable filter for specific transient objects such as microlenses . at present we are performing searches for regular periodic variables in order to calibrate the performance of the pipeline and , ultimately , catalog target populations such as rr lyrae stars . using dia we should be able to extend the sensitivity of traditional photometry offered by the vdfs catalogs and detect much fainter variables and transients , as well as variable sources that are blended and thus missing from the casu catalogs . dia provides only _ relative _ photometry for variable light curves it does not tie the photometry onto a calibrated magnitude scale . however , absolute calibration for a dia photometry time - series reduces to calibrating just one epoch as all other epochs are then calibrated through the dia relative photometry . we calibrate our dia photometry using vdfs aperture photometry catalogs for one of the best - seeing epochs in the time series . we refer to this epoch as our _ photometric reference _ , which is generally different to the reference image used for image subtraction discussed in as we have seen , the image subtraction reference image is typically one of poor seeing , in order to maximize sampling of the psf and ensure good image subtraction . for photometric calibration , however , we choose a good - seeing image , as this provides the best signal - to - noise photometry and minimizes blending effects . in order to tie in the dia photometry catalogs to the vdfs photometry catalogs we have to cross - match them . prior to cross - matching the catalogs are first filtered for unsaturated objects which are classified as `` stellar '' by the vdfs pipeline . the ra and dec coordinates for objects in the dia source lists are obtained using the world coordinate system information in the dia reference image . each object is then cross - matched to the vdfs object with the smallest angular separation from the dia position , within a maximum radius of @xmath42 , where @xmath43 is the dia reference image seeing . the factor of 2.5 was chosen by inspection of a histogram of first- and second - nearest matches for some of the most crowded fields , assuming the second - nearest matches are tracers of potential contaminants . figure [ fig : xmatchhist ] shows an example histogram of radial distances for first- and second - nearest vdfs catalog matches to dia objects within the four galactic center tiles . is given above the plot . the cross - match maximum radius is shown as a dashed line . the first- and second - nearest matches are shown as the blue and red histograms , respectively . the radial distances , @xmath44 , are given in units of @xmath45 , where @xmath43 is the dia reference image seeing . the fraction of potential contaminants ( second - nearest matches ) within the maximum radius , @xmath46 , is shown beneath the legend.,scaledwidth=80.0% ] the vdfs pipeline at casu performs crowded - field aperture photometry @xcite . for each object , flux counts are obtained for a series of aperture sizes , and used to build a curve - of - growth model . this model is used as an approximation to a psf profile to calculate the necessary corrections for the conversion of aperture flux to object magnitude . for each vvv dia object and cross - matched vdfs catalog object , it is necessary to select the largest aperture which contains the majority of the psf while minimizing the risk of contaminating flux from blended stars . the largest vdfs aperture useful for point - source measurements has a radius of 4 arcsec , which is large enough to guarantee encircling at least 99% of the total flux under all expected seeing conditions . however , within highly crowded fields such an aperture may be strongly affected by blending , and under typical seeing 99% of the flux will be encircled by much smaller apertures . within the range of 6 vdfs apertures up to this largest aperture , we select the largest aperture for which blending effects remain small and we perform this selection for each object individually . to this end we compute the forward difference magnitude between neighbor vdfs apertures ( which have curve - of - growth aperture corrections applied to them ) , @xmath47 , and we then determine the aperture @xmath48 , at which @xmath49 exceeds some limit . this is currently set to @xmath50 mag , based on an inspection of the aperture magnitude series of a few thousand objects in the galactic center tiles , but we are seeking a more robust way of determining this limit . having established the dia object magnitude at the photometry reference epoch , @xmath51 , we can then transform the difference flux measurements for all epochs onto a calibrated magnitude scale . the magnitude @xmath52 at each epoch @xmath39 is calibrated from the difference flux , @xmath53 , and the difference flux at the photometry reference epoch , @xmath54 . taking these together with the ( optimal ) aperture flux ( corrected to the full psf flux via the curve - of - growth model ) , @xmath55 , the magnitude zp , @xmath56 , and exposure time , @xmath57 , from the vdfs cross - matched object , we have : @xmath58 where @xmath59 @xmath60 and @xmath61 the random error on each individual epoch magnitude is simply the error on its difference magnitude @xmath62 . the systematic error in the baseline is obtained from the quadrature sum of the errors in the photometric reference epoch magnitude and difference magnitude : @xmath63 -band light curves for two example cepheids ( nos . 10 , _ top panel _ , and 26 , _ bottom panel _ ) in @xcite . the gray line is a smoothed radial basis function approximation to the vvv dia data ( black points ) , with the light gray region representing the maximal systematic offset error in the light curve baseline ( equation [ sys ] ) . the @xmath4 light curve of @xcite ( red points ) is also shown , offset - corrected to the vvv magnitude calibration for ease of comparison.,title="fig:",scaledwidth=75.0% ] -band light curves for two example cepheids ( nos . 10 , _ top panel _ , and 26 , _ bottom panel _ ) in @xcite . the gray line is a smoothed radial basis function approximation to the vvv dia data ( black points ) , with the light gray region representing the maximal systematic offset error in the light curve baseline ( equation [ sys ] ) . the @xmath4 light curve of @xcite ( red points ) is also shown , offset - corrected to the vvv magnitude calibration for ease of comparison.,title="fig:",scaledwidth=75.0% ] the errors @xmath64 , @xmath65 , and @xmath62 are all obtained from equations ( [ eq : photmag]-[eq : epochdmag ] ) through standard propagation of errors . figure [ fig : matsunaga ] shows calibrated vvv dia light curves for two cepheids found by @xcite , with their @xmath4-band photometry also shown for comparison . we typically find an offset between the calibrated vvv dia photometry and the @xcite calibrated photometry , though it is within our computed maximal systematic uncertainty ( determined from equation [ sys ] and shown by the light gray region ) . in figure [ fig : matsunaga ] we have subtracted off these offsets and we have folded the vvv photometry to the @xcite periods for direct comparison . the gray lines represent smooth radial basis function approximations to the vvv dia data ( black points ) . figure [ fig : matsunaga2 ] , in turn , compares vvv dia photometry with vdfs aperture photometry ( aperture number 3 ) for the same @xcite objects as in figure [ fig : matsunaga ] . vdfs aperture photometry also typically shows an offset ( subtracted off in figure [ fig : matsunaga2 ] ) within the systematic uncertainty of the dia , for reasons which we are still investigating . the random errors from dia compare favorably with those from aperture photometry . -band light curves for the same cepheids as in figure [ fig : matsunaga ] , compared with light curves obtained directly from vdfs aperture photometry ( aperture number 3 ; blue points ) and offset - corrected . the dia light curves are identical to those in figure [ fig : matsunaga ] . , title="fig:",scaledwidth=75.0% ] -band light curves for the same cepheids as in figure [ fig : matsunaga ] , compared with light curves obtained directly from vdfs aperture photometry ( aperture number 3 ; blue points ) and offset - corrected . the dia light curves are identical to those in figure [ fig : matsunaga ] . , title="fig:",scaledwidth=75.0% ] the vvv survey aims at reaching a highly complete census of variable stars that can be used as population tracers , such as radially pulsating stars and eclipsing binary systems . while the old and metal - poor rr lyrae stars will be used to trace the 3d structure of the bulge , classical cepheids and eclipsing binaries will be employed to map the spiral arm structure on the ( largely unexplored ) far side of the galaxy , both in the disk region and behind the bulge . in order to get an unbiased picture on galactic structures traced by these stars , both the rate and precision in the detection of periodic signals have to be sufficiently high . the time - domain coverage at the present stage of the vvv survey is not yet sufficiently large for conducting variability searches at high completeness , therefore our current investigations are limited to already known samples of distance indicators ( @xcite ) . however , we are already in the position to give estimates on the future signal detection performance in vvv based on the time - series data of the 8 vvv bulge fields with more than 60 epochs , which have a partial overlap with the highly complementary optical time - domain survey ogle - iii @xcite . in the following , we present a simple and preliminary assessment of signal detection rates based on the data for the ab - type ( i.e. , fundamental - mode ) rr lyrae stars in this region that are known from ogle - iii . we performed a positional cross - matching procedure between the ogle - iii catalog of rr lyrae stars @xcite and the vdfs _ pawprint _ photometric catalogs , including data up to 2012 november ( 61 epochs ) . figure [ fig : ksep ] ( _ top panel _ ) shows the density plot of the angular separations between the best - matching sources as a function of the average @xmath4 magnitude , for all the images . the average cross - matching accuracy is @xmath66 , and more than @xmath67 of the data points are concentrated in a small locus with separations less than @xmath68 , which indicates that both the precision and accuracy in the astrometry of both surveys is very high ( the average vircam pixel size is @xmath8 ; see [ sec : over ] ) . the small clusters of points represent complete light curves with poorer cross - matching accuracy , while the points with more diffuse distribution are due to intermittent cross - matching inaccuracies due to , e.g. , source merging at poor seeing , elongated sources , spurious signal contamination close to saturated stars , etc . in total , 1558 out of 1832 rrab stars were successfully cross - matched with near - ir counterparts with separations not larger than @xmath69 , meaning @xmath70 completeness in on - chip detections with respect to ogle - iii . the main limiting factor for the vdfs photometry is source crowding , since many close objects can not be separated even using small apertures . psf photometry and dia ( see [ dia ] ) are expected to significantly improve this figure . figure [ fig : ksep ] ( _ bottom panel _ ) shows the cumulative distribution of average @xmath71 magnitudes of those rrab stars where near - ir counterparts were not found in vdfs catalogs within the above constraints . the distribution is close to uniform , which means that a similar fraction of objects is missing at bright and faint magnitudes . thus the limiting magnitude does not affect our completeness compared to ogle - iii . on the contrary , for stars lying close to the galactic plane , where extinction is severe in optical bands , we expect to have a much higher completeness in the census of variable stars , due to the advantage of the employed near - ir wavebands in penetrating high - extinction regions . to evaluate the signal detection rate , we compute the lomb - scargle periodograms @xcite of the vvv @xmath4 light curves of successfully cross - matched rrab stars that have at least 50 epochs ( 1076 objects ) . then , we characterize the completeness in signal detection by the cumulative detection efficiency ( cde ) : @xmath72 where @xmath73 is the number of successfully detected rrab stars , and @xmath74 is the total number of rrab stars , respectively both including all stars with magnitude @xmath75 up to a certain magnitude @xmath4 . we consider a detection successful if the signal is within the @xmath76 significance level evaluated on white noise , and its frequency , refined by non - linear least - squares fitting , does not differ from the one reported by ogle - iii by more than the nominal frequency resolution ( i.e. , the inverse of the baseline , or @xmath77 ) . the result is shown in figure [ fig : cdp ] , compared with detection efficiencies computed from synthetic data . for the latter , a typical @xmath4-band rrab light curve ( similar to sw and ; see @xcite , and also fig . [ fig : felipe ] below ) with low and high amplitude ( @xmath78 , @xmath79 mag ; and @xmath80 , @xmath81 mag , respectively ) was sampled randomly within the visibility periods of the bulge in a vvv - like scenario , random noise was added based on the scatter diagram shown in figure [ fig : sigma - ks ] , and the cde was computed from a few thousand realizations . we conclude that a signal detection rate of @xmath82 can already be reached on vvv data for rrab stars in the bulge , in light curves sampled with @xmath83 epochs . this number will be pushed significantly higher by raising the number of epochs to @xmath84 ( i.e. , increasing the spectral @xmath85 and resolution , as well as decreasing the effect of aliasing ) , and improving the photometric precision by either profile - fitting photometry or dia . another important aspect of signal detection efficiency is the relative number of significant signals with incorrect periods , e.g. , due to aliasing . ideally , the number of such cases should not exceed a few percent because , for instance , wrong periods imply wrong distances through the period - luminosity ( pl ) relations , and if these errors are too frequent , they can blur the 3d structures that we wish to trace . the relative frequency of signals with incorrectly recovered periods can be quantified by the cumulative false detection rate ( cfr ) , defined analogously to cde ( eq . [ cde ] ) : @xmath86 where @xmath87 is the cumulative number of detections where the frequency is incorrect , and @xmath88 is the total number of light curves with significant signals , both again including sources up to a certain magnitude . the top panel of figure [ fig : cdp ] shows the cfr measured in our rrab sample , once again in comparison with the values computed from synthetic data ( see above ) . the cfr is about @xmath89 for bulge rrab stars , which is already sufficient to rather sharply trace the 3d structure of the underlying stellar population , and we expect to reach @xmath13 by the end of the survey . we note that the peak close to 12 mag is produced by very few stars , and is caused by heavy aliasing introduced by the intermittent saturation of these bright objects . current estimates based on the analysis of the available vista datasets as delivered by the casu vdfs pipeline have suggested that the final number of variable stars observed in the vvv survey may be in the range between 10@xmath90 and 10@xmath91 stars . these large numbers of objects require new approaches for the data analysis and selection , including artificial intelligence algorithms . machine learning techniques applied to variable star classification have become particularly popular in recent years . for instance , in @xcite the authors explored several classification techniques , quantitatively comparing performance ( e.g. , in terms of computational time ) and final results ( e.g. , in terms of accuracy ) of different classifiers with their corresponding learning algorithms . more recently , a few other studies have focused on specific methodologies , with the implicit goal of finding the best compromise between robustness and speed . as an example , @xcite and @xcite have independently presented tree - based methods for the automated classification of hipparcos and ogle variable stars ; @xcite have employed machine learning techniques to detect quasi - stellar objects in the macho and eros-2 databases ; while @xcite opted for algorithms based on multivariate bayesian statistics in order to explore the variable star content of the tres lyr1 sky field . whichever specific algorithms preferred , the general idea behind these supervised machine learning methodologies is to create a function , the _ classifier _ , able to infer the most probable _ label _ of an object ( in our case , the variability class to which an unclassified variable star belongs to ) on the basis of what is learned by the analysis of inputs ( light curve features ) from a _ training set _ ( a collection of high - quality light curves of previously classified variable stars ) . in the most general framework , the sequence of steps to be performed can be thought as : 1 ) build a training set ( template database ) ; 2 ) determine the input feature representation ( e.g. , periods and harmonics , as derived from fourier analysis ) of the learned function ; 3 ) determine the nature of the classifier with its corresponding learning algorithm ( e.g. , artificial neural network , support vector machines ( svms ) , tree - based methods , etc . ) ; 4 ) run the classifier on the gathered training set , using the information on well - known variables stored in the training set for searching and labeling unknown variable stars in the test set , i.e. , in the data archive that one is dealing with ; 5 ) finally , evaluate the accuracy of the learned function , i.e. , evaluate the fraction of correctly classified variable stars ( e.g. , through a so - called confusion matrix ; * ? ? ? * ) . in what follows , we describe in some detail the first three points of this proposed working strategy . until now , stellar variability in the near - ir has been a relatively ill - explored research field : in particular , the number of high - quality light curves was very limited and , even worse , many variability classes have not yet been observed in a sufficiently extensive way in the near - ir , so that good light curves are entirely lacking for some such classes . since vvv is the first ever large survey dedicated to stellar variability in the near - ir , the first problem we had to face has thus been the construction of a proper training set , i.e. , a database of high - quality ( `` template '' ) near - ir light curves taken to be representative of the different variability classes under study . our effort in building such a template database has included four main routes , which are described in the following . * _ a ) near - ir light curves from the literature . * firstly , we have extensively explored the available literature , searching for papers that presented high - quality near - ir time - series data . when these data were not published in machine - readable form and we could not obtain the data directly from the authors , we digitized the data tables and/or plots , and used optical character recognition software to convert those into ascii files . among the types of variables for which data could be retrieved in this way are rr lyrae , classical cepheids , miras , eclipsing binaries , and wolf - rayet stars , among others . examples of high - quality data obtained in this way are shown in figure [ fig : felipe ] . * _ b ) near - ir light curves from public archives . _ * we have also searched publicly available archival databases for high - quality near - ir data , finding the 3.8 m united kingdom infrared telescope ( ukirt ) wide - field camera ( wfcam ; @xcite ) calibration archive ( wfcamcal ) especially appropriate for our purposes . wfcamcal s current data release ( dr8 ) contains data from 364,905 individual pointings on both the northern and southern hemispheres , spread over nearly half of the sky . the majority of the fields are observed repeatedly , with a rather irregular sampling that has however provided for many sources a reasonably extended time coverage . the selection of variable sources from the wfcamcal catalog and the corresponding light curves will be presented in ferreira lopes et al . ( 2013 , in preparation ) ; an example of a light curve derived in this way is provided in figure [ fig : wfcam ] . * _ c ) vvv templates observing project . * in addition to data from the literature and from public archives , we have also carried out time - series near - ir observations of our own . in this way , we have monitored hundreds of ( optically well - studied ) variable stars in the @xmath2 , @xmath3 , and @xmath4 bands , using several different facilities located at different observatories across the globe ( see table 3 in @xcite for a listing of all the telescopes and instruments used ) . an example of a light curve obtained in this way is shown in figure [ fig : wysco ] ; additional examples can be found in @xcite . a full description of the project will be provided in angeloni et al . ( 2013 , in preparation ) . * _ d ) vvv light curves . _ * last but not least , vvv itself has started delivering light curves of variable stars that had already been classified by previous optical surveys , most notably macho @xcite and ogle @xcite . as an example , figure [ fig : vvv_blher ] shows a bl her ( type ii cepheid ) light curve , obtained using ogle - iii @xmath71-band data @xcite and vvv @xmath4 data . as more and more light curves of this quality become available , our template database will be augmented accordingly , thus enabling increasingly more accurate classification of previously unclassified vvv variable sources . the problem of automated classification of variable stars using temporal series has been around for some time . for example , @xcite have successfully implemented algorithms for automated classification which are based on extensive light curve databases , containing thousands of entries , from hipparcos , ogle , and other projects all providing light curves in the visible bandpasses only . in the case of near - ir projects like vvv , special care must be taken both in the feature extraction process and in the implementation of classification algorithms , because the number of training examples ( i.e. , high - quality light curve templates ) that is available at the outset is not as large . how does one optimally identify and extract the most informative features that best describe a temporal series ? the usual model for a light curve of a given variable star , where @xmath92 represents the magnitude as a function of time @xmath93 ( usually julian days ) , is @xmath94 where @xmath95 here @xmath48 and @xmath96 represent the intercept and slope respectively of the linear trend in the time series , @xmath97 is the harmonic sum ( which tries to describe the star s actual variability ) , @xmath98 is the number of harmonics for a given frequency @xmath99 , @xmath100 is the number of frequencies obtained from the light curve , and @xmath101 is the ( possibly correlated ) noise . from here , obtaining features of the light curve is straightforward : the frequencies could be estimated via the `` classical '' lomb - scargle periodogram @xcite , first detecting and subtracting the linear trend , then detecting and subtracting the frequencies one at a time ( i.e. , performing prewhitening ) , and finally performing a full least - squares fit with the model in equation ( [ fullfit ] ) . from here , one can decide to extract some useful features for classification . following the work of @xcite , these features are ( i ) the parameters of the linear trend ; ( ii ) the amplitudes , which are given by @xmath102 ( iii ) the ( time - invariant ) phases , given by @xmath103 } \,\ , , \ ] ] where @xmath104 ( with @xmath105 ) ; and ( iv ) the ratio @xmath106 of two variances . @xmath107 corresponds to the variance of the raw photometric measurements , while @xmath108 is the variance from the residual _ after _ the first frequency fit ( with the corresponding @xmath109 harmonics of this first frequency ) . the above prescribed procedure , though somewhat `` standard , '' has nonetheless some issues that must be carefully addressed . the first issue with which we must be careful is whether classical periodogram searches , such as the lomb - scargle method , constitute the optimum solution for the frequency / period searches . for example , @xcite claimed that the classical periodogram method can be thought of as trying to fit sine curves ( with the corresponding phase ) of different frequencies , i.e. , a fit of the form @xmath110 where the periodogram is just a function of the amplitude of this sinusoid for each frequency @xmath111 . according to these authors , this procedure does not take into account the fact that the sinusoid may be `` floating '' around a different mean value than that of the linear trend in equation ( [ fullfit ] ) . they claim , in addition , that this procedure does not weigh the data points according to their respective variances , as they should in order to have unbiased estimators for the amplitudes in the presence of uncorrelated noise . to solve these issues , @xcite add an extra term to the function @xmath112 above to account for this `` floating mean , '' and write the least - squares solution assuming a diagonal covariance structure for the data , where each entry of this diagonal covariance matrix corresponds to the variance on each data point . @xcite , on the other hand , disagree with this procedure . they actually claim that performing least - square fits to sinusoids is different from calculating the periodogram , which in essence is the squared amplitude of the discrete fourier transform of the data . moreover , they claim that their formalism is capable of dealing with all types of colored noise . the second issue that one has to take care of is aliasing , i.e. , spurious peaks due to sampling . for example , @xcite have shown that the problem of aliasing can trick not only period - finding algorithms , but also the human eye , misidentifying a `` real '' peak in the periodogram by a fake peak produced by sampling ( ir)regularities . this is a very serious and critical problem for classification purposes : if we fail to identify the true period of a periodic variable star , then we lose perhaps its most important feature . basically , @xcite showed that it is not always true that the highest peak in the periodogram is actually a ( physical ) period of the object of study . the third issue is how to select @xmath100 , the number of frequencies in the harmonic fit and , given that number , how to select @xmath98 , the number of harmonics in each frequency fit . even further , one may ask whether or not the number of harmonics should be the same for all the frequencies . physically , there is no a priori reason to assume that the number of harmonics should be the same , nor that all variability types are characterized by the same number of frequencies . as can be seen , the issue of feature identification and extraction remains subject to considerable debate . we are currently conducting extensive tests using vvv data and our templates database ( [ sec : templates ] ) , in order to obtain an optimum strategy for the purpose of classifying vvv light curves . as a first approach to solve some of the issues mentioned above , we propose to initially over - fit the model in equation ( [ fullfit ] ) , using more features than possibly necessary to fit the light curves . in more detail , we force our model to fit for each light curve up to five frequencies and up to five harmonics for each frequency to obtain the features with which we represent the light curves . therefore , each light curve can now be represented as a vector @xmath113 containing the frequencies , amplitudes , phases , slope and intercept from the linear trend , and ratio of variances . in a second step , we fit a binary classifier , @xmath114 , for each class @xmath115 which can learn how to discriminate between classes and perform feature selection _ simultaneously_. this classifier is called lasso . the lasso classifier estimates the parameters @xmath116 by minimizing a cost function subject to a constraint on the size of the @xmath117 norm of the parameter vector given by @xmath118 . this constraint on the size of the parameter vector shrinks the parameter estimates towards zero , and the use of the @xmath117 norm forces some of the parameters to be equal to zero . the features associated with parameters that are estimated as being equal to zero are irrelevant for the classifier . therefore , this procedure allows us to have a different set of features for each classifier , and is convenient in the present setting in which we need to classify variable stars which can be mono - periodic , multi - periodic , or non - periodic . for example , mono - periodic light curves should need fewer frequencies in the harmonic fit than multi - periodic stars , which implies that fewer features are needed for the classifier and this is borne out naturally by lasso . in order to improve our classifiers as new light curve data become available , we plan to augment our templates training set ( [ sec : templates ] ) by selecting the most informative light curves through an active learning methodology . active learning procedures are very helpful to efficiently select the most informative instances to be included in training sets @xcite . in the astronomical context , @xcite recently used active learning and showed its benefits for handling astronomical data . these authors show that active learning techniques can reduce the bias of the training process and increase the classification accuracy . most active learning models are composed of two phases . the first is the exploration phase , where the model explores the most informative instances to select . the second is the exploitation phase , where the model includes the feedback after the new selected instance and updates itself in order to repeat the exploitation phase @xcite . usually the exploration phase requires a lot of computations , in most cases passing through the data many times , making it very difficult to be directly applied to large astronomical catalogs . the exploitation phase may be less costly , depending on the model used . as in @xcite , most active learning models assume the existence of an oracle , an entity that can correctly classify any query instance . unfortunately , such an oracle may not be available , or / and may be too expensive to implement . trying to deal with the absence of an oracle , semi - supervised active learning techniques attempt to use only the available labels @xcite . these models use expectation maximization to estimate the best prediction for the missing labels using the current ones , but unfortunately most of these models can not handle huge datasets because of the computational cost of the algorithm . recent work introduced active learning for large datasets @xcite , based on a similar approach as in @xcite but using hashing techniques to speed up the process . this provides a very interesting approach to the large datasets " problem , but still assumes the existence of an oracle . in this context , we are currently developing an active learning framework for large datasets , modelling the partial absence of the oracle . after developing this semi - supervised active learning framework , we will iterate between automated classification of the available vvv light curves and increasing the size of the training set by incorporating new template - quality light curves ( i.e. , light curves with extremely high classification probability , as judged by the code and/or the oracle , in this case a vvv astronomer ) to the latter . this will allow us to refine the classification procedure , and then re - run the whole process with increased confidence as additional data are incorporated into the main vvv light curve database . in this way , our final vvv light curve database will contain not only periods and magnitudes , but also the variability classes to which the stars are associated which in some cases could / should be confirmed a posteriori using additional data , such as spectra and photometric observations in other bandpasses . one of the main scientific goals of the vvv survey is to complete the census of classical radially pulsating variable stars , such as rr lyrae stars and cepheids , in the galactic bulge and disk . these stars provide very important means for the study of the 3d structure of the milky way because they follow precise pl relations in the near - ir ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , thus they can be used as standard candles , and employed to infer the spatial distribution of their underlying stellar populations . this is particularly efficient if photometric data are available in multiple bands , because in this case the interstellar reddening can be computed on a star - by - star basis , by comparing the measured color index to the intrinsic one , as predicted by the pl relations in different bandpasses . this information can be used both for the distance determinations and for mapping the extinction , and even to trace the large - scale variations in the reddening law in the bulge that has been reported by various studies ( see , e.g. , * ? ? ? * ; * ? ? ? the vvv near - ir data on pulsating stars is particularly powerful in combination with optical ( e.g. , @xmath71-band ) light curves of complementary time - domain surveys such as ogle - iii , because the accurate mean magnitudes and the large wavelength separation allow a very precise determination of the absolute extinction from the color excess , and the result will be highly independent from the value of the selective - to - total extinction ratio @xmath119 . in @xcite , we used vvv near - ir photometry of known bulge rrab stars in combination with optical light curves from ogle - iii , to study the 3d structure of the bulge . our results showed that the spatial distribution of the rr lyrae stars is significantly different form the x - shaped distribution of the red clump stars @xcite ; rather , it is rather spheroidal , and does not show a strong bar . this finding implies that the milky way may have retained a classical bulge component , with a high fraction of stars in non - cylindrical stellar orbits . the new rr lyrae stars discovered by the vvv survey will allow us to extend our analysis to a much larger sample . based on our signal detection tests discussed in [ sec : cross ] , we can conclude that the vvv survey is capable of yielding a highly complete census of fundamental - mode rr lyrae stars in the bulge . figure [ fig : rrab ] shows one of the several thousand new rr lyrae stars discovered so far by vvv in bulge areas that are not covered by any other time - domain surveys . the high - quality photometry will allow us to unambiguously classify the majority of the new objects , even relatively faint stars , based on their light - curve features ( see [ sec : class ] ) . by greatly expanding the areal coverage as compared to previous optical surveys , and reaching much deeper in highly obscured areas , the vvv survey will provide unique data that are essential for understanding the structure and formation history of the milky way . the same can be expected for classical cepheids either behind the bulge or in the disk region , since these stars have much larger amplitudes . these objects will provide important means to trace the spiral arm structure on the far side of the galaxy , which has been out of the reach of optical surveys due to high interstellar extinction , and thus present vast uncharted territories of the milky way . as we have seen , one of the main goals of the vvv survey is to describe in detail the inner milky way structure , which is already being accomplished using distance indicators such as rr lyrae stars and red clump giants ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , in addition to rr lyrae and cepheid variables , long - period variables ( lpvs ) are also expected to be detected in large numbers . since lpvs are very bright and also follow families of pl relations ( see * ? ? ? * ; * ? ? ? * for recent reviews and extensive references ) , they are potentially very powerful distance indicators as well . lpv stars , including miras and semi - regular variables ( srvs ) , have been studied previously in the galactic bulge by different teams ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , these were mostly limited to bright objects ( @xmath120 mag ) . vvv represents a significant progress in the search and study of lpvs in the inner bulge , due to its higher spatial resolution and variability campaign carried out in the near - ir , which allows one to pierce deeply into the most crowded and highly obscured regions of the galaxy , reaching up to @xmath121 mag in most observed fields @xcite . a search for miras and srvs in the inner galactic bulge is already in progress , using vvv @xmath4-band data . in this first analysis , we selected 12 vvv fields , namely b327 to b338 ( see fig . [ fig : area ] ) , covering about 18 square degrees in the inner bulge . in a massive variability survey such as vvv , miras are relatively easy to find , due to the large amplitude of their light curves , which exceed 0.35 mag @xcite and are on average 0.6 mag @xcite in the @xmath37-band , and periods in the range @xmath122 . about 100 reliable mira candidates were identified from a total of @xmath123 million light curves that were produced , with 23 - 32 data points per object . each light curve covers the first two years of observations ( 2010 - 2012 ) with different cadence depending on the vvv field ( see [ sec : status ] , [ sec : var - over ] ) . in order to find the period and amplitude of each variable , the light curves of the mira candidates were fitted by applying both fast @xmath124 @xcite and lomb - scargle methods @xcite . an illustrative example of our results is shown in figure [ fig : lpv ] ; naturally , the derived periods will become better defined as more data covering a longer baseline become available . a first catalog will be presented in gran et al . ( 2013 , in preparation ) . in addition to miras and srvs , this includes a large number of variable agb stars ( previously identified as oh / ir or maser sources , see @xcite ) , besides young stellar objects and other still unclassified sources . the majority of pre - main sequence ( pms ) stars have measurable variability , which is usually attributable to magnetic activity . hot spots formed at the base of accretion funnel flows can also cause variability , though this would be more important at blue and near - uv wavelengths . some pms stars , e.g. kh15d @xcite , have variable extinction , which can lead to high - amplitude variability even in the ir . however , the highest amplitudes are observed in eruptive pms variables , where the mechanism is believed to be large variations in the accretion rate which directly change the luminosity of the star and the inner parts of the circumstellar disk . eruptive pms variables are usually divided into fu orionis types ( fuors ) and ex lupi types ( exors ) , which show occasional variability in excess of 2 to 6 magnitudes on timescales of months ( exors ) or years to decades ( fuors ; for a review , see @xcite ) . fuors show a rapid rise in luminosity followed by a slow decay over decades . many fuors have associated molecular outflows or jets @xcite in which the effects of historical eruptions on the outflow rate can sometimes be seen @xcite . fewer than 20 eruptive variables are known in each of the fuor and exor classes , but it is possible that this type of variability is common amongst pre - main sequence stars , though intermittent in nature with long intervals of quiescence . if it is common , this would be important for two reasons . firstly , it may explain the commonly observed scatter in hertzsprung - russell diagrams of pms clusters , a phenomenon that hampers the assignation of masses to pms stars with evolutionary tracks , with consequences for measurements of the initial mass function . secondly , they may explain the long - standing `` luminosity problem '' ( e.g. , * ? ? ? * ) , which consists of the fact that low - mass pms stars are typically less luminous than expected for objects that should be above the main sequence while descending a hayashi track . as far as eruptive pms variables go , our work on the vvv data has the potential to ( i ) precisely quantify the incidence of exors and ( ii ) unveil variability in optically - obscured protostars , an area where only a small amount of work has been done to date . we might expect the high - amplitude variability to be more common in these obscured , generally younger systems , in which the average accretion rate is higher . we are also beginning to explore the potential of the common low - amplitude variability as a method for tracing dispersed pms stellar associations . this may allow us to investigate the duration of pms evolution ( which will be mass - dependent ) , by comparing the number of variables with the number of stars with disks . our first searches for pms variables were based on vvv data release 1 ( dr1 ; * ? ? ? * ) , which had a typical time baseline of only a few months in 2010 and only 5 epochs of @xmath4 photometry . we searched in the disk region of vvv ( @xmath125 , see fig . [ fig : area ] ) for candidate variables with @xmath126 mag , which is the approximate upper limit for magnetic variability . we also did a more general search for variables with rms @xmath4 variability @xmath127 mag , in order to see whether that would pick out star formation regions ( sfrs ) . both searches of the catalogs produced a large number of false positive candidates requiring visual inspection , but their number was reduced by requiring a detection in the _ spitzer _ glimpse legacy survey @xcite , and in particular by focusing on the subset of 18,949 very red sources ( @xmath128-[8.0]>1 $ ] mag ) identified in glimpse by @xcite . most of the robitaille sources are candidate protostars . the @xmath126 mag search identified 1881 candidates in vvv , of which 47 are detected in glimpse and 12 were known as red robitaille sources . these 12 were all confirmed as genuine variables by inspection of the images . they have @xmath129 between 1.02 and 1.76 mag , and the majority are undetected in the @xmath2 and @xmath3 passbands , although they are well above the @xmath4 detection limit . a low - resolution spectrum was obtained for one of these red variable sources ( g314 , see fig . [ fig : g314 ] ) with magellan / fire @xcite in march 2012 . the spectrum in figure [ fig : g314 ] is based on a preliminary reduction , but it clearly shows several very strong emission lines of h@xmath130 , which are marked with vertical lines . moreover , the emission was spatially extended by a few arcseconds along the slit , demonstrating the presence of a large - scale outflow . an additional search was made for lower - amplitude variables with rms magnitude variations @xmath127 mag in the dr1 data . this returned 24,798 candidates , of which 703 are in glimpse and 41 objects are robitaille red sources ( including the 12 with @xmath126 mag ) . we cross - matched these candidate variables which have _ not _ been visually inspected with the @xcite catalog of sfrs to see whether there was a significant spatial association . we found that 4% ( 937/24,798 ) of the candidate variables are within 10 arcminutes of a known sfr and the histogram of candidate to sfr separations shows a rising number of matches with decreasing separation . this indicates a clear asociation of at least a significant minority of vvv variables with sfrs ( see fig . [ fig : sfr ] ) . similarly , we found that 30% ( 214/703 ) of the glimpse sources are within 10 arcminutes of a known sfr , and the separation histogram has a similar trend . to establish the true fraction of vvv variables that are pms stars will require visual inspection of a large sample and careful checking of the photometry to avoid the errors that can arise in crowded fields . subsequently , we undertook a search for very high - amplitude variables in the combined 2010 to 2012 datasets for the disk region , focusing on the region at @xmath131 ( contreras pea et al . 2013 , in preparation ) . following visual inspection of a few hundred candidates , a total of 77 genuine variables were identified with @xmath132 mag , which is the level more typically associated with eruptive variability when long time baselines are available . spectroscopy from 0.8 to 2.5 @xmath133 m has now been obtained for 6 of these systems in april 2013 , using magellan / fire in echelle mode at @xmath134 . priority was given to sources ( i ) with a likely association with clusters or sfrs of known distance ; ( ii ) with highest amplitude ; ( iii ) with signs of a recent outburst in the light curve ; ( iv ) bright enough for quick observation . a further 4 systems with slightly lower amplitude in the g305 star forming complex were also observed , and g314 was re - observed . a few of these 11 systems also showed strong , spatially extended h@xmath130 emission , similar to that seen in g314 . data reduction and analysis is currently ongoing . eclipsing binary systems in general , but detached systems especially , are extremely important objects in astrophysics , for they provide one of the most robust means of deriving stellar radii , masses , and even ages ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , thus providing a `` royal road '' to stellar astrophysics ( * ? ? ? * ; * ? ? ? * and references therein ) . the vvv survey will provide a huge number of eclipsing systems of all types , and we have already started to travel along this `` royal road '' using vvv data . examples of vvv light curves for two eclipsing binary systems can be found in figure [ fig : ebs ] . as an example of their use in astrophysics , a method to use the vvv data for tracing the structure of the milky way with detached eclipsing binaries ( debs ) was introduced in @xcite . the idea behind the method is that one can calculate the distance by comparing the derived absolute magnitudes from the model with the observed ones . this normally requires spectroscopic observations to calculate radial velocities of both components , used for calculating the masses and orbital parameters , crucial for obtaining the true absolute values of the stellar parameters ( like the radii ) . one also needs accurate theoretical atmosphere models and/or calibrations to infer surface brightness on the basis of colors , line ratios or other observables @xcite . with the increasing availability of extensive databases of state - of - the - art stellar evolution models , it is now possible to derive a complete set of physical parameters of a binary s components without time - consuming spectroscopy , from their light curves only . in our approach we implemented two codes , which can analyze large numbers of deb light curves in a short time . the first one the detached eclipsing binary light curve fitter ( debil ) is a program which rapidly fits deb light curves to a simple , geometric model @xcite . the second code method for eclipsing component identification ( meci ; * ? ? ? * ; * ? ? ? * ) fits a physical model to each deb using readily available photometric data only . it is designed to work from the debil model as a starting point , building an improved physical model of the deb therefrom . meci assumes that the binary s stellar components formed together and evolved along their respective evolutionary tracks , without any mass transfer . if observed magnitudes in different bands are given , meci returns the absolute magnitudes , so the distance may be calculated directly . finally , to deal with the strong and variable interstellar reddening , we implemented the idea of reddening - free indices , introduced in @xcite . these indices are combinations of magnitudes in 3 bands , in the form of @xmath135 where @xmath136 are the apparent magnitudes in the available bands , and @xmath115 is a multiplication coefficient dependent on the extinction law assumed . the coefficient @xmath115 is given in such way that , within a given extinction law , defined in terms of the ratios of extinction values in the given bands @xmath137 , the following equation is true : @xmath138 where the `` 0 '' subscripts indicate quantities corrected for reddening , and thus @xmath139 , @xmath140 , @xmath141 . combining equations ( [ eq : mx ] ) and ( [ eq : mx0 ] ) , we have : @xmath142 = m_{1,0 } - c(m_{2,0 } - m_{3,0 } ) . \label{eq : mxd1}\ ] ] thus , @xmath143 naturally , the definition in equation ( [ eq : mx ] ) is also applicable to _ absolute _ magnitudes , and so @xmath144 where the capitalized @xmath145 s denote absolute magnitudes . on this basis , we obtain , for the apparent distance modulus , @xmath146 - \left[m_1 - c(m_2 - m_3)\right ] , \label{eq : mx - mx}\ ] ] which implies @xmath147 - \left[m_1 - c(m_2 - m_3)\right ] . \label{eq : mx - mxd2}\ ] ] this can be rewritten as @xmath148 - \left[m_1 - c(m_2 - m_3)\right ] + \left[a_1 - c ( a_2 - a_3)\right ] . \label{eq : mx - mxd3}\ ] ] thus , @xmath149 , \label{eq : mx - mxd4}\ ] ] and so , using equation ( [ eq : cdef ] ) , @xmath150 where @xmath151 is the distance in pc . therefore , _ reddening - free magnitudes are ideally suited for the calculation of distances , without the need for prior knowledge of extinction values towards individual stars_. the vvv variability campaign is still ongoing ( [ sec : var - over ] ) , and a catalog of eclipsing binaries has thus not yet been prepared . therefore , we used the eclipsing binaries from the ogle - ii variable stars catalog @xcite , for which @xmath71-band light curves are available . @xcite presented models for about 10,000 of them , obtained with the debil code , and identified 3170 of them as `` detached . '' to build the reddening - free indices we used @xmath152 photometry from vvv , in addition to @xmath153 data from ogle - ii , and assumed a canonical extinction law with @xmath154 . for consistency , we also built a set of reddening - free isochrones , based on the padova models @xcite . the procedure just described successfully yielded physical parameters , ages , and distances to 23 deb systems . in figure [ fig_2.4_lc ] we present examples of the utilized ogle - ii light curves , superimposed on the best - fitting meci models . in figure [ fig_2.4_rec ] we present the positions of the 16 closest systems in the galactic plane @xmath155 , with the sun located at ( 0,0 ) and the galactic center at ( 8,0 ) kpc . they are plotted over a reconstruction of the milky way from @xcite . the locations of at least 13 of the studied targets coincide well with major structures of the milky way , such as the bulge and spiral arms , including scutum - centaurus , norma , far 3 kpc , and perseus . this shows that our approach is suitable for tracing the structure of the milky way . it is notable how many objects were found in a poorly studied area behind the bulge , which proves that the combination of near - ir vvv data with optical photometry from other sources can be a powerful tool for studying this part of the milky way . another 7 systems ( not shown on the reconstruction ) were found at larger distances , corresponding to the sagittarius stream a structure of stars related to the sgr dsph galaxy ( e.g. , * ? ? ? these sagittarius deb candidates , the first to be identified in that galaxy s stream , will be the subject of detailed spectroscopic follow - up by our team . cataclysmic variable stars ( cvs ) , including novae and dwarf novae , consist of tight binary systems in which one of the components the so - called _ primary _ is a wd star , and the other component the _ secondary _ is a low - mass , k- or m - dwarf ( i.e. , main sequence ) star . in cv systems , the dwarf star has filled its roche lobe , and is transferring matter onto the primary . cvs are thus eruptive binary systems , with the outbursts being detectable over a wide wavelength range , including the near - ir . however , the quiescent light curves of dwarf novae and related objects are dominated by the orbital motion , and their behavior in the near - ir differs from that seen at optical wavelengths . while in the optical the variations are produced by the hotter parts of the system , namely the primary star and/or the accretion disk , in the near - ir the light curve is dominated by the emission of the cool , late - type secondary star , whose spectra peak at longer wavelengths ( e.g. , * ? ? ? * ) . in close binaries the secondary star will be distorted by tidal effects when close to or filling its roche lobe . the orbital motion thus causes ellipsoidal variations which are more prominent in near - ir light curves . this modulation enables us to determine the orbital parameters of the system and even to map the surface brightness distribution of these stars ( e.g. , * ? ? ? * ; * ? ? ? the recent discovery of dozens of new dwarf novae cvs by the ogle team demonstrates the ability of large surveys to search for new variable sources even in the most crowded regions of the galaxy @xcite . long - period symbiotic systems , in which the secondary star is a late - type giant rather than a main - sequence star , have also been discovered in both the ogle and macho data @xcite . in like vein , we expect large numbers of interacting binary systems to show up in the vvv data . indeed , in the case of novae , preliminary results have already been presented in @xcite . there are about 400 known novae in the galaxy , with @xmath156 of them falling in the vvv survey area . interestingly , the spatial distribution of novae shows a `` zone of avoidance , '' with just a few objects belonging to the innermost regions . moreover , the comparison with nearby galaxies suggests that we lose many nova eruptions every year ( * ? ? ? * and references therein ) . not surprisingly , the regions with a lack of objects are the regions most heavily obscured by dust , beyond the capabilities of the current searches for novae , which are mostly carried out in the optical . @xcite provide @xmath12 data for 93 galactic novae . for some of these objects , colors have been reported for the first time in a homogeneous dataset , since a large fraction of these novae were beyond the detection limit of previous near - ir surveys ( e.g. , 2mass ; * ? ? ? * ) . the low novae rate observed in the galaxy encouraged us to start a search for the hidden novae in the aforementioned `` zone of avoidance '' region . we thus selected 24 vvv fields covering about 36 square degrees towards the inner galactic bulge . in this first attempt , we made use of the first variability tables available , limited to @xmath157 epochs in the inner bulge at the time of writing . even with a limited number of epochs , the vvv data allowed us to discover two new galactic novae candidates ( vvv - nov-001 and vvv - nov-002 ; * ? ? ? * ; * ? ? ? * ) , and a few more objects are currently under study . vvv light curves of vvv - nov-001 and vvv - nov-002 are presented in figure [ fig : lcurve ] . spectra taken with the soar telescope show the presence of emission lines which are typical of novae in vvv - nov-001 ( saito et al . 2013c , in preparation ) , while optical data from ogle confirmed vvv - nov-002 as a d - class nova @xcite . while vvv - nov-001 was discovered on the galactic plane , with @xmath158 deg , vvv - nov-002 is one of the closest novae to the galactic center known , with @xmath159 deg . a search for novae will be extended to the high - extinction regions of the inner disk in the near future . gravitational microlensing events can be of enormous astrophysical significance ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . models of the spatial dependence of the microlensing optical depth @xmath160 @xcite show that ir surveys like vvv can be very efficient in the search for microlensing events , and can probe directly the mass distribution contained in the inner regions of the galaxy . unfortunately , existing optical microlensing searches based on ccd detectors do not cover the whole bulge or the plane ( although ogle - iv already represents an important step forward in this regard ) , and in particular , they miss the inner regions where this optical depth is higher , thus poorly constraining the models . therefore , one of the main goals of the vvv survey is to search for microlensing events in the inner milky way . we are especially interested in looking for rare events in our data , such as : * very reddened events ; * short - timescale events , due to planetary or brown dwarf microlensing . the advantage of using microlensing to search for planets is that there is no preference for nearby objects or bright stars , contrary to what occurs with other techniques . thus , microlensing allows us to further probe the planet parameter space , searching , for instance , for planets with periods that are too long to be detected by other techniques , or not sufficiently close to the star as to produce detectable doppler shifts in their spectra @xcite ; * very long - timescale events , due to massive black holes . such a long - term variability study is possible with our @xmath161-year - long survey , adequately covering the baseline for these long - timescale microlensing events ; * binary microlensing events ; * parallax events ; * high - magnification events in obscured dense fields ; * microlensing of source stars in the sgr dsph galaxy ( e.g. , * ? ? ? * ) . so far we have concentrated our searches mostly on the regions that have not been previously searched for microlensing . these fields are located in the innermost regions of the bulge , where the microlensing optical depth is expected to be high . a quick search has so far revealed about two dozen bulge microlensing events with high amplification . the light curves and the parameters fitted show a range of timescales from a few days to several months , consistent with previous ( complementary ) results from the ogle survey @xcite . some examples of microlensing events are provided in figure [ fig : ml ] . the galactic bulge is known to contain a significant number of chromospherically active and rotating variable stars , including stars showing ellipsoidal light curve modulation , fk com stars , and rs cvn systems @xcite . we have cross - matched the @xcite catalog , which is based on macho observations , with the vvv photometric catalogs as provided by the casu vdfs 1.2 pipeline , successfully obtaining @xmath4-band light curves for numerous stars previously classified as rotating variables . some examples of the corresponding vvv light curves are provided in figure [ fig : rscvn1 ] . these light curves confirm that vvv near - ir data can be successfully used even to study systems whose light curve amplitude does not exceed @xmath162 mag in @xmath4 , thus also helping properly classify objects whose previous classification may be unclear or ambiguous . a significant number of galactic star clusters fall inside the area surveyed by the vvv : 36 known globular clusters ( gcs ) and 355 known open clusters ( ocs ; * ? ? ? * ) , in addition to newly discovered gc @xcite and oc @xcite candidates . the inner galactic star systems are generally deeply buried behind a curtain of dust and gas that hides them in optical observations , and so our knowledge of their variable star populations in particular is at present very incomplete ( e.g. , * ? ? ? fortunately , the highly diminished extinction in @xmath4 ( @xmath163 ) will help us unveil the variable stars that are present in these poorly studied objects . also fundamental in our variability studies is vvv s temporal coverage , with @xmath164 epochs over a 5-year period ; the conditions of the observations , with most of them taken with seeing under 1 arcsec ; the excellent spatial resolution of the vircam camera , @xmath8 per pixel ; and a complete spatial coverage of the inner regions of the galaxy . vvv data thus place us in a privileged position , as far as the study of variable stars in heavily obscured star clusters goes , allowing us to probe deep into the centers of these objects and out to their tidal radii and beyond . we have already started the search for variable stars in ocs and gcs alike , focusing initially on the vvv fields that lie in the bulge region of the survey , since the latter have been observed more often than the disk fields ( [ sec : var - over ] ) . in what follows , we describe some of our initial results , as far as the gc variability search is concerned ( see also * ? ? ? gcs contain large numbers of rr lyrae stars , which show a tight relation between periods and absolute magnitude in the near - ir @xcite , thus helping constrain the vvv gcs distances and extinctions . for many of these clusters , especially the faintest ones , the other photometric technique extensively used to derive their parameters , the study of their color - magnitude diagrams ( cmd ) , is highly complicated by the presence of elevated field stellar contamination and differential reddening , even at near - ir wavelengths ( see fig . [ fig_cmd_ter ] ) . when present , rr lyrae variables can provide us with better means to obtain these physical parameters . unfortunately , rr lyrae analysis in the near - ir has its own difficulties : the amplitude of the rr lyrae decreases towards the near - ir when compared to the optical , and the shape of the rrab light curves becomes more sinusoidal ( see fig . [ fig_var_6441 ] ) . even so , the quality of our light curves will allow us to look for these variables and to properly characterize their amplitudes , periods , mean magnitudes , and colors . in @xcite , we present our first attempts to tackle the study of variable stars in the vvv gcs , and show our efforts to characterize , in the near - ir , variables that had already been previously identified in optical studies ( as in the case of ngc 6441 ) , or to discover new variables in extremely obscured gcs where optical observations are very complicated or unfeasible ( as in the cases of terzan 10 or 2mass - gc02 ) . an intriguing characteristic that only galactic gcs seem to show is the so - called oosterhoff dichotomy ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) , i.e. , galactic gcs divide themselves in two main groups according to the mean period of their rrab variables , oosterhoff i systems having shorter periods ( @xmath165 days ) and oosterhoff ii systems with longer periods ( @xmath166 days ) . in the milky way , but not in nearby satellite galaxies , very few systems are found in the so - called `` oosterhoff gap '' zone , with @xmath167 . interestingly , the periods of the rr lyrae candidates we found in 2mass - gc02 and in terzan 10 make these two gcs outliers in the established picture , with terzan 10 being an oosterhoff ii gc but seemingly having too high a metallicity ( @xmath168 } = -1.0 $ ] dex ; @xcite , feb . 2010 update ) to belong to the group , and 2mass - gc02 falling in the almost empty oosterhoff gap region @xcite . terzan 10 could thus be a less extreme example of the new `` oosterhoff iii '' group proposed by @xcite ( @xcite ; see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which so far contains exclusively bulge gcs with even higher [ fe / h ] and longer @xmath169 , namely ngc 6388 and ngc 6441 . any new rr lyrae stars that we may be able to find in the inner gcs , and especially those with very few or no rr lyrae known , will be particularly useful to further explore and understand the oosterhoff dichotomy , and thus help us place bulge gcs in the wider picture of milky way formation ( * ? ? ? * and references therein ) . in addition to the many different types of well - classified variable stars discussed in the previous sections , the vvv survey data have already resulted in the detection of various transient objects @xcite , some of which appear to defy commonly adopted classification schemes . one example is provided by vvv - wit-01 @xcite , an extreme transient event whose light curve is shown in figure [ fig : wit01 ] . its color , as shown by the vvv multi - color data , is also extremely red , with @xmath170 mag . as discussed by @xcite , the source could be an eruptive pre - main sequence star , a reddened luminous blue variable , a nova , or even a highly obscured supernova event or a previously unknown kind of variable star . like vvv - wit-01 , many additional transient events are expected to be detected in the vvv data , thus opening an exciting path towards the discovery and analysis of energetic events towards the innermost and most heavily obscured regions of the milky way . at the other extreme , long - term photometric monitoring of ultra - cool dwarf ( ucd ) stars can also be carried out with the vvv survey . indeed , the variability of ucds is key to understand the atmospheric conditions and cloud formation in sub - stellar objects . several groups have monitored ucds on a daily basis , over timescales ranging from a few days to a few months ( e.g. , * ? ? ? * ; * ? ? ? using vvv data we will be able not only to discover new ucds , but also probe a new regime of variability for sub - stellar objects , thus helping to constrain long - term atmospheric changes in these very low - mass objects . in figure [ fig : bd ] is given an example of the long - term behavior of the newly discovered brown dwarf @xcite . as discussed by @xcite , the available data do not present signs of periodicity , and any variability is restricted to amplitudes @xmath171 mag . the vvv eso public survey provides a treasure trove of scientific data that can be exploited in numerous different scientific contexts . in terms of stellar variability , the project will provide up to several million calibrated @xmath4-band light curves for genuinely variable sources , including pulsating stars , eclipsing systems , rotating variables , cataclysmic stars , microlenses , planetary transits , and even transient events of unknown nature . at the present point in time , with the data - gathering phase of the vvv survey having just crossed its half - way mark , we are really just taking the first steps in what will certainly be a long and exciting journey , during which it will be possible to address a myriad of time - domain astronomical applications , including not only research on variable stars as such but also their use as distance indicators and tracers of galactic structure , origin , and evolution . vvv is a public survey , and so the data will quickly be made available to the entire astronomical community as we move along , thus opening the door to many additional applications and synergies with other ongoing and future projects that target the same fields as those covered by vvv . * acknowledgements . * the vvv survey is supported by the european southern observatory , the basal center for astrophysics and associated technologies ( pfb-06 ) , and the chilean ministry for the economy , development , and tourism s programa iniciativa cientfica milenio through grant p07 - 021-f , awarded to the milky way millennium nucleus . we gratefully acknowledge the support provided by fondecyt through grants # 1110326 ( m.c . , j.a .- g . ) , 1120601 ( j.b . ) , 1130140 ( r.k . ) , 3130320 ( r.c.r ) , and 3130552 ( j.a .- g . ) . c. navarrete acknowledges grant conicyt - pcha / magster nacional/2012 - 22121934 . , s.e . , and k.p . acknowledge additional support by project vri - puc 25/2011 . alonso - garca , j. , dkny , i. , catelan , m. , contreras ramos , r. , & minniti , d. 2013 , in fifty years of wide field studies in the southern hemisphere : resolved stellar populations in the galactic bulge and the magellanic clouds , asp conf . , in press ( arxiv:1307.0419 )
|
the vista variables in the va lctea ( vvv ) eso public survey is an ongoing time - series , near - infrared ( ir ) survey of the galactic bulge and an adjacent portion of the inner disk , covering 562 square degrees of the sky , using eso s vista telescope .
the survey has provided superb multi - color photometry in 5 broadband filters ( @xmath0 , @xmath1 , @xmath2 , @xmath3 , and @xmath4 ) , leading to the best map of the inner milky way ever obtained , particularly in the near - ir .
the main variability part of the survey , which is focused on @xmath4-band observations , is currently underway , with bulge fields having been observed between 31 and 70 times , and disk fields between 17 and 36 times .
when the survey is complete , bulge ( disk ) fields will have been observed up to a total of 100 ( 60 ) times , providing unprecedented depth and time coverage .
here we provide a first overview of stellar variability in the vvv data , including examples of the light curves that have been collected thus far , scientific applications , and our efforts towards the automated classification of vvv light curves .
[ 1996/06/01 ] 2@xmath5 2c ii 4c iv 2fe ii 3fe iii 1 mg i 2 mg ii 2si ii 4si iv 2al ii 3al iii 1o
i 1n i 1h i =
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
exascale computing @xcite is expected to revolutionize computational science and engineering by providing 1000x the capabilities of currently available computing systems , while having a similar power footprint . the total performance of the 500 systems in the 44th top500 list ( 18 nov 2014 , http://top500.org/ ) is about 0.3 exaflops . the hpc community @xcite is now working towards the development of the first exaflop computer , expected around 2020 , after reaching the petaflop milestone in 2008 . however , only a few hpc applications are so far able to fully exploit the capabilities of petaflop systems @xcite . examples of typical scalability for commonly used hpc applications in our organizations are provided in table [ tab : hpcapps ] . as the existing hpc applications are the major hpc asset , it is important and challenging to increase their scalability and lifetime by making them exascale - ready before 2020 . [ cols="<,<,<,>",options="header " , ] [ tab : hpcapps ] the major challenge for preparing hpc applications for exascale is that there is no exascale system available yet . currently all we have are assumptions about exascale systems . therefore the commonly used measurement - based approaches for reasoning about performance issues are not applicable . pre - exascale systems ( known as _ summit _ and _ sierra _ ) that ibm @xcite is developing for the u.s . department of energy will exceed 100 petaflops and may provide hints about the extreme - scale architectures of the future . this paper argues that efforts for preparing hpc applications for exascale should start before such systems become available . we identify challenges that need to be addressed and recommend solutions in areas that are relevant for porting hpc application to future exascale computing systems , including formal modeling , static analysis and optimization , runtime analysis and optimization , and autonomic computing . we suggest that porting of hpc applications should be made by successive , stepwise improvements based on the currently available assumptions and data about exascale systems . this approach should support application improvement each time new information about future exascale systems becomes available , including the time when the application is actually deployed and runs on a concrete exascale system . a high - level application representation that captures key functional and non - functional properties in conjunction with the abstract machine model will enable programmers and tools to reason about and perform application improvements , and will serve as input to runtime systems to handle performance and energy optimizations and self - aware fault management . a tunable abstract machine model encapsulates current assumptions for future exascale systems and enables a priori application improvement before the concrete execution platform is known . at runtime , the model is a posteriori tuned to support activities such as feedback - oriented code improvement or dynamic optimization . major contributions of this paper include , * identification of challenges and recommendation of solutions in formal modeling ( section [ subsec : modeling ] ) , static analysis and optimization ( section [ subsec : static ] ) , runtime analysis and optimization ( section [ subsec : runtime ] ) , autonomic computing ( section [ subsec : autonomic ] ) ; * a conceptual framework for preparing hpc applications for exascale that supports _ a priori _ application improvements before the concrete execution platform is known as well as _ a posteriori _ optimization at runtime ( section [ sec : approach ] ) ; * a discussion of the related work ( section [ sec : related - work ] ) . in this section we identify challenges and recommend solutions in formal modeling , static analysis and optimization , runtime analysis and optimization , and autonomic computing . our goal is to adapt hpc application code to exascale execution platforms to achieve good utilization of resources . for this , we need to address questions such as : 1 . what would happen if we change application or hardware layout ? 2 . what would happen if we change some parameters of the execution platform ? 3 . what would happen if we use a different execution platform ? unfortunately , answering these questions can not be done experimentally at the concrete level because such platforms do not yet exist . an alternative is to address these questions at an abstract level , focusing only on _ relevant information _ without actually executing the program . we believe that relevant information in this context is not what the code aims to achieve ( the result of the computation ) but its corresponding _ resource footprints _ , that is , how computational tasks communicate and synchronize , the amount of resources ( such as memory and computing time ) these tasks require , and how they access and move data . in order to adapt the hpc code to a particular architecture we need to capture such resource footprints of software modules at different levels of granularity ( e.g. , program statements , blocks in procedure bodies and whole procedures ) , and be able to compare different task compositions . consequently , the modeling language must feature massively parallel operators over such task - level resource footprints @xcite . a similar notion of resource footprints and composition can be used to express the properties of the architecture in a machine model to capture the resources that the architecture can make available to the code . working with resource footprints can be supported by an abstract behavioral specification language @xcite , in which models describe both tasks and deployments . these models can be used to predict the non - functional behavior of code before it is deployed , and to compare deployments using formal methods . this requires a formal semantics for the specification language that can be used to devise static analysis techniques . when developing code from scratch using a model - based approach , the resource footprints can be specified in tandem with the standard model in a model - driven development @xcite . however , when building such models from existing hpc code , monitoring profiles of low- and medium - scale systems can be used to extract resource footprints that approximate the resource consumption in terms of probabilistic distributions . the application of formal methods to parallel programs for analyzing functional properties , such as safety and liveness , has a long tradition . for _ non - functional properties _ , such as execution time and energy consumption , most performance analysis approaches use _ monitoring _ and present statistical information to the user . these approaches are helpful to improve hpc application code , but they also have some shortcomings : 1 . due to non - determinism , different program executions might lead to different observations . as a consequence , these methods are not able to provide reliable probabilistic information about average or worst - case execution times . they are based on execution on a real platform , thus they can not be used to predict performance on exascale computers , which are not available yet . 3 . these methods can be used to identify execution bottlenecks , but they can not explain the reasons for these bottlenecks , and thus they do not offer any concrete support for code improvement . we expect that _ formal methods _ can address these limitations to provide performance analysis tools that considerably go beyond the state - of - the - art . a major step in this direction will be the usage of resource footprints which describe both hpc applications and execution platforms as abstract probabilistic models . formal analyses can be applied to these models to predict their probabilistic behavior . while a range of techniques are available for non - probabilistic programs , the analysis of parallel probabilistic programs still need development effort . to achieve a reasonable balance between scalability and precision for challenging hpc applications , it seems fundamental to use _ hybrid approaches _ @xcite that combine techniques such as static analysis , dynamic analysis , simulation , ( parametric ) model checking @xcite , counterexample - guided abstraction refinement @xcite , deductive approaches , etc . to deliver the envisaged performance analysis tools , we face the following challenges : ( 1 ) determining the computation of cost properties that are given by means of probabilistic distributions ; ( 2 ) the inference of average cost in addition to the traditional worst - case cost ; ( 3 ) take into account the underlying platform through a set of probabilistic parameters ; ( 4 ) deal with massive and heterogeneous parallelism @xcite which is challenging for program analysis in general ; and ( 5 ) develop multi - objective resource usage analyses and optimizations . formal modeling and static analysis should be enhanced with analysis of measurements at runtime . plenty of tools ( for instance , http://www.vi-hps.org ) have been developed for performance measurement and analysis of hpc applications at runtime . however , these tools will experience several issues when applied to exascale . the collection rates and the overall volume of monitoring data in an exascale computing environment will exceed the scalability of present performance tools . therefore , throttling the data volume will have to be applied online in order to store as less data as possible and as much as necessary for later _ post mortem _ analysis . however , simple profiling will not be sufficient due to loss of temporal information , thus a hybrid approach will have to be applied that performs on - the - fly trace analysis in order to discard irrelevant data , while retaining the same amount of information . the metric classification should be based on the formal model ( see section [ subsec : modeling ] ) . such an approach will provide a generic insight into the performance of an hpc application that can be used for detecting performance bottlenecks . the instrumentation and hardware counter monitoring should follow a similar procedure where source code probing should be applied automatically by using tools such as opari @xcite . while many tools for collecting metrics of computing performance have been developed , very few analysis tools exist for energy consumption metrics in adequate accuracy and time resolution necessary for the runtime performance analysis @xcite . currently , common approaches ( see for example prace best practices @xcite ) for optimizing hpc applications require per - case inspection of runtime performance measurement data , such as profiling and tracing data . after the critical region has been determined , diverse heuristic approaches , such as `` _ _ trial and error _ _ '' , `` _ _ educated guess _ _ '' or `` _ _ rule of thumb _ _ '' , are applied to make changes in the affected source code sections . the most significant limitation of these heuristics and knowledge - based approaches is that , 1 . all changes are made directly and manually in the source code , and 2 . the effect of the changes does not always lead to an improved performance which makes necessary the repeating of all steps several times . moreover , exascale computers pose a multi - objective optimization problem , weighing out the effects of several sometimes incongruent requirements . therefore , a systematic and automatic approach for the optimization problem is essential to find the optimal solution . another problem is that the critical section in an application typically changes with the optimization iterations and/or with upscaling , due to the law of diminishing returns , which makes the manual analysis and source code changes even more laborious and inefficient , even if done by an experienced hpc developer . thus instrumentation , collection / measurement and analysis steps should be automated , for example based on high - level scalable tools @xcite , and integrated into a feedback loop ( see section [ subsec : autonomic ] ) . during the execution of an application , failures may occur or the application performance may be below the expectation . these issues are addressed typically by programmers in a `` _ _ trial and error _ _ '' manner , i.e. by manually changing and adapting their code to handle the failures and improve the performance . our proposed framework ( section [ sec : approach ] ) provides means for model - based failure handling or performance improvement based on _ autonomic computing_. autonomic computing addresses self - managing characteristics of distributed computing resources with the facilities to adapting to unpredictable changes while hiding management complexity to operators and users @xcite . among the explored categories , _ advanced - control based methods _ and more specifically _ distributed controllers _ are the first candidates to realize autonomic computing in exascale systems . we propose to devise methodologies to efficiently collect runtime information balancing the amount and cost for storage of monitoring data with the quality of monitored data necessary to make deductions about the application behavior ( e.g. trace analysis ) . the goal is thereby to define methodologies to scale current monitoring tools to exascale , balancing between quality and volume of monitoring data . combining the information from both static code analysis and runtime analysis , as outlined above , we will iteratively apply objective - oriented transformations to legacy application code at a formal level based on the exascale dsl model ( see section [ subsec : modeling ] ) . to this end , we will automate the analysis , optimization and transformation processes by implementing a generic feedback loop independent of the concrete programming language , algorithms used and target hardware architecture . a feedback loop driver enables to link the static analysis tool , the runtime analysis tool , the knowledge database and multi - parameter multi - objective optimization . as output , a set of rules ( policies ) is generated which is then applied to transform the formal application model and to adapt the runtime environment parameters ( cf . figure [ fig : diagram ] ) . after the transformations a new application executable is built and started in the adapted runtime environment . this described loop is iterated until convergence of the optimization . in this section we propose a conceptual framework for porting hpc applications to exascale computing systems . furthermore , we highlight benefits of our conceptual framework in the context of exascale computing . our proposed approach for preparing hpc applications for exascale is depicted in figure [ fig : diagram ] . the usage of a _ domain - specific exascale language ( dsel ) _ facilitates the programmer to express non - functional aspects ( like required time to solution , resilience or energy - efficiency ) of the execution of scalable parallel hpc codes . dsel has a formal operational semantics that enables the formal analysis of the code . the aim of the _ scalable model - based analyzer ( sma ) _ is to address non - functional properties of hpc codes , with a particular focus on scalability while complying with the crucial dimensions of resource consumption for exascale computing : time , energy , and resilience . the sma is responsible for analyzing resource consumption in terms of time , energy , and resilience , based on developed dsels . the _ exascale runtime data collector ( erdc ) _ is responsible for scalable monitoring to _ extract important monitoring data _ through the utilization of various techniques like _ filtering _ , _ streaming _ , or _ data mining_. the runtime information is used to verify or to tune the model of the code via the _ autonomous feedback loop ( afl)_. to endow the system with self - adaption , control - theoretical concepts are incorporated in autonomic computing paradigm . based on the autonomic technology for application optimization , programmers will be less dependent on the currently used `` _ _ trial - and - error _ _ '' approach . our approach considers optimization opportunities during the application life cycle comprising improvements based on static code analysis , deployment - time optimization , and run - time optimization . the developed models are used to identify the potential for improvement of the scalability for hpc applications under study and suggest application modifications that may lead to better scalability . exascale computing is not simply the continuation of a computational capability trend that has been proven true for the last five decades . first , while clock rate scaling is limited , complex multicore architectures and parallel computing still follow moore s law . second , exascale computing capability will finally allow complex real - life simulations and data analytics . the latter will greatly expand the horizons of scientific discovery and enable the new data - driven economy to become a reality . however , the exascale promise faces a series of obstacles , with the most difficult being energy , scalability , reliability and programmability . our proposal is to develop a holistic , unifying and mathematically founded framework to systematically attack the roots of these problems . that is , instead of attacking these problems separately , we propose a holistic approach to study them as a multi - parametric problem which will allow us to deeply understand their interplay and thus make the right decisions to navigate in this complex landscape . the benefits are targeting the full spectrum of actors and beneficiaries . system developers will have a much better path to design , while end users and application developers will benefit from increased scalability , performance , reliability and programmability . hpc centers will see a great increase in overall system usability and an energy budget that is affordable . this in turn has the potential to greatly limit and contain the overall impact of high end hpc to the environment . in a prospective analysis of issues with extreme scale systems @xcite , the importance of concurrency , energy efficiency and resilience of software , as well as software hardware co - design has been elucidated . focusing on energy - aware hpc numerical applications , the exa2green project ( http://exa2green-project.eu ) has developed energy - aware performance metrics @xcite , as well as energy - aware basic algorithm motifs such as linear solvers @xcite . further work will strongly benefit from these results . different power measurement interfaces available on current architecture generations have been evaluated and the role of the sampling rate has been discussed @xcite . the autotune approach @xcite employs the periscope tuning framework @xcite to automate performance analysis and tuning of hpc applications with the goal to improve performance and energy efficiency . therein , both performance analysis and tuning are performed automatically during a single run of the application . the deep project @xcite has developed a novel exascale - enabling supercomputing architecture with a matching software stack and a set of optimized grand - challenge simulation applications . the goal of the deep architecture is to enable unprecedented scalability and with an extrapolation to millions of cores to take the deep concept to an exascale level . the follow - up deeper project ( http://www.deep-er.eu ) is mainly focusing on i / o and resiliency aspects . the cresta project ( http://www.cresta-project.eu ) has adopted a co - design strategy for exascale , including all aspects of hardware architectures , system and application software . a major asset from the cresta project is the score - p measurement system @xcite on which an integration and automation of performance analysis tools ( cf . section [ subsec : runtime ] ) can be based . in addition , efforts have been made on developing a domain - specific language for expressing parallel auto - tuning specifications and an adaptive runtime support framework . exascale computing will revolutionize high - performance computing , but the first exascale systems are not expected to appear before 2020 . in this paper we have argued that the effort for preparing hpc applications for exascale should start now . we have proposed that porting of hpc applications should be made by successive , stepwise improvements based on the currently available assumptions and data about exascale systems . this approach should support application improvement each time new information about future exascale systems becomes available , including the time when the application is actually deployed and runs on a concrete exascale system . we have identified challenges that need to be addressed and recommended solutions in key areas of interest for our approach including : formal modeling , static analysis and optimization , runtime analysis and optimization , and autonomic computing . our future research will address the development of a framework that supports the conceptual framework presented in this paper . e. dhollander , j. dongarra , i. foster , l. grandinetti , and g. joubert , eds . , _ transition of hpc towards exascale computing _ , ser . advances in parallel computing.1em plus 0.5em minus 0.4emios press , 2013 , vol . 24 . department of energy selects ibm `` data centric '' systems to advance research and tackle big data challenges , '' 2014 . [ online ] . available : http://www-03.ibm.com/press/us/en/pressrelease/45387.wss s. pllana , i. brandic , and s. benkner , `` a survey of the state of the art in performance modeling and prediction of parallel and distributed computing systems , '' _ international journal of computational intelligence research ( ijcir ) _ , vol . 4 , no . 1 , pp . 1726 , january 2008 . e. b. johnsen , r. hhnle , j. schfer , r. schlatte , and m. steffen , `` abs : a core language for abstract behavioral specification , '' in _ 9th international symposium on formal methods for components and objects ( fmco10 ) _ , ser . lecture notes in computer science , vol . 6957.1em plus 0.5em minus 0.4emspringer , 2011 , pp . 142164 . e. arkin , b. tekinerdogan , and k. m. imre , `` model - driven approach for supporting the mapping of parallel algorithms to parallel computing platforms , '' in _ model - driven engineering languages and systems _ , lecture notes in computer science.1em plus 0.5em minus 0.4em springer , 2013 , vol . 8107 , pp . 757773 . s. pllana and t. fahringer , `` on customizing the uml for modeling performance - oriented applications , '' in _ uml _ , ser . lecture notes in computer science , vol . 2460.1em plus 0.5em minus 0.4emspringer , 2002 , pp . 259274 . s. pllana , s. benkner , e. mehofer , l. natvig , and f. xhafa , `` towards an intelligent environment for programming multi - core computing systems , '' in _ euro - par 2008 workshops - parallel processing _ , lecture notes in computer science , vol . 5415.1em plus 0.5em minus 0.4emspringer , 2008 , pp . 141151 . t. fahringer , s. pllana , and j. testori , `` teuta : tool support for performance modeling of distributed and parallel applications , '' in _ computational science - iccs 2004 _ , ser . lecture notes in computer science.1em plus 0.5em minus 0.4emspringer berlin heidelberg , 2004 , vol . 3038 , pp . 456463 . s. pllana , s. benkner , f. xhafa , and l. barolli , `` hybrid performance modeling and prediction of large - scale parallel systems , '' in _ parallel programming , models and applications in grid and p2p systems _ , ser . advances in parallel computing.1em plus 0.5em minus 0.4emios press , 2009 , vol . 17 , pp . 5482 . n. jansen , f. corzilius , m. volk , r. wimmer , e. brahm , j .- katoen , and b. becker , `` accelerating parametric probabilistic verification , '' in _ quantitative evaluation of systems ( qest14 ) _ , ser . lecture notes in computer science , vol . 8657.1em plus 0.5em minus 0.4emspringer , 2014 , pp . 404420 . c. dehnert , n. jansen , r. wimmer , e. brahm , and j .- katoen , `` fast debugging of prism models , '' in _ international symposium on automated technology for verification and analysis ( atva14 ) _ , ser . lecture notes in computer science , vol . 8837.1em plus 0.5em minus 0.4emspringer , 2014 , pp . 146162 . s. benkner , s. pllana , j. traff , p. tsigas , u. dolinsky , c. augonnet , b. bachmayer , c. kessler , d. moloney , and v. osipov , `` peppher : efficient and productive usage of hybrid computing systems , '' _ micro , ieee _ , vol . . 5 , pp . 2841 , sept 2011 . c. kessler , u. dastgeer , s. thibault , r. namyst , a. richards , u. dolinsky , s. benkner , j. traff , and s. pllana , `` programmability and performance portability aspects of heterogeneous multi-/manycore systems , '' in _ design , automation test in europe conference exhibition ( date ) , 2012 _ , march 2012 , pp . 14031408 . m. sandrieser , s. benkner , and s. pllana , `` using explicit platform descriptions to support programming of heterogeneous many - core systems , '' _ parallel computing _ , vol . 38 , no . 1 - 2 , pp . 5256 , january 2012 . j. dokulil , e. bajrovic , s. benkner , s. pllana , m. sandrieser , and b. bachmayer , `` efficient hybrid execution of c++ applications using intel(r ) xeon phi(tm ) coprocessor , '' _ corr _ , vol . abs/1211.5530 , 2012 . [ online ] . available : http://arxiv.org/abs/1211.5530 p. alonso , r. badia , j. labarta , m. barreda , m. dolz , r. mayo , e. quintana - orti , and r. reyes , `` tools for power - energy modelling and analysis of parallel scientific applications , '' in _ 2012 41st international conference on parallel processing ( icpp12)_.1em plus 0.5em minus 0.4emieee , 2012 , pp . 420 429 . a. bohra and v. chaudhary , `` vmeter : power modelling for virtualized clouds , '' in _ 2010 ieee international symposium on parallel distributed processing , workshops and phd forum ( ipdpsw10)_.1em plus 0.5em minus 0.4emieee , 2010 , pp . 1 8 . m. geimer , f. wolf , b. wylie , and b. mohr , `` scalable parallel trace - based performance analysis , '' in _ recent advances in parallel virtual machine and message passing interface _ , lecture notes in computer science . 1em plus 0.5em minus 0.4emspringer , 2006 , vol . 4192 , pp . 303312 . b. mohr and f. wolf , `` kojak : a tool set for automatic performance analysis of parallel programs , '' in _ euro - par 2003 parallel processing _ , ser . lecture notes in computer science.1em plus 0.5em minus 0.4em springer , 2003 , vol . 2790 , pp . 13011304 . f. wolf and b. mohr , `` automatic performance analysis of hybrid mpi / openmp applications , '' in _ eleventh euromicro conference on parallel , distributed and network - based processing_.1em plus 0.5em minus 0.4emieee , 2003 , pp . 1322 . c. bekas and a. curioni , `` a new energy aware performance metric , '' _ computer science - research and development _ , vol . 25 , no . 3 - 4 , pp . 187195 , 2010 . [ online ] . available : http://dx.doi.org/10.1007/s00450-010-0119-z p. klavk , a. c. i. malossi , c. bekas , and a. curioni , `` changing computing paradigms towards power efficiency , '' _ philosophical transactions of the royal society a _ , vol . 372 , no . 2018 , 2014 . [ online ] . available : http://dx.doi.org/10.1098/rsta.2013.0278 m. e. m. diouri , m. f. dolz , o. glck , l. lefvre , p. alonso , s. cataln , r. mayo , and e. s. quintana - ort , `` assessing power monitoring approaches for energy and power analysis of computers , '' _ sustainable computing : informatics and systems _ , vol . 4 , no . 2 , pp . 68 82 , 2014 r. miceli , g. civario , a. sikora , e. csar , m. gerndt , h. haitof , c. navarrete , s. benkner , m. sandrieser , l. morin , and f. bodin , `` autotune : a plugin - driven approach to the automatic tuning of parallel applications , '' in _ applied parallel and scientific computing _ , ser . lecture notes in computer science.1em plus 0.5em minus 0.4emspringer , 2013 , vol . 7782 , pp . 328342 . s. benedict , v. petkov , and m. gerndt , `` periscope : an online - based distributed performance analysis tool , '' in _ tools for high performance computing 2009_.1em plus 0.5em minus 0.4emspringer , 2010 , pp . n. eicker , t. lippert , t. moschny , and e. suarez , `` the deep project - pursuing cluster - computing in the many - core era , '' in _ 2013 42nd international conference on parallel processing ( icpp13 ) _ , 2013 , pp . 885892 . [ online ] . available : http://dx.doi.org/10.1109/icpp.2013.105 x. aguilar , j. doleschal , a. gray , a. hart , d. henty , t. hilbrich , d. lecomber , s. markidis , h. richardson , and m. schliephake , `` the exascale development environment : state of the art and gap analysis ( cresta white paper ) , '' 2013 , accessed 2015 - 02 - 04 . [ online ] . available : http://cresta-project.eu/
|
while the hpc community is working towards the development of the first exaflop computer ( expected around 2020 ) , after reaching the petaflop milestone in 2008 still only few hpc applications are able to fully exploit the capabilities of petaflop systems . in this paper
we argue that efforts for preparing hpc applications for exascale should start before such systems become available .
we identify challenges that need to be addressed and recommend solutions in key areas of interest , including formal modeling , static analysis and optimization , runtime analysis and optimization , and autonomic computing .
furthermore , we outline a conceptual framework for porting hpc applications to future exascale computing systems and propose steps for its implementation .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
radio - quiet quasars ( rqq ) represent more than half the known agn at high redshift ( @xmath3 ) , yet information on the galaxies that host these active nuclei is currently extremely limited . at lower redshift , the luminous rqq are found primarily in giant ellipticals with luminosities of several times a @xmath4 galaxy at @xmath5 ( e.g. mclure et al . 1999 ) . in addition , the luminosity of the host seems to correlate roughly with that of the nucleus ( mcleod & rieke 1995 ; mclure et al . 1999 , mcleod , rieke , & storrie - lombardi 1999 ) . similarly , almost every nearby bulge - dominated galaxy contains a supermassive black hole candidate whose mass is roughly proportional to the mass of its bulge ( magorrian et al . 1998 ; van der marel 1999 ) , implying a strong link between the formation and evolution of galaxies and those of the quasars and their hosts . semi - analytic hierarchical clustering models of galaxy formation have been applied by kauffman & haehnelt ( 1999 ) to this question , and they have made some specific predictions about the evolution of the nuclear magnitude host relation for radio - quiet quasars . to test such theories we have made hst nicmos observations of radio - quiet quasar hosts near the epoch of the peak quasar density ( @xmath02 3 ) . we have chosen a sample of quasars whose nuclei are faint enough to provide a good comparison sample to those of the well - studied low-@xmath6 quasars . throughout this paper we use @xmath7 = 50 km s@xmath8 mpc@xmath8 and @xmath9 = 1 . we selected 5 quasars from the faint quasar survey of zitelli et al . ( 1992 ) with 21.6@[email protected] at @xmath0 23 . the exact redshift range was constrained to avoid emission lines falling in the nicmos h filter passbands ; this resulted in a sample of 3 objects at @xmath01.8 and 2 objects at @xmath02.7 . their nuclear @xmath11 are in the range @xmath1222 to @xmath13 , making them comparable ( and somewhat fainter ) in absolute magnitude to many low-@xmath6 quasar samples ( e.g. bahcall et al . 1997 , mcclure et al . 1999 , mcleod et al . 1999 ) the observations were made using the nic2 aperture of hst s nicmos camera , which has a field size of 192 @xmath14 192 , at a scale of 0075 pixel@xmath8 . to achieve emission - line - free imaging , the @xmath15 objects were imaged in the f165 m filter , resulting in a restframe wavelength of @xmath2v , and the @xmath16 objects in the f160w filter , corresponding to rest - frame @xmath2b . we chose a nearby star for each of the 5 quasars , and observed this star in the same visit as the quasar using the identical dither pattern in order to characterize the point spread function ( psf ) . the dither pattern included half - pixel offsets to improve resolution by adequately sampling the hst psf at these wavelengths . we observed each of the 5 quasars ( and its corresponding psf star ) with two visits separated by several months , resulting in observations of each field at significantly different position angles on the sky relative to the psf pattern this allows an independent check on the reality of any residual emission seen . the final fwhms achieved were 014 016 . observing times per frame were on the order of 1500 s per quasar . total exposures for the quasars were on the order of 6000 s per visit . however , due to variable saa cr persistence problems , sky noise levels in the final images differ . we recalibrated the raw nicmos data using a modified version of the standard pipeline process . we have combined each of the two visits to each object separately , by using a simple method that determines the locations of the bad pixels in the initial frames and creates bad pixel masks using crrej , resamples the corrected frames to double the linear dimensions , and combines these using standard rejection . we achieved the cleanest subtraction with a psf star observed during the same visit as the quasar . we have made a simple , direct psf subtraction through an iterative method that varies the centering and scaling of the psf versus the quasar , subtracts the two images , and measures the chi - square of the residuals . finding an unambiguous relative centering is simple ; determining the best scaling is more subjective . here , we require flatness of the residual across some central region ( generally within a radius of @xmath2015 ) . this will still likely result in an oversubtraction , depending on how peaked the real host galaxy is within this inner radius . in these images , the psf residuals seem to dominate within a radius of @xmath201 . within this region , we do not expect to recover morphological information ; we can however find real excess flux . we have clearly resolved excess flux around 4 out of the 5 quasars relative to the psf stars , and also relative to a star with an apparent magnitude similar to those of the quasars found within the field of mzz 9592 . this field star provides a good check that our results are not the product of some difference in the way we have observed and reduced the psf stars versus the quasars . we have thus also treated this star as if it were a quasar and applied the same psf subtraction techniques to it ; we find no significant flux in the residual . in figure 1 , we show the results of these analyses for both visits of mzz 9592 , separately . we have rotated the results of the second visit to match the orientation of the first visit . in figure 2 , we give the psf - subtracted combined results for 4 of the quasars , gaussian - smoothed . this shows the range from the brightest to faintest of the residual host galaxies . to provide the simplest basis for cross - comparison between samples , we have first calculated simple aperture magnitudes . we have used an aperture of 2 ( corresponding to @xmath216 kpc ) to include most of the flux expected from a host galaxy , while excluding most nearby discrete companions . to estimate errors , we have also calculated the host magnitudes for the two visits separately . in most cases the differences were @xmath170.3 mag . we give in table 1 the results of our magnitude analysis ; we find the detected hosts vary from @xmath2l@xmath1 to 4 l@xmath1 , using @xmath18 and @xmath19 at @xmath5 from the field galaxy luminosity function of loveday et al . the faintest residual host ( around mzz 4935 ) we may barely detect at a flux @xmath20.4l@xmath1 . these fluxes will generally need a correction for flux lost from the psf subtraction process which will vary depending on how compact the intrinsic , underlying host is . in the next section we discuss simple models to estimate the amount of this correction . .mzz quasar nuclear and host properties [ cols="<,^,^,^,^,^,^ " , ] the morphologies of these hosts are quite compact , generally less than @xmath205 ( @xmath24 kpc ) effective radius . we also find in 2 cases galaxies close to the quasar in projection ( @xmath17 10 kpc ) , and in mzz 9592 an off - center host residual . ( 350,400 ) ( -80,-170 ) ( -25,360 ) ( 194,360 ) ( -25,130 ) ( 194,130 ) * figure 1 . * mzz 9592 , psf subtractions and tests . _ a. _ visit one , psf - subtracted , using the observed bright psf star from the same visit . the same image with a different scaling is shown in the lower left inset , demonstrating that the residual host is not centered on the nucleus . the upper right inset shows the unsubtracted quasar . _ b. _ visit two , psf - subtracted , rotated to match the orientation of visit one . extra sky noise comes from unremoved low - level cr residuals . _ c. _ visit one , psf - subtracted with the star that falls within the field , demonstrating that the residual is not an artifact of the mismatch between the observational strategies applied to the quasar fields and the bright psf stars . _ d. _ the field star minus the psf star : no residual flux . ( 350,500 ) ( -80,-100 ) ( -25,430 ) ( 194,430 ) ( -25,200 ) ( 194,200 ) * figure 2 . * psf - subtracted mzz quasar hosts , gaussian - smoothed with a kernel of 006 . each panel is 57 square ( or roughly 45 kpc ) , n up , e left . _ a. _ mzz 9592 , @xmath20 2.7 , with an inset of the central region ( unsmoothed ) . there is an off - center residual host component . _ b. _ mzz 9744 , @xmath0 1.8 _ c. _ mzz 1558 , @xmath21 2.7 _ d. _ mzz 4935 , @xmath201.8 , host not detected . note the apparent close ( @xmath2 10 kpc ) companion galaxies in panels _ b _ and _ c_. to compare the results of our study with studies of lyman break galaxies and radio galaxies at similar redshifts , we have made some simple simulations of the effect the presence of the quasar nucleus and the psf - subtraction process have on the derived magnitudes and morphologies . we have used for the first set of these simulations hst nicmos observations of a number of spectroscopically identified @xmath0 2 3 lyman break galaxies from the hdf north ( dickinson et al . 1999 ) , which were observed in the f160w filter to a much greater depth than our observations . for the observed galaxies that were at the same redshift as our quasars , we first resampled these imaging data to match our final pixel scale , added `` quasar nuclei '' of varying magnitudes at the center of the galaxy by scaling and adding the mzz9592 field star , then added poissonian noise to the appropriate level . we then used one of the observed psf stars to run the same automatic psf - subtraction process that we used on the quasars . we have found that the subtraction process in most cases on these lyman break galaxies resulted in a loss of flux of @xmath20.3 magnitude , giving us an indication of the amount to correct our derived hosts . even after such a correction , most of our quasar hosts have relatively moderate total magnitudes of @xmath2l@xmath1 , and these magnitudes and compact sizes are basically consistent with the magnitudes and compactness of star - forming galaxies at similar epochs ( dickinson et al . 1999 ) . to enable a comparison with previous ground - based infrared imaging work on radio - loud galaxies , we adjust our fluxes to match those of eales et al . 1997 , who studied 6c radio galaxies up to @xmath03 . extrapolating our h band fluxes to the k band gives k band magnitudes of 22.0 , 20.3 , and 20.5 for the @xmath201.8 quasar hosts , and 20.1 and 20.8 for the @xmath02.7 quasar hosts ( after correction to the larger aperture used by eales et al . ) . they found that @xmath22 6c galaxies have a median @xmath23 magnitude of @xmath218.5 , while our 3 quasar hosts are at least a magnitude fainter than this , even allowing for a half magnitude correction to our fluxes lost in the subtraction process . our @xmath02.7 quasar hosts are also a magnitude fainter at k than the 6c objects . as the quasars in our sample have nuclear magnitudes that are comparable to those of the well - studied samples of low redshift quasars , we can make a direct comparison , and find that these @xmath22 3 rqq hosts are of similar or fainter m@xmath24 as those of the sample of bahcall et al . ( 1997 ) and mclure et al . ( 1997 ) , for example . however , if the host galaxies of luminous radio - quiet quasars evolve passively into giant ellipticals today , then at high redshift they should have had similar host magnitudes to the radio galaxies , rather than the moderately faint hosts we have found . our results are consistent with other recent results on some brighter , lensed quasars ; rix et al . 1999 find that their sample of lensed @xmath22 radio - quiet quasars ( de - magnified @xmath25 @xmath2 24 28 ) also had hosts with comparably faint magnitudes . though inconsistent with passive evolution , our finding l@xmath1 hosts at @xmath22 3 agrees fairly well with the bottom - up hierarchical galaxy formation models of kauffman & haehnelt ( 1999 ) ; they predict median host luminosities that are somewhat below present day l@xmath1 for quasars at @xmath26 ( and even fainter at @xmath27 ) , for quasars with the nuclear magnitudes of our sample . these hosts might therefore still be undergoing major mergers which would allow them to evolve into the present day ges associated with low-@xmath6 quasars ; it is unclear in this interpretation , however , why the radio galaxies do not undergo similar mergers and evolution . conversely , these high-@xmath6 quasar hosts could be l@xmath1 galaxies that will not significantly evolve in luminosity . bahcall , j. , kirhakos , s. , saxe , d. , & schneider , d. 1997 , apj , 479 , 642 dickinson et al . 1999 , in preparation . eales , s. , rawlings , s. , law - green , d. , cotter , g. , & lacy , m. 1997 , mnras , 291 , 593 kauffmann , g. & haehnelt , m. 1999 , mnras , submitted ( astro - ph/9906493 ) loveday , j. , peterson , b. , efstathiou , g. , maddox , s. 1992 , apj , 390,338 . magorrian j. , et al . , 1998 , aj , 115 , 2285 mclure r.j . , dunlop j.s . , kukula m.j . , baum s.a . , odea c.p . , hughes d.h . , aj , submitted ( astro - ph/9809030 ) mcleod , k. , & rieke , g. 1995 , apj , 454 , l77 mcleod , k. , rieke , g. , & storrie - lombardi , l. 1999 , apjl , 511 , l67 van der marel r.p . , 1999 , aj , 117 , 744 zitelli , v. , mignoli , m. ; zamorani , g. ; marano , b. ; boyle , b. j. , 1992 , mnras , 256 , 349
|
we will present the results of a nicmos h - band imaging survey of a small sample of @xmath02 3 radio - quiet quasars .
we have resolved extension in at least 4 of 5 objects and find evidence for a wide range in the morphologies and magnitudes of these hosts .
the host galaxy luminosities range from sub - l@xmath1 to about 4 l@xmath1 , with most of the hosts having luminosities about l@xmath1 .
these host galaxies have magnitudes and sizes consistent with those of the ly break galaxies at similar redshifts and at similar rest wavelengths , but are about a magnitude fainter than the comparable 6c radio galaxies .
one residual host component is not centered on the quasar nucleus , and several have close companions ( within @xmath210 kpc ) , indications that these systems are possibly in some phase of a merger process .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent work @xcite ( see also @xcite ) we have analysed some aspects of the cauchy problem for the einstein equations with data on a characteristic cone in all dimensions @xmath0 , see @xcite and references therein for previous work on the subject . in this note we apply the results derived in @xcite to present an existence theorem for this problem , with initial data which approach rapidly the flat metric near the tip of the light cone , see theorem [ t23vii.1 ] below . the reader s attention is drawn to @xcite , where sets of unconstrained data on a light - cone centered at past timelike infinity are given in dimension @xmath1 . it is well known that by using normal coordinates centred at @xmath2 the characteristic cone @xmath3 of a given lorentzian metric can be written , at least in a neighbourhood of @xmath2 , as a cone in minkowski spacetime whose generators represent the null rays . it is therefore no geometric restriction to assume that the characteristic cone of the spacetime we are looking for is represented in some coordinates @xmath4 , @xmath5 , @xmath6 of @xmath7 by the equation of a minkowskian cone with vertex @xmath2 , @xmath8 the parameter @xmath9 is an affine parameter when normal coordinates are used , and it is also going to be an affine parameter in the solutions that we are going to construct . coordinates @xmath10 as above , which moreover satisfy the wave - equation , @xmath11 , will be called _ normal - wave _ coordinates . given a smooth metric , such coordinates can always be constructed ( see @xcite or @xcite ) by solving the wave equation in the domain of dependence of @xmath3 , with initial data the normal coordinates on @xmath3 . given a smooth metric , one obtains a coordinate system near the vertex , which suffices for our purposes . the coordinates @xmath12 will be normal - wave coordinates for the solution which we aim to construct . given the coordinates @xmath13 we can define coordinates @xmath14 on @xmath7 by setting @xmath15 with @xmath16 local coordinates on @xmath17 . the null geodesics issued from @xmath2 have equation @xmath18 , @xmath19constant , so that @xmath20 is tangent to those geodesics . on @xmath21 ( but not outside of it in general ) the spacetime metric @xmath22 that we attempt to construct takes the form ( we put an overbar to denote restriction to @xmath3 of spacetime quantities ) @xmath23 where @xmath24 are respectively an @xmath25-dependent scalar , one - form , and riemannian metric on @xmath17 . the symbol @xmath9 will be used interchangeably with @xmath26 . to avoid ambiguities , we will write @xmath27 for the components of a metric tensor in the @xmath28 coordinates , and @xmath29 for the components in the coordinate system @xmath10 . this convention will be used regardless of whether @xmath22 is defined on the space - time , or only on the light - cone . the analysis in @xcite uses a wave - map gauge , with minkowski target @xmath30 , with the light - cone of @xmath30 being the image by the wave - map @xmath31 of the light - cone of the metric @xmath22 that one seeks to construct . quite generally , a metric @xmath22 on a manifold @xmath32 will be said to be _ in @xmath33-wave - map gauge _ if the identity map @xmath34 is a harmonic diffeomorphism from the spacetime @xmath35 onto the pseudo - riemannian manifold ( @xmath36 . recall that a mapping @xmath37 is a harmonic map if it satisfies the equation , in abstract index notation , @xmath38 in a subset in which @xmath31 is the identity map defined by @xmath39 , the above equation reduces to @xmath40 , where the _ wave - gauge vector _ @xmath41 is given in arbitrary coordinates by the formula @xmath42 where @xmath43 are the christoffel symbols of the _ target _ metric @xmath33 . see @xcite for a more complete discussion . there are various ways of choosing free initial data for the cauchy problem for the vacuum einstein equations on the light - cone @xmath3 . in this work we choose as initial data a one - parameter family , parameterized by @xmath9 , of conformal classes of metrics @xmath44 $ ] on @xmath17 , thus @xmath45 is assumed to be conformal to @xmath46 , and where @xmath9 will be an affine parameter in the resulting vacuum space - time . the initial data needed for the evolution equations are the values of the metric tensor on the light - cone , which will be obtained from @xmath46 by solving a set of wave - map - gauge constraint equations derived in @xcite , namely equations ( [ 3v.1 ] ) , ( [ 5vi.13 ] ) , ( [ xiabis])-([cafinal2 ] ) and ( [ 19viii.1 ] ) below . the main issue is then to understand the behaviour of the fields near the vertex of the light - cone , making sure that the @xmath47-coordinates components of the metric @xmath48 , obtained by solving the wave - map - gauge constraints , can be written as restrictions to the light - cone of sufficiently smooth functions on space - time , so that the pde existence theorem of @xcite can be invoked to obtain the vacuum space - time metric . it is convenient to start with some notation . for @xmath49 and @xmath50 we shall say that a tensor field @xmath51 , of valence @xmath52 , defined for @xmath53 , is @xmath54 if there exists a constant @xmath55 such that @xmath56 where @xmath57 are coordinate components of @xmath58 , in a coordinate system which will be made clear as needed . [ t23vii.1 ] let @xmath60 , @xmath61 , @xmath62 , @xmath63 , @xmath64 , @xmath65 $ ] . suppose that there exist smooth tensor fields @xmath66 on @xmath17 , @xmath67 , so that , in local charts on @xmath17 , the coordinate components @xmath68 satisfy @xmath69 then : 1 . there exist functions @xmath70 such that @xmath71 if there exists @xmath72 so that @xmath73 for @xmath74 , then @xmath75 for @xmath76 . if moreover @xmath77 then there exists a @xmath78 lorentzian metric defined in a neighbourhood of the vertex of @xmath3 , with @xmath79 $ ] , which is vacuum to the future of @xmath2 . roughly speaking , the index @xmath80 is the final sobolev differentiability of the solution . the ranges of indices above are only needed for the second part of the theorem , and arise from the fact that dossa s existence theorem @xcite requires initial data which are of @xmath81 differentiability class in coordinates regular near the vertex . there is a loss of three derivatives when going from the free data @xmath68 to the full initial data @xmath82 , which brings the threshold up to @xmath83 . finally , the existence argument invokes the bianchi identity , which in its natural version requires a @xmath84 metric ; together with the sobolev embedding , this leads to the restriction @xmath62 . straightforward taylor expansions at @xmath2 show that a smooth metric on a space - time @xmath85 will lead to the form ( [ 30vi.5x ] ) of @xmath44 $ ] , with @xmath80 , @xmath86 and @xmath52 which can be chosen at will , and with @xmath87 . so ( [ 30vi.5x ] ) is necessary in this sense . in view of our theorem above , to obtain a complete solution of the problem at hand it remains to provide an exhaustive description of those @xmath44 $ ] s that lead to ( [ 23vii.5 ] ) . we conjecture that ( [ 23vii.5 ] ) will hold for all @xmath46 s arising from the restriction of a smooth metric to a light - cone , where @xmath9 is an affine parameter . a similar problem arising on the null cone at past infinity in dimensions @xmath1 has been solved by friedrich in @xcite , but no details have been presented . proof of theorem [ t23vii.1 ] : we need to analyze the behaviour of the solutions of the wave - map - gauge constraint equations near the vertex of the cone . we start by noting that , for some smooth functions @xmath95 on @xmath17 , @xmath96 where @xmath97 is the matrix inverse of @xmath68 . further , for some smooth tensors @xmath98 on @xmath17 , @xmath99 note that @xmath100 is the @xmath59trace - free part of @xmath101 . this leads to , for some functions @xmath102 on @xmath17 , @xmath103 where @xmath104 has been defined in the left - hand side of . let @xmath105 where @xmath106 is the divergence of @xmath3 ( sometimes denoted by @xmath107 in the literature ; cf . , e.g. , @xcite ) : @xmath108 in terms of @xmath12 , the vacuum raychadhuri equation @xmath109 , where @xmath110 is a null tangent to the generators of @xmath3 , reads @xmath111 using known arguments ( compare @xcite and ( * ? ? ? * lemma 8.2 ) ) , there exist functions @xmath112 such that @xmath113 we rewrite this as @xmath114 so that @xmath115 using this last formula , it is easy to establish existence of functions @xmath116 such that @xmath117 let us write @xmath118 from @xmath119 there exist functions @xmath120 such that : @xmath121 equation ( [ 30vi.8 ] ) implies that there exist smooth tensor fields @xmath122 on @xmath17 such that @xmath123 subsequently , @xmath124 for some tensor fields fields @xmath125 . it is a consequence of the affine - parameterization condition and of the wave - gauge conditions that the function @xmath126 solves the equation ( see @xcite for details ) @xmath127 set @xmath128 then ( [ 5vi.13 ] ) integrates to @xmath129 where @xmath130 and @xmath131 we find successively @xmath132 for some @xmath133 . closer inspection shows that @xmath134 . integrating ( [ 30vi.12 ] ) , we conclude that there exist functions @xmath135 such that @xmath136 if @xmath137 the sum from four to @xmath138 is understood as zero . one has a similar formula for @xmath139 . we pass now to a vector @xmath140 , defined as @xmath141 here the @xmath142 s are the christoffel symbols of the canonical metric @xmath59 on @xmath17 , @xmath143 is as in ( [ wavegauge0 ] ) , the @xmath144 s are the christoffel symbols of the @xmath145-dimensional metric @xmath146 , with associated covariant derivative operator @xmath147 , and @xmath148 , while @xmath149 in terms of @xmath150 the equation @xmath151 in wave - map gauge reads @xmath152 where @xmath153 . from what has been proved so far this last equation can be written as @xmath154 for some @xmath155 . the unique solution of this equation with the relevant behaviour at the origin satisfies @xmath156 for some @xmath157 . viewing ( [ xiabis ] ) as an equation for @xmath158 , we obtain @xmath159 for some @xmath160 . we finally need to integrate the third wave - map - gauge constraint , @xmath161 , where @xmath162 is the einstein tensor : @xmath163 we need the formula , for some functions @xmath164 , @xmath165 and the result , for a minkowski target , @xmath166 with @xmath167 . hence @xmath168 for some functions @xmath169 . a straightforward analysis of the remaining terms in ( [ 19viii.1 ] ) shows that solutions with the required asymptotics satisfy , for some functions @xmath170 , @xmath171 in order to apply dossa s existence theory from @xcite to the wave - map gauge reduced einstein equations , we transform the metric to coordinates @xmath172 , regular near the tip of the light - cone , defined as @xmath173 where @xmath174 . in these coordinates our wave - map - reduced einstein equations become the usual harmonically - reduced equations , which have the right structure for the existence results in @xcite . one finds that there exist functions @xmath175 such that @xmath176 this proves the first part of theorem [ t23vii.1 ] . now , for general @xmath66 s the sum @xmath177 will _ not _ be the restriction of a polynomial to the light - cone . assume , however , that this is the case . then the remainder term in the difference @xmath178 will be of the differentiability class @xmath179 with @xmath180 , as required by dossa for existence @xcite , provided that @xmath181 the solution @xmath182 is then in dossa s space @xmath183 for some @xmath184 , which embeds into @xmath185 leading to , for small @xmath186 , @xmath187 in the coordinates @xmath172 the harmonicity vector can be calculated using the usual formula , @xmath188 which has bounded components in the @xmath172 coordinate system near the tip of the cone . the bianchi identity ( which , in its simplest version , requires a @xmath84 metric ; this raises the differentiability threshold to @xmath62 ) together with the arguments in @xcite show then that @xmath189 and thus the solution of the wave - map reduced einstein equations , obtained from dossa s theorem , is also a solution of the vacuum einstein equations . @xmath190 acknowledgements : ptc and ycb are grateful to the mittag - leffler institute , djursholm , sweden , for hospitality and financial support during part of work on this paper . they acknowledge useful discussions with vincent moncrief , as well as comments from roger tagn wafo . ycb wishes to thank thibault damour for making available his detailed manuscript calculations in the case @xmath191 leading to equations ( 22 ) of @xcite . jmm thanks oxpde for hospitality . rendall , _ the characteristic initial value problem for the einstein equations _ , nonlinear hyperbolic equations and field theory ( lake como , 1990 ) , pitman res . notes math . ser . , vol . 253 , longman sci . harlow , 1992 , pp .
|
we prove an existence theorem for the cauchy problem on a characteristic cone for the vacuum einstein equations .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it has recently been pointed out @xcite that many phenomena taking place in the sun , especially in its corona and in its magnetic field , are far from being completely understood . the production by the sun of axion like particles ( alps ) and their subsequent interactions in the solar environment could provide a key to interpreting the physical mechanisms underlying these phenomena . these , and other considerations led to starting a search with the cast magnetic helioscope @xcite for hypothetical alps emitted by the sun in the energy range below 100 ev . the first step of this search , which will be reported here , has involved looking for 2 - 4 ev photons produced in the cast magnet bore by the primakoff @xcite conversion into photons of solar alps in the latter energy range . the short term objective was to efficiently couple a detector system sensitive in the ev energy range to a cast magnet bore and evaluate its background in normal operating conditions . the long term objective of the effort is attempting to detect , using sensors with the appropriate spectral sensitivity and good enough background , `` low''-energy ( tens of ev s ) photons generated in the cast helioscope by possible interactions of low - energy solar alps . we will briefly describe the detector system , which has been developed for this purpose under the barbe project financed by the italian istituto nazionale di fisica nucleare ( infn ) , along with the coupling of this system to the cast magnet . finally , the data taking campaigns will be discussed and a summary of the data presented . the starting idea of the barbe project is to begin with readily available photon detectors sensitive in the visible range , test them in ideal laboratory conditions , and then design and build an optical system to couple the detectors to one of the bores of the cast magnet . the devices used in this initial phase were a photomultiplier tube ( pmt ) ( model 9893/350b made by emi - thorn ) and an avalanche photodiode ( apd ) ( model id100 - 20 made by idquantique ) . the pmt had an active area dia . of 9 mm , with peak sensitivity at 350 nm . the apd had an active area dia . of 20 @xmath0 m and peak sensitivity at 500 nm . for both detectors the dark count rate ( dcr ) measured in the laboratory during preliminary tests was about 0.4 hz . to measure the dcr , the pmt was biased at 1950 v and instrumented with an electronic readout chain consisting of nim standard modules . the apd readout module gave a 2 v , 10 ns wide , output signal which was trasformed into a ttl pulse via a custom circuit . both detectors were cooled by means of their built in peltier coolers , operating at -20 @xmath1c , and were illuminated by a suitably attenuated blue led source . to demonstrate single photon operation , the count histogram for each detector was fitted with a poissonian distribution with average equal to 1 . the dcr was then obtained after cutting the electronic noise pedestal using the fitted curve . the basic elements of the coupling system were a galileian telescope , a 40 m multimode optical fiber complete with input collimator and an optical switch ( mod . 1x2 made by leoni ) . the galileian telescope consisted of a 2 inch dia . , f = 200 mm , convex lens and of a 1 inch dia . , f = - 30 mm , concave lens and was designed to optically couple the 40 mm dia . bore of the cast magnet into the 9 mm dia . input collimator of the multimode fiber . the telescope was mounted directly onto one of the sunset side ports of the cast magnet . the detectors were placed far away from the magnet in the cast experimental hall . the optical switch , which can be triggered by external ttl pulses , was used in order to share the light coming from the fiber between the pmt and the apd : each detector could then look at the magnet bore for 50% of the time and at the background for the remaining 50% . figure [ fig : figure1 ] shows a block - diagram of the layout of the system as mounted on cast . the overall light collection efficiency was about 50% for the pmt and less than 1% for the apd . the low efficiency of the apd channel was due to unresolved focussing difficulties . since the apd data are of inferior quality , only the pmt measurements are considered . the total number of counts in each measurement is affected by afterpulses generated either by the pmt itself or by its readout electronic chain . it was found that afterpulses account for 11% of total counts . to eliminate the effect of the afterpulses the mean rate of counts is calculated , fot both light and dark counts by solving for @xmath2 the equation @xmath3 , where x is the channel number , a is the total number of occurences in all channels , @xmath4 is the number of occurences in the x - th channel measured experimentally . in this way occurrences in channel 0 are not affected by afterpulses . two measurement campaigns were conducted . the first one in november 2007 with the telescope attached to the v2 port of the cast magnet , while the second one was in march 2008 with the telescope on the v1 port . r0.75 in the first run each detector , pmt and apd , looked at the magnet bore for 50% of the time and at the background for the other 50% . environmental checks and background measurements in different magnet positions and with field on and off were performed . a total of 45000 s of `` live '' data with the magnet on were taken , 10000 s of dummy solar tracking data and 35000 s of actual sun tracking data . the second run was conducted using the pmt only , but keeping the switching system in order to have again the detector share its live time equally between signal and background . in this case 45000 s of live data were taken , of which 5000 s of dummy solar tracking data , 20000 s of actual solar tracking data and 20000 s of data with the magnet pointing off the sun center ( 10000 s pointing 0.25@xmath1 to the right and 10000 s pointing 0.25@xmath1 to the left ) . figure [ fig : figure2 ] shows a plot of the difference of the measured average count rates between `` light '' , when the pmt was looking at the magnet bore , and `` dark '' , when the optical switch was toggled on the other position . the abscissa axis refers to the different conditions in which data were taken . taking into account the 1 @xmath5 error bars reported in the plot , no statistically significant difference is found between `` light '' and `` dark '' count rates . the average background count rate measured for 3 - 4 ev photons during solar tracking was 0.35 @xmath6 0.02 hz for a total of 75000 s of data . the measurement runs conducted with the barbe detector system demonstrated that it is possible to couple two detectors to the cast magnet via an optical fiber , while preserving a reasonable light collection efficiency ( 50% in the pmt case , that corresponds to 10% of overall system efficiency when taking into account the pmt spectral response curve ) , and without introducing additional noise sources . in 12 data sets taken during solar tracking ( including 2 sets pointing off center ) the background count rate for 3 - 4 ev photons was 0.35 @xmath6 0.02 hz and no significative excess counts over background were observed . this is the first time such a measurement has been done with an axion helioscope . the challenge is now to progress to a new detector(s ) with lower intrinsic background , possibly extending the spectral sensitivity to other regions of the energy interval below 100 ev . in the case of visible photons , one could also envision enclosing the cast magnet bore in a resonant optical cavity in order to enhance the axion - photon conversion probability @xcite . this would however require solving rather complex compatibility problems with the rest of the cast apparatus . three types of detectors have at this moment been considered for future developments , a transition edge sensor ( tes ) @xcite , a silicon sensor with depfet readout @xcite and an apd cooled to liquid nitrogen temperatures . the tes sensor promises practically zero background , spectroscopic capability and sensitivity from less than 1 ev up to tens of ev s . it however requires operation at 100 mk and it has a small sensitive area ( about @xmath7 ) . the depfet sensors could reach a very low background if used in the repetitive non destructive readout ( rndr ) mode , however they also have a small sensitive area . finally , the cooled apd could be operated relatively easily if one accepts afterpulsing events , which should not pose a problem in a low expected rate environment . on the other hand , the sensitive area would only be about 300 @xmath8 and the spectral sensitivity limited to the visible region . the present plan is to initially pursue all three possibilities in the hope of identifying the one where progress is faster . funding for the barbe project has been provided by infn ( italy ) . the support by the gsrt in athens is gratefully acknowledged . this research was partially supported by the ilias project funded by the eu under contract eu - rii3-ct-2004 - 506222 . special thanks to m. karuza , v. lozza and g. raiteri of infn trieste for their excellent detector work , and to the entire cast collaboration for wonderful support and assistance . we also warmly thank the cern staff for their precious help . 99 see for instance k. zioutas , `` the enigmatic sun : a crucible for new physics '' , cern courier , may 2008 . s. andriamonje _ et al . _ [ cast collaboration ] , jcap * 0704 * ( 2007 ) 010 [ arxiv : hep - ex/0702006 ] ; kuster , m. _ et al . _ , new j. phys . , * 9 * ( 2007 ) 169 ; abbon , p. _ et al . _ , new j. phys . * 9 * ( 2007)170 ; autiero , d. _ et al . _ , new j. phys . * 9 * ( 2007 ) 171 ; zioutas , k. _ et al . _ [ cast collaboration ] , phys . * 94 * ( 2005 ) 121301 . see for instance g. raffelt and l. stodolsky , phys . d * 37 * ( 1988 ) 1237 - 1249 . p. sikivie , d. b. tanner and k. van bibber , phys . * 98 * ( 2007 ) 172002 [ arxiv : hep - ph/0701198 ] . b. cabrera , j low temp phys * 151 * ( 2008 ) 82 - 93 ; d. bagliani et al . , j low temp phys * 151 * ( 2008 ) 234 - 238 . j. kemmer and g. lutz , nucl . instrum . meth . a * 253 * ( 1987 ) 365 ; p. fischer _ et al . _ , nucl . instrum . a * 582 * ( 2007 ) 843 .
|
we have started the development of a detector system , sensitive to single photons in the ev energy range , to be suitably coupled to one of the cast magnet ports .
this system should open to cast a window on possible detection of low energy axion like particles emitted by the sun .
preliminary tests have involved a cooled photomultiplier tube coupled to the cast magnet via a galileian telescope and a switched 40 m long optical fiber .
this system has reached the limit background level of the detector alone in ideal conditions , and two solar tracking runs have been performed with it at cast .
such a measurement has never been done before with an axion helioscope .
we will present results from these runs and briefly discuss future detector developments .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the munu experiment was designed to study @xmath1 scattering at low energy , probing in particular the magnetic moment of the neutrino . the detector was set up near a nuclear power reactor in bugey ( france ) serving as antineutrino source . we report here on an analysis of data corresponding to 66.6 days of live time reactor - on and 16.7 days reactor - off . many experiments now show that neutrinos have masses , and that the weak eigenstates @xmath2 are superpositions of the mass eigenstates @xmath3 : @xmath4 intense work is being done to determine more precisely the mixings @xmath5 and the masses @xmath6 . besides masses , neutrinos can have magnetic moments . as shown in ref . @xcite , the fundamental magnetic moments are associated with the mass eigenstates , in the basis of which they are represented by a matrix @xmath7 ( @xmath8 ) . dirac neutrinos can have both diagonal and off - diagonal ( transition ) moments , while off diagonal moments only are possible with majorana neutrinos . astrophysical considerations put strong constraints on dirac neutrinos , and less strong ones on majorana neutrinos ( see @xcite ) . upper limits applying in both cases from stellar cooling of order @xmath9 have been derived . magnetic moments large enough would lead to spin - flavor precession in the toroidal magnetic field in the interior of the sun , as discussed for instance in @xcite , further complicating the oscillation pattern of solar neutrinos resulting from masses and mixings . these astrophysical bounds on magnetic moments , however , depend on various assumptions and , to a certain extent , are model dependent . direct measurements are under much better control . so far the best limits stem from experiments studying @xmath10 or @xmath0 scattering , and looking for deviations of the measured cross - section from the one expected with weak interaction alone . the cross - section is given by @xmath11 \nonumber\\ & & \mbox { } + \frac{\pi \alpha^{2}\mu_{e}^{2}}{m_{e}^{2}}\frac{1-t / e_{\nu } } { t } \end{aligned}\ ] ] with the contribution of the weak interaction in the first two lines , and that from the magnetic moment @xmath12 in the last one @xcite . here @xmath13 is the incident neutrino energy , @xmath14 the electron recoil energy , and the couplings are given by @xmath15 the relative contribution of the magnetic moment term increases with decreasing neutrino and electron energies . therefore it is essential to have a low electron detection threshold , looking for neutrinos from a low energy source . so far the sun and nuclear reactors have been used . future experiments with radioactive sources are planned . the measured squared magnetic moment @xmath16 depends on the mixings and , in case of a large distance @xmath17 from the source to the detector , on the propagation properties of neutrinos . for vacuum oscillations it is given by : @xmath18 with @xmath19 the momentum of @xmath20 with mass @xmath21 , for a given neutrino energy @xmath13 . in the case of matter enhanced oscillations the dependence of the propagation eigenstates on the local density must be taken into account @xcite . here the source to detector distance is long , oscillations are relevant and the measured quantity is denoted @xmath22 . super - kamiokande has measured with a very high statistical accuracy the electron recoil spectrum from the scattering of solar @xmath23b neutrinos @xcite . the total rate is reduced because of neutrino oscillations , but the shape is seen to be in good agreement , within statistics , with that expected assuming weak interaction alone . beacom and vogel @xcite looked to what extent excess counts at the low energy end from a magnetic moment can be ruled out . the assumption is made that such an excess is not compensated by a distortion due to oscillations . the limit @xmath24 @xmath25 at 90 % confidence level ( cl ) was derived . nuclear reactors are strong sources of @xmath26 with energies ranging up to about 8 mev . these are essentially produced in the beta decay of fission fragments , with however a significant contribution at low energy from nuclei activated by neutrons . above 1.5 - 2.0 mev the integral beta spectrum of the fission fragments of the isotopes in which fission is predominantly induced by thermal neutrons ( @xmath27u , @xmath28pu , @xmath29pu ) was measured . these isotopes dominate . a simple procedure leads to the corresponding neutrino spectra , which are thus known well , with a precision of order 5 % or better @xcite . in @xmath30u , fission is induced by fast neutrons , and no such measurements are available . less precise calculations ( @xmath3110 % ) only can be used . but this isotope contributes only about 6 - 7 % in a conventional reactor . the sum spectrum above 2 mev can be reconstructed with a precision of order @xmath325 % , knowing the relative contributions of the various fissile isotopes at a given reactor . studies of @xmath33 scattering at reactors @xcite , which sample the antineutrino spectrum above 2.5 mev , show good agreement with these predictions . below 1.5 - 2 mev the neutrino spectrum can only be reconstructed from inherently less precise calculations as shown in ref . @xcite . in that work it is mentioned that neutron activations of the fissile isotopes , leading in particular to @xmath28u , @xmath28np and @xmath34u , and of the fission products , contribute . the yields have been estimated . in reactor experiments the source to detector distance is short compared to oscillation lengths , and the magnetic moment searched for @xmath35 is given by eq . ( [ mudef ] ) with l set to zero . the irvine group was the first to observe @xmath0 scattering @xcite . the experiment was performed at the savannah river reactor , with a 16 kg plastic scintillation counter surrounded by a nai veto at 11 m from the reactor core . a reactor - on minus reactor - off signal was seen , with a threshold at 1.5 mev . analyzing the electron recoil data with the most recent knowledge of the weinberg angle and the reactor spectrum , vogel and engel @xcite found a slight excess of events which can be explained by a magnetic moment @xmath35 of order @xmath36 @xmath25 . also @xmath0 scattering was studied at the rovno reactor by a group from saint petersburg . the detector consisted of a stack of silicon sensors , with a total mass of 75 kg . a reactor - on minus reactor - off signal was seen above 600 kev , with a signal to background ratio of order 1:100 . the limit @xmath37 @xmath25 at 95(68 ) % cl was reported @xcite . more recently the texono collaboration installed an ultra low background high purity germanium detector , with a fiducial mass of 1.06 kg , near the kuo - sheng reactor in taiwan @xcite . here the approach is somewhat different . the threshold on the electron recoil ( 12 kev ) is extremely low . the reactor - on and reactor - off spectra were found to be identical within statistical errors . from that the limit @xmath38 @xmath25 at 90(68 ) % cl was derived . in the aforementioned experiments the energy of the recoil electron candidate only was measured . the munu collaboration @xcite has built a detector of a different kind , in which the topology of events is recorded . this allows a better event selection , leading to a lower background . moreover , in addition to the energy , the initial direction of an electron track can be measured . a second parameter , the electron scattering angle , can therefore be reconstructed . this allows to look for a reactor signal by comparing forward electrons , having as reference the reactor to detector axis , with the backward ones . the background is measured on - line , which eliminates problems from detector instabilities , as well as from a possible time dependence of the background itself . the detector is described in details in ref . @xcite , and we only present here the essential features . the detector , made from radiochemically clean materials , was installed at 18 m from the core of a commercial reactor in bugey ( france ) with a power of 2750 mwth . it emits neutrinos from the fission fragments of @xmath27u ( 54 % on average over an annual reactor cycle ) , @xmath28pu ( 33 % ) , @xmath29pu ( 6 % ) , and @xmath30u ( 7 % ) . the central component of the munu detector consists of a time projection chamber ( tpc ) filled with 3 bar of cf@xmath39 gas ( figure [ fi : munu0 ] ) , acting as target and detector medium for the recoil electron . cf@xmath39 was chosen because of its relatively low z , leading to reduced multiple scattering , its absence of protons , which eliminates backgrounds from @xmath40 scattering , its good drifting properties and its high density . as shown in figure [ fi : munu0 ] the gas is contained in a cylindrical acrylic vessel of 1 m@xmath41 volume ( 90 cm in diameter , 162 cm long , total cf@xmath39 mass 11.4 kg ) , which is entirely active . the drift field parallel to the tpc axis ( z - axis ) is defined by a cathode on one side , a grid on the other side , and field shaping rings at successive potentials outside the acrylic vessel . the voltages were set to provide a homogeneous field leading to a drift velocity of 2.14 @xmath42s@xmath43 . an anode plane with 20 @xmath44 m wires and a pitch of 4.95 mm , separated by 100 @xmath44 m potential wires , is placed behind the grid , to amplify the ionization charge . the integrated anode signal gives the total energy deposit . a pick - up plane with perpendicular x and y strips ( pitch 3.5 mm ) behind the anode provides the spatial information in the x - y plane perpendicular to the z - axis . the spatial information along the z - axis is obtained from the time evolution of the signal . to reduce systematics when comparing forward and backward events , the tpc was positioned orthogonally to the reactor - detector axis , which moreover coincides with the bisecting line between x and y strips on the pick - up plane . the anode wires are rotated by 45@xmath45 with respect to the x - y plane . the tpc is thus absolutely symmetric between backward and forward directions with regards to the reactor - detector axis . the imaging capability of the tpc is illustrated in figure [ fi : munu_track ] which shows an electron track . [ fi : munu_track ] the energy resolution and calibration is obtained by comparing spectra measured using various @xmath46 sources ( @xmath47cs ( 662 kev ) , @xmath48mn ( 835 kev ) and @xmath49na ( 1274 kev ) ) with simulations . the relative energy resolution is found to be 8 % ( 1 @xmath50 ) at 1 mev , correcting for small variations across the anode plane . it scales with the power 0.7 of the energy , rather than with the square root . we think that this is because the electron attachement in cf@xmath39 in the strong field around the anode wires is rather high , affecting the statistics in the avalanche @xcite . the gain stability is monitored regularly throughout the data taking period , with sources and by measuring the spectrum of cosmic muons crossing the tpc . the data are corrected for small instabilities . the angular resolution is around 10@xmath45 ( 1 @xmath50 ) at 1 mev , for tracks scanned visually , as derived from monte - carlo simulations , with a slight angular dependence @xcite . the energy dependence is only weak above 700 kev . the acrylic vessel is immersed in a steel tank filled with liquid scintillator and viewed by photomultipliers , 24 on each side . it acts as a veto counter against the cosmics and as an anti - compton detector to reduce the background from @xmath46 s . the scintillator also sees the light produced by the avalanche around the anode wire , providing an additional measurement of the ionisation charge . the primary light of heavily ionizing particles such as @xmath51 s is seen as well . a few @xmath51 s are observed from remaining surface contaminations on the cathode . the primary light of minimum ionizing particles confined in the gas volume however is below detection threshold . the tpc threshold is set at 300 kev , and that in the scintillator at 100 kev , with a minimum of 5 photomultipliers hit . we note that the singles rates in the photomultipliers show no azimuthal dependence , which could result from hot spots . in normal data taking tpc events above threshold are read out , provided they are not in coincidence with a 200 @xmath44s signal started by scintillator pulses above 22 mev . this eliminates direct cosmic hits , or neutrons associated with them . the 48 photomultipliers are read out first . the light from the photomultipliers on the anode side and the cathode side is compared . reading proceeds only if the relative difference is less than @xmath3230 % . events corresponding to real tracks inside the gas volume were found to always fall within these limits @xcite . discharges however can result in a larger imbalance , in which case they are readily eliminated . this helps reducing the average readout time . good events are single electrons contained in the tpc volume . the selection of neutrino scattering events proceeds in two steps . first an automatic filtering eliminates obviously bad events , namely events * identified as @xmath51 s or discharges from their topology ( high ionization in a small volume ) , * or in delayed coincidence ( 80 @xmath44s , corresponding to the tpc length ) with a signal in the scintillator , * or events not contained in a fiducial volume of 42 cm radius , * and finally events with a fast rise time , due to particles crossing the amplification gap between the pick up plane plane and the grid , in either direction . here the avalanche light signal is used , as described in ref . @xcite . the precise live time is derived in this process on a daily basis . it is found to fluctuate around 65 % . it is limited primarily by the total veto time of the scintillator ( 11 % ) and the dead time of the tpc itself , caused by the relatively long data read - out and data transfer time ( 24 % ) . then a final scan is performed . in a separate publication , we reported first results from a crude automatic procedure @xcite , carried out with a pattern recognition program . here we present an analysis based on a visual scan of events , which has the advantage of cleaner event selection . also the risk of misidentifying the beginning of a track is much reduced . this approach is time consuming , however . to minimize the work load , that analysis was restricted to energies above 700 kev . results from an improved version of the automatic scanning procedure will be discussed in a subsequent section . the data sets corresponds to 66.6 days of live time reactor - on and 16.7 days reactor - off . in this second scan both the @xmath52 and @xmath53 projections of an event are scrutinized , as well as the evolution in time of the anode and scintillator signals . continuous electron tracks only are retained . the end of the track is identified from the increased energy deposition , due to the higher stopping power , as exemplified in figure [ fi : munu_track ] . events with a second high deposition along the track , in particular near the other end , are discarded . this improves the background suppression , at the cost of events with a delta electron , which however contribute negligibly . then the tangent at the start of the track is determined by eye . from that the angles @xmath54 with respect to the reactor core - detector axis is determined , as well as the angle @xmath55 with respect to the tpc axis . an excess of events from the anode side was observed . it is presumably due to additional activities , resulting from the greater complexity of the readout system , and to a larger inactive volume in the scintillator because of the stronger and thicker acrylic lid . for that reason only electrons emitted in the half sphere from the cathode side ( @xmath56 ) are accepted . this reduces the acceptance by a factor 2 , but leads to a better signal to background ratio . in figure [ fi : theta_ron ] we show the distribution of @xmath57 of single contained electrons , for both reactor - on and reactor - off . the slightly non linear angular response of the tpc , and its geometry , explain the accumulation of events at @xmath57 around 1 , -1 and 0 for reactor - off . the distribution is however identical in forward ( @xmath58 ) and backward ( @xmath59 ) directions . the reactor - on spectrum shows a clear excess of events in forward direction from @xmath0 scattering . uncertainties from instabilities of the gain , the veto rate , the live time , cancel out in this forward minus backward comparison . for each electron event the neutrino energy @xmath13 is reconstructed from the electron recoil energy @xmath14 and from the scattering angle , taken as @xmath54 . forward event candidates are defined as those with positive neutrino energy : @xmath60 . to select backward events the neutrino energy which must be positive is that calculated using @xmath61 as scattering angle . this procedure has an acceptance close to 100 % above 700 kev , as determined by monte - carlo simulations . it takes into account that electrons are emitted in an narrower cone whith increasing energies , and is somewhat more sophisticated than the application of a crude cut on @xmath57 . in practice , however , with a threshold of 700 kev , it produces almost the same result as the application of a cut @xmath62 or @xmath63 . the energy distributions of both forward and backward events are displayed in figure [ fi : fbe ] . a clear excess of forward events ( 458 in total ) over backward events ( 340 ) is seen . the total forward minus backward count rate above 700 kev is thus [email protected] day@xmath43 . the background , given by the backward events , shows a steep low energy component ending at about 1.2 mev . above it is seen to be fairly low , and much flatter . the background was observed to increase slightly during the course of the experiment , by about 10 % in 6 months , possibly because of outgassing . a small drop in efficiency of the anti - compton , even without an observable reduction in count rate , could also explain this . as a cross check the reactor - off recoil spectra , both forward and backward , measured after the reactor - on period , were reconstructed as well , and are displayed in figure [ fi : fbeoff ] . they are indeed identical within statistics , the integrated forward minus backward rate above 700 kev being @xmath64 day@xmath43 . the difference of the forward minus backward reactor - on spectra is shown in figure [ fi : fminusbe ] . the expected event rate was calculated using the best knowledge of the reactor spectrum as described above , and taking into account the known activations of fissile isotopes and fission products . the uncertainty is around 5 % above 900 kev , as discussed previously , and larger below . the various acceptances of the event selection procedure were determined from measurements with sources , and by monte carlo simulations using the geant3 code . data sets including recoil electrons from neutrino scattering were produced , filtered and scanned just as the real data . the containment efficiency in the 42 cm fiducial radius for recoil electrons was found to vary from 63 % at 700 kev , 50 % at 1 mev to 12 % at 2 mev . the relative uncertainty is of order 2 - 3 % . some tracks with weird topologies can not be reconstructed , reducing the acceptance by 4 % , relatively speaking . similarly the remaining acceptances together , including that of the @xmath60 cut , lead to an additional 6 % reduction . neutrino interactions in the copper cathode give rise to electrons which can escape into the gas volume . these can not be vetoed . taking into account all cuts they increase the expected rate between 700 kev to 2 mev by some 3 % . the relative uncertainty on the global acceptance is of order 7 % , leading to a total uncertainty of 9 % on the total expected rate . the total expected rate above 700 kev assuming a vanishing magnetic moment was found to be [email protected] day@xmath43 , to be compared with [email protected] day@xmath43 measured , as mentioned above . there is thus a certain excess of measured events , which has however only a small statistical significance . the calculated energy distribution is shown in figure [ fi : fminusbe ] . the excess counts are seen to be in the region below 900 kev . to be more quantitative , a @xmath65 was calculated using 100 kev bins as shown in figure [ fi : fminusbe ] from 700 kev to 1400 kev , and then a bin from 1400 to 2000 kev . gaussian statistics applies then to all bins . the error on the expected rate is small in comparison to the statistical uncertainties , and turns out to be negligible . the @xmath65 was calculated this way for several values of the squared magnetic moment @xmath66 , constraining it to the region @xmath67 @xmath68 . this is consistent with a vanishing magnetic moment . renormalizing to the physical region @xmath69 we find the limit @xmath70 @xmath71 at 90(68)% cl . nevertheless even the best fit is not very satisfactory . the @xmath65 ( 9.9 for 7 degrees of freedom ) is on the high side . more troublesome , as seen in figure [ fi : fminusbe ] , the inclusion of a magnetic moment in the calculated spectrum improves the agreement with the data in the two first bins from 700 to 900 kev , but makes it worse in the upper ones . these two first bins are solely responsible for the high total rate . we note that such an excess is also visible , unfortunately also with limited statistical precision , and appearing at somewhat smaller energies , in the data of ref . this excess may well result not from a magnetic moment , but instead from sources not taken into account when evaluating the reactor neutrino spectrum . as mentioned above , the low energy part , which contributes significantly to the electron spectrum below 0.9 mev , is not so well known . additional neutron activations may contribute , beyond the known ones of the fissile isotopes and the fission products . in that sense it seems safe at this stage to restrict the analysis to electron energies above 900 kev . there the measured event rate is @xmath72 day@xmath43 , in good agreement with the expected one @xmath73 day@xmath43 . from the event rate and the energy distribution , using the same prescription as above , the allowed range @xmath74 @xmath68 is found . the best @xmath65 is 1.51 for 5 degrees of freedom . this yields the limit @xmath75 somewhat more stringent than the previous one . extending the visual scan to energies below 700 kev is not feasible because of the rapid increase of the background below that energy , leading to an unmanageable workload . but the automatic scanning procedure mentioned in @xcite has been improved , and used on the same data set @xcite . the program first identifies electron tracks , searches the vertex , and then fits the beginning of the track . electron tracks as identical as possible to real tracks were produced by monte - carlo . the electronics noise was included . the tracks were analyzed with the same program to determine the global acceptance , as well as the angular resolution . the angular resolution is inferior to that obtained in the visual scanning , varying from 31@xmath45 at 300 kev , to 18@xmath45 at 700 kev and finally 12@xmath45 at 2 mev . this reduces the acceptance of the @xmath60 cut in particular at low energy . this , combined with poorer electron track identification , leads to an acceptance of 20 % at 300 kev , increasing to 54 % at 1 mev and remaing constant above that energy . events with @xmath76 were taken . the other cuts and acceptances are the same as in the visual scanning . the background suppression is inferior by a factor 3 or more , so that the statistical precision is less . but the method can be applied to energies extending down to 300 kev . we show in figure [ fi : f - be_auto ] the forward minus backward spectra for reactor - on . it is in good general agreement with the spectra from the eye scan . the high bin at 750 kev is reproduced . but , admittedly with relatively large statistical errors , a catastrophic rise when going to lower energies can be ruled out . applying the same procedure as above to the data one obtains the limit @xmath77 @xmath71 at 90 % cl . it does not change with the threshold in the range 300 to 800 kev , the better sensitivity at low energy being offset by the larger statistical uncertainties . the munu experiment studied @xmath0 scattering at low energy near a nuclear reactor . for electron recoil energies above 900 kev , good agreement is seen with expectations assuming weak interaction alone . from this the following limit on the magnetic moment of the neutrino can be derived : @xmath78 @xmath71 at 90(68 ) % cl , limited primarily by statistics . in any event this bound rules out the possible indication for a relatively large magnetic moment from the savannah river experiment @xcite , and improves on the limit from ref . @xcite and @xcite . it is somewhat more stringent than the limit on @xmath79 from solar neutrinos @xcite , which however does not apply to exactly the same quantity . 99 b. kayser , phys . d26 ( 1982 ) 1662 r. e. shrock , nucl . b206 ( 1982 ) 359 j. f. beacom and p. vogel , phys . 83 ( 1999 ) 5222 g. g. raffelt , _ stars as laboratories for fundamental physics _ , university of chicago press , chicago 1996 the munu collaboration ( c. amsler et al . ) , nucl . inst . and meth . a396 ( 1997 ) akhmedov , a. lanza and s.t . petcov , phys . b303 ( 1993 ) 85 e. kh . akhmedov and j. pulido,_solar neutrino oscillations and bounds on neutrino magnetic moment and solar magnetic field _ , hep - ph/0209192 , 2002 p. vogel and j. engel , phys . d39 ( 1989 ) 3378 the superkamiokande collaboration ( y. fukuda et al . 82 ( 1999 ) 2430 g. zacek et al . d34 ( 1986 ) 2621 b. achkar et al . , b534 ( 1995 ) 503 v.i . kopeikin et al . phys . of atomic nuclei vol . 60 ( 1997 ) 172 f. reines , h.s . gurr and h.w . sobel , phys . 37(1976)315 a.i . derbin et al . , jetp lett . , 57 ( 1993 ) 768 the texono collaboration ( h.b.li et al . ) , phys . ( 2003 ) 131802 the munu collaboration ( m.avenier et al . ) , nucl . inst . and meth . a482 ( 2002 ) 408 j. lamblin , thesis , institut national polytechnique de grenoble , 2002 z. daraktchieva , thesis , university of neuchtel , in preparation o. link , thesis , university of zurich , 2003 the munu collaboration ( c. amsler et al . ) , phys . lett.b 545 ( 2002 ) 57
|
the munu experiment was carried out at the bugey nuclear power reactor .
the aim was the study of @xmath0 elastic scattering at low energy .
the recoil electrons were recorded in a gas time projection chamber , immersed in a tank filled with liquid scintillator serving as veto detector , suppressing in particular compton electrons . the measured electron recoil spectrum is presented .
upper limits on the neutrino magnetic moment were derived and are discussed .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
photoionization of atoms and molecules is one of the most fundamental quantum processes . it played a key role in the early days of quantum mechanics and has ever since been paving the way towards an improved understanding of the structure and dynamics of matter on a microscopic scale . today , kinematically complete photoionization experiments allow for accurate tests of the most sophisticated ab - initio calculations . besides , photoionization studies in a new frequency domain are currently becoming feasible by the availability of novel xuv and x - ray radiation sources @xcite , giving rise to corresponding theoretical developments ( see , e.g. , @xcite ) . various photoionization mechanisms rely crucially on electron - electron correlations . prominent examples are single - photon double ionization as well as resonant photoionization . the latter proceeds through resonant photoexcitation of an autoionizing state with subsequent auger decay . in recent years , a similar kind of ionization process has been studied in systems consisting of two ( or more ) atoms . here , a resonantly excited atom transfers its excitation energy radiationlessly via interatomic electron - electron correlations to a neighbouring atom leading to its ionization . this auger - like decay involving two atomic centers is commonly known as interatomic coulombic decay ( icd ) @xcite . it has been observed , for instance , in noble gas dimers and water molecules @xcite . in metal oxides , the closely related process of multi - atom resonant photoemission ( marpe ) was also observed @xcite . we have recently studied resonant two - center photoionization in heteroatomic systems and shown that this ionization channel can be remarkably strong @xcite . in particular , it can dominate over the usual single - center photoionization by orders of magnitude . besides , characteristic effects resulting from a strong coupling of the ground and autoionizing states by a relatively intense photon field were identified . also resonant two - photon ionization in a system of two identical atoms was investigated @xcite . we note that photoionization in two - atomic systems was also studied in @xcite and @xcite . the inverse of two - center photoionization ( in weak external fields ) is two - center dielectronic recombination @xcite . in the presence of an external laser field and two neighbouring atoms @xmath1 and @xmath2 . apart from the direct photoionization of @xmath0 there are interatomic channels via resonant photoexcitation of the `` molecular '' system @xmath1-@xmath2 and subsequent icd . , scaledwidth=45.0% ] in the present contribution , we extend our investigations of electron correlation - driven interatomic processes by considering photoionization of an atom @xmath0 in the presence of _ two _ neighbouring atoms @xmath1 ( see figure 1 ) . all atoms are assumed to interact with each other and with an external radiation field . we show that the photoionization of atom @xmath0 via photoexcitation of the system of two neighbouring atoms @xmath1 and subsequent icd can be by far the dominant ionization channel . moreover , we reveal the characteristic properties of the process with regard to its temporal dependence and photoelectron spectra . in particular , by comparing our results with those for photoionization in a system of two atoms @xmath0 and @xmath1 , we demonstrate the influence which the presence of the second atom @xmath1 may have . atomic units ( a.u . ) are used throughout unless otherwise stated . let us consider a system consisting of three atoms , @xmath0 , @xmath1 and @xmath2 , where @xmath1 and @xmath2 are atoms of the same element and @xmath0 is different . we shall assume that all these atoms are separated by sufficiently large distances such that free atomic states represent a reasonable initial basis set to start with . let the ionization potential @xmath3 of atom @xmath0 be smaller than the excitation energy @xmath4 of a dipole - allowed transition in atoms @xmath1 and @xmath2 . under such conditions , if our system is irradiated by an electromagnetic field with frequency @xmath5 , the ionization process of this system ( i.e. , essentially of the atom @xmath0 ) can be qualitatively different compared to the case when a single , isolated atom @xmath0 is ionized . indeed , in such a case @xmath0 can be ionized not only directly but also via resonant photoexcitation of the subsystem of @xmath1 and @xmath2 , with its consequent deexcitation through energy transfer to @xmath0 resulting in ionization of the latter . in the following , we consider photoionization in the system of atoms @xmath0 , @xmath1 and @xmath2 in more detail . for simplicity , we suppose that the nuclei of all atoms are at rest during photoionization . denoting the origin of our coordinate system by @xmath6 , we assume that the nuclei of the atoms @xmath1 and @xmath2 are located on the @xmath7-axis : @xmath8 and @xmath9 . the coordinates of the nucleus of the atom @xmath0 are given by @xmath10 . the coordinates of the ( active ) electron of atom @xmath11 with respect to its nucleus are denoted by @xmath12 , where @xmath13 . the total hamiltonian describing the three atoms embedded in an external electromagnetic field reads @xmath14 where @xmath15 is the sum of the hamiltonians for the noninteracting atoms @xmath0 , @xmath1 and @xmath2 . we shall assume that the ( typical ) distances @xmath16 between the atoms are not too large , @xmath17 , where @xmath18 is the speed of light , such that retardation effects in the electromagnetic interactions can be ignored . if transitions of electrons between bound states in atoms @xmath1 and @xmath2 are of dipole character , then the interaction between each pair of atoms @xmath19 ( with @xmath20 ) can be written as @xmath21 where @xmath22 and @xmath23 is the kronecker symbol . note that in ( [ inter_atomic ] ) a summation over the repeated indices @xmath24 and @xmath25 is implied . in ( [ inter_atomic ] ) , @xmath26 denotes the interaction of the atom @xmath11 with the laser electromagnetic field . the latter will be treated as a classical , linearly polarized field , described by the vector potential @xmath27 , where @xmath28 , @xmath29 is the angular frequency and @xmath30 is the field strength . the interaction @xmath26 then reads @xmath31 where @xmath32 is the momentum operator for the electron in atom @xmath33 . our treatment of photoionization will be based on the following points : oscillator strengths for dipole - allowed bound - bound transitions can be very strong . this means that , provided that the distances between all the atoms in our system are of the same order of magnitude , the interaction between atoms @xmath1 and @xmath2 is much more effective than the interaction between atoms @xmath0 and @xmath1 ( or @xmath0 and @xmath2 ) . besides , atoms @xmath1 and @xmath2 will , in general , couple much more strongly to a resonant laser field than atom @xmath0 . in what follows , we shall assume that the intensity of the laser field is relatively low such that the interaction between atoms @xmath1 and @xmath2 changes the states of the system more substantially than the coupling of these atoms to the laser field . therefore , we shall begin with building states of the @xmath1-@xmath2 subsystem in the absence of the field . the second step of our treatment will be to include the interaction of the @xmath1-@xmath2 subsystem with the laser field and , in the third step , we complete the treatment of ionization by considering the interaction of atom @xmath0 with both the laser field and the field - dressed subsystem of atoms @xmath1 and @xmath2 . i. we denote the ground and excited states of the undistorted atoms @xmath1 and @xmath2 by @xmath34 , @xmath35 and @xmath36 , @xmath37 , respectively . let the corresponding energies of these states be @xmath38 and @xmath39 . the state @xmath40 of the @xmath1-@xmath2 subsystem can be expanded into the `` complete '' set of undistorted atomic states represented by the configurations ( i ) @xmath41 , ( ii ) @xmath42 , ( iii ) @xmath43 and ( iv ) @xmath44 . in the approximation , which neglects the interatomic interaction , the configurations @xmath42 and @xmath43 are characterized by exactly the same value of the ( undistorted ) energy @xmath45 . the latter , in turn , strongly differs from the energies @xmath46 and @xmath47 which are characteristic for the configurations @xmath41 and @xmath44 , respectively . therefore , provided that the distance between the atoms is not too small , the interaction @xmath48 will strongly mix the configurations ( ii ) and ( iii ) only , while the other configurations ( i ) and ( iv ) will be affected only very weakly . taking this into account , it is not difficult to find the states of the subsystem of interacting atoms @xmath1 and @xmath2 which read @xmath49 these two - atomic states are normalized and mutually orthogonal . they posses energies given by @xmath50 , @xmath51 , @xmath52 and @xmath53 , respectively , where @xmath54 . note that , for definiteness , @xmath55 has been assumed to be real and negative here , as will always be the case in our examples below ( see section 3 ) . let us now consider two interacting atoms @xmath1 and @xmath2 embedded in a resonant laser field . one can look for a state of such a system by expanding it into the new set of states given by eq . ( [ states_of_bb ] ) , @xmath56 inserting the expansion ( [ bb_in_laser_1 ] ) into the corresponding wave equation , we obtain a set of coupled equations for the unknown time - dependent coefficients @xmath57 , @xmath58 , @xmath59 and @xmath60 : @xmath61 the system of equations ( [ bb_in_laser_2 ] ) can be greatly simplified by noting the following . first , all transition matrix elements of the interaction with the laser field , which involve the asymmetric state @xmath62 , are equal to zero and , thus , only the remaining three states can be coupled by the field . second , if we suppose that the frequency of the laser field is resonant to the transitions @xmath63 and that the field is relatively weak such that the non - resonant transitions @xmath64 are much less effective than the above resonant ones , the system ( [ bb_in_laser_2 ] ) effectively reduces to @xmath65 which can be readily solved by using the rotating wave approximation . assuming that the field is switched on suddenly at @xmath66 , we obtain two solutions @xmath67 \ , \varphi_{g } \nonumber \\ & + & \frac{w_{+,g}}{z_{2 } - z_{1 } } \left ( { \rm e}^{-i z_{2 } t } - { \rm e}^{-i z_{1 } t } \right ) { \rm e}^{- i \omega_0 t } \ , \varphi_{+ } \label{bb'_in_laser_4}\end{aligned}\ ] ] and @xmath68 { \rm e}^{- i \omega_0 t } \ , \varphi_{+}. \label{bb'_in_laser_5}\end{aligned}\ ] ] in the above equations , we have introduced @xmath69 where @xmath70 is the rabi frequency , @xmath71 and @xmath72 . the two solutions in ( [ bb_in_laser_5 ] ) correspond to two different initial conditions : at @xmath66 the system is either in the state @xmath73 or in @xmath74 . they are orthogonal to each other and form a `` complete '' set of field - dressed states of the subsystem @xmath1-@xmath2 . note also that we have neglected the spontaneous radiative decay of the excited state @xmath74 which , in our case , is justified as long as @xmath75 , where @xmath76 is the radiative width of @xmath74 . now , as the last step , we shall add atom @xmath0 to our consideration . let @xmath77 and @xmath78 , where @xmath79 is the electron momentum , be the ground and a continuum state of a single , isolated atom @xmath0 . the wavefunction of the total system @xmath0@xmath1-@xmath2 can be expanded into the following `` complete '' set of states @xmath80 here , the initial conditions are given by @xmath81 , @xmath82 and @xmath83 . the coupling of atom @xmath0 to both the subsystem @xmath1-@xmath2 and the laser field involves bound - continuum transitions which are normally much less effective than the bound - bound ones . for this reason , we may assume that the interactions of @xmath0 with the laser field and the @xmath1-@xmath2-subsystem is weak and consider ionization of atom @xmath0 in the lowest order of perturbation theory in these two interactions . as a result , by inserting the expansion ( [ ioniz_of_a_1 ] ) into the corresponding schrdinger equation we obtain @xmath84 where @xmath85 is the energy of the electron in the initial state @xmath77 of atom @xmath0 and @xmath86 is the electron energy after the emission . the probability for ionization of the three - atomic system , as a function of time , then reads @xmath87 note that equations ( [ ioniz_of_a_2 ] ) are readily solved analytically . however , the resulting expressions are somewhat lengthy and will not be given here . based on the results obtained in the previous section , let us now turn to the discussion of some aspects of photoionization in a system consisting of one lithium and two helium atoms . we suppose that in our three - atomic system the positions of the lithium and helium atoms are given by the vectors @xmath88 , @xmath89 and @xmath90 , respectively . our system is initially ( at time @xmath66 ) in its ground configuration and is irradiated by a monochromatic laser field . the field is linearly polarized along the @xmath7-axis and its frequency is resonant to the @xmath91 - @xmath92 transition in the he - he subsystem , i.e. , @xmath93 . choosing @xmath94 a.u . we obtain that the energy spitting @xmath95 between the states @xmath74 and @xmath62 of the he - he subsystem is @xmath96 ev . assuming a field strength of @xmath97 a.u . , the corresponding rabi frequency amounts to @xmath98 ev which is much less than @xmath99 . in figure [ time - develop ] , we present the probability for ionization of our system as a function of time . the probability shows a non - monotonous behaviour in which time intervals , when the ionization probability rapidly increases , are separated by intervals , when the probability remains practically constant , reflecting oscillations of the electron populations with the rabi frequency @xmath100 between the ground and excited states of the he - he subsystem in a resonant electromagnetic field . for comparison , we also show in figure [ time - develop ] results for ionization of a single ( separated ) li atom and for ionization in a two - atomic li - he system . in the latter case , the lithium atom is located at the origin ( @xmath88 ) and the coordinates of the helium atom are @xmath101 . the frequency of the laser field is assumed to be resonant to the @xmath102@xmath103 transition frequency of the corresponding bound states of a single he atom . in contrast to the single - atom ionization , in both the two- and three - atomic cases the ionization probability demonstrates a step - wise temporal development in which time intervals of rapid probability growth are followed by intervals of almost constant probability . we point out that in the three - atomic case , however , the size of these time intervals is shorter by a factor of @xmath104 . compared to ionization of a single li atom , ionization in the two - atomic system is very strongly enhanced @xcite . when the three - atomic system is irradiated , the enhancement increases even further . in the range of small values of @xmath105 , where all ionization probabilities still increase monotonously , this additional enhancement is equal to a factor of @xmath106 . at larger @xmath105 , however , when the two ionization probabilities exhibit step - wise behaviours , this additional enhancement due to the presence of the second he atom is reduced to a factor close to @xmath107 on average , as can also be seen in figure [ time - develop ] . all the above features can be understood by noting the following : \i ) for the chosen set of parameters of our two- and three - center systems , the indirect channels of ionization , which involve two- or three - atomic correlations , are substantially stronger than the direct one . therefore , these correlations have a dominating effect on the ionization . \ii ) at small @xmath105 , ionization in the two- and three - atomic systems is basically a two - step process : the first step is photoexcitation in the he or he - he subsystem and the second step is a consequent energy transfer to li . in each case , both these steps are described by basically the same dipole transition matrix element of the subsystem . since , compared to a single he atom , this dipole element in he - he is by @xmath104 larger than in he , one obtains a factor of 2 for the enhancement in the ionization amplitude , leading to a factor of @xmath106 in the ionization probability ( see also @xcite ) . \iii ) at larger @xmath105 , when rabi oscillations show up , the second step `` saturates '' in the sense that the averaged probability to find the corresponding subsystem in the excited state becomes equal to 50% . therefore , the ionization probability in the three - atomic system is now larger ( on average ) by a factor of @xmath107 only . \iv ) the origin of the step - wise behaviours of the ionization probabilities for the two- and three- atomic systems lies in the oscillations of the population between the ground and excited states in the he atom ( for the two - atomic case ) or in the he - he subsystem ( for the three - atomic case ) . the scale of these oscillation is set by the rabi frequency and , because in the he - he subsystem the latter is larger by a factor of @xmath104 , the corresponding time intervals are shorter by the same factor . additional information about the ionization process can be obtained by considering the energy spectrum of emitted electrons . such a spectrum is shown in figure [ spectra ] for the same systems and parameters as in figure [ time - develop ] and for a pulse duration of @xmath108 ps . in panel ( a ) , we compare the energy spectra of electrons emitted in the process of photoionization of li - he - he and li - he systems . in both cases , the main feature is the presence of three pronounced maxima . the origin of these peaks is similar to the splitting into three lines of the energy spectrum of photons emitted during atomic fluorescence in a resonant electromagnetic field @xcite . in such a field , the ground and excited levels of the he and he - he subsystems split into two sub - levels , which differ by the corresponding rabi frequency @xmath109 . as a result , the resonant electronic correlations between these subsystems and the li atom lead to an energy transfer to the li which peaks at @xmath110 and @xmath111 . since , as was already mentioned , the rabi frequencies of these subsystems differ by a factor of @xmath104 , the magnitude of the separation between the corresponding maxima in panel ( a ) of figure [ spectra ] also differs by this factor . note also that the widths of these main maxima as well as the appearance of additional multiple maxima , seen in the figure , are related to the finiteness of the pulse duration ; the distance between the latter is roughly given by @xmath112 . the distinct influence , which the interatomic electron - electron correlations exert on the shape of the photoelectron spectra , is further highlighted in panel ( b ) of figure [ spectra ] . it compares the energy spectra of photoelectrons emitted from our li - he - he system and an isolated li atom . in the latter case , there is only one main maximum , while the two main side peaks are missing , as one would expect ( the additional multiple maxima are related again to the finiteness of the pulse duration ) . we have studied resonant photoionization in a system @xmath0-@xmath1-@xmath2 consisting of three atoms , with two atoms @xmath1 of the same element and one different atom @xmath0 . we have shown that the mutual correlations among the atoms can largely enhance the ionization probability and distinctly modify also other properties of the process in a characteristic manner . in particular , as compared to the case of resonant photoionization in a two - atom system @xmath0-@xmath1 , it has been demonstrated that the presence of a second atom @xmath1 can ( i ) further enhance the photoionization process , ( ii ) change the time dependence of the ionization probability and ( iii ) move the side peaks in the photoelectron spectrum further apart . a.b.v . acknowledges the support from the extreme matter institute emmi . 10 rudenko a _ et al _ 2008 _ phys . rev . lett . _ * 101 * 073003 cederbaum l s , zobeley j and tarantelli f 1997 _ phys . lett . _ * 79 * 4778 for recent reviews on icd , see averbukh v _ et al _ 2011 _ j. electron spectrosc . relat . phenom . _ * 183 * 36 ; hergenhahn u 2011 _ j. electron spectrosc . relat . phenom . _
|
modifications of photoionization arising from resonant electron - electron correlations between neighbouring atoms in an atomic sample are studied .
the sample contains atomic species @xmath0 and @xmath1 , with the ionization potential of @xmath0 being smaller than the energy of a dipole - allowed transition in @xmath1 .
the atoms are subject to an external radiation field which is near - resonant with the dipole transition in @xmath1 .
photoionization of an atom @xmath0 may thus proceed via a two - step mechanism : photoexcitation in the subsystem of species @xmath1 , followed by interatomic coulombic decay . as a basic atomic configuration , we investigate resonant photoionization in a three - atomic system @xmath0-@xmath1-@xmath1 , consisting of an atom @xmath0 and two neighbouring atoms @xmath1 .
it is found that , under suitable conditions , the influence of the neighbouring atoms can strongly affect the photoionization process , including its total probabilty , time development and photoelectron spectra . in particular , by comparing our results with those for photoionization of an isolated atom @xmath0 and a two - atomic system @xmath0-@xmath1 , respectively , we reveal the characteristic impact exerted by the third atom .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of chromospheric and coronal activity has progressed impressively in the last two decades , thanks to the launch of x - ray satellites . coronal sources have been detected in x rays all over the cool part of the h - r diagram ( vaiana et al . 1980 ) and active binaries of the rs cvn type have soon been recognized as powerful x - ray emitters ( walter et al . 1980 ) . with the advent of the rosat satellite and its all - sky survey , a complete x - ray study of known active binaries has been possible ( dempsey et al . 1993a , b ) . a large number of previously unknown coronal active stars is becoming available ( see e.g. metanomski et al . 1998 ) , with interesting consequences on our understanding of the population of young stars in the solar neighborhood ( guillout et al . 1998 ) . while the general framework of dependence of x - ray emission on stellar rotational velocity ( and then stellar age ) has been established long time ago ( pallavicini et al . 1981 ) , there still remain many uncertainties about which stellar parameters affect the coronal emission in late - type stars . we know , for instance , that old stars may preserve a high level of x - ray emission if they are in binary systems , where a high rotational velocity can be maintained through tidal interaction , but it is not clear which role stellar mass , radius , orbital period and eccentricity play in determining the level of activity . one of the most relevant limitations is that , since most studies are performed in field samples , it is difficult to determine precisely the characteristics of the studied objects . stars with different ( but poorly determined ) masses , ages and possibly evolutionary histories are often compared . the study of clusters , where ages , masses , and evolutionary status of the counterparts of x - ray sources can be well established , is a necessary step forward . in this framework , not only rosat observations of young clusters have provided new insight to the discussion on the age - activity relationship ( see e.g. randich and schmitt 1995 ) , but for the first time it has also been possible to detect coronal sources in old clusters , i.e. with ages comparable with the sun . with the rosat observations of m 67 and other intermediate age clusters , belloni et al.(1993 , 1996 , 1997 ) allowed to extend the study of the evolution of coronal activity to systems with age up to @xmath0 6 gyrs . by examining these coeval samples , these observations allow the study of active coronae from a different , unique perspective : it is possible to investigate which stars ( or stellar systems ) show the highest activity level in samples so old that the emission from single , solar - type stars is expected to be very low . the rosat observations require a follow up at different wavelengths , in order to : * confirm the identification of the optical counterparts . in these clusters the stellar density is rather high , and more than one source may be contained in the rosat error box . as an example of the relevance of this issue , the reader can compare the results of the present work ( summarized in table 1 ) with those of belloni , verbunt & schmitt ( 1993 ) , based only on x - ray and general optical information , but not on a detailed follow up . * study in more detail the characteristics of the sources , their binary nature , x - ray and chromospheric emission . * compare the observed characteristics with those of field stars . in this work , we present the identification and the study of the rosat counterparts of the old open cluster m 67 ( belloni , verbunt & schmitt 1993 ) . a second , longer rosat pointing to m 67 has been performed ( belloni , verbunt & mathieu in preparation ) : where possible , the results from this observation have been included in the present paper . m 67 is a very interesting target , not only because its age and metallicity are similar to those of the sun , but also because the cluster has been subject to several photometric and spectroscopic studies which have led to detailed membership determinations ( sanders 1977 , girard et al . 1989 ) , a large amount of data on binary stars ( latham et al . 1992 ) , photometric variables , w uma candidates ( gilliland et al . 1991 ) , and blue stragglers ( mathys 1991 ) . this large body of ancillary data makes m 67 one of the best studied clusters ; one am her system and a hot white dwarf belonging to the cluster have already been confirmed within this identification project ( pasquini et al . the observations were carried out at eso , la silla , over the period 1992 - 1995 , using a variety of instruments and telescopes . we stress that , being mainly interested in the coronal counterparts of rosat detections , we did not attempt a complete identification of all the sources , but only of the possible coronal cluster members . this implies that , when only optically faint candidates are within the rosat error box , observations were not always pursued further . first , observations were carried out at the eso 1.52 m telescope equipped with the b&ch spectrograph ( turatto 1997 ) : with a resolution of 2 / pixel and a 2048 pixel ccd , the range 3700 - 7600 was covered . all objects within the reach of the telescope - spectrograph combination were observed , within @xmath0 40@xmath1 from the nominal rosat source in order to be confident that no possible counterpart would be missed . a posteriori , we found this radius exceedingly large and we could confirm the good accuracy of the rosat error box . bright stars within fields with no obvious counterpart have been observed even if located at a comparatively large distance from the x - ray position . the low - resolution spectra were inspected to derive ( or check ) spectral types or spectral anomalies , but mostly to find signatures of high chromospheric emission , like filling - in or emission of the balmer lines . most of the stellar candidates were selected in this way . additional observations at low resolution needed for the identification of some of the fainter counterparts were performed using the 3.6 m telescope with the efosc spectrograph ( benetti et al . 1997 ; see also pasquini et al . 1994a ) . finally , the 3.6 m telescope with the caspec spectrograph ( randich and pasquini 1996 ) and the ntt with the emmi spectrograph were used to obtain intermediate- and high - resolution spectroscopy of the pre - selected candidates . this last step is required to allow a firm identification of the targets and to derive absolute chromospheric fluxes at the stellar surface . the observations were centered in the ca ii h and k region , and in the h@xmath2 region . caspec spectra have a resolving power r=18000 . the emmi spectra were obtained in dichroic mode : blue and red spectra were recorded simultaneously . blue spectra were acquired with a holographic grating at r=6000 ( pasquini et al . 1994b ) , while red spectra were obtained with a resolution r=20000 or r=3000 , depending on the stellar apparent magnitude . the spectra have been reduced using the midas package ( banse et al . 1988 ) . when a candidate showed enhanced chromospheric activity and acceptable positional coincidence , it was accepted as counterpart . however , for some of the fields no star fulfilled both conditions . for these fields , we obtained low resolution spectroscopy of fainter candidates , and when no acceptable alternatives were found , the original stars were accepted as ( possible ) candidates . belloni , verbunt & schmitt ( 1993 , hereafter bvs ) list 22 sources detected in the central 20 arcmin of a rosat pspc field centered on a position @xmath3 east of the center of m 67 . the rosat sources with firm or possible optical counterparts are summarized in table 1 : the numbering scheme is the same as in bvs . source number 11 was recognized by bvs as a blend of a hard and a soft source . in the analysis of the new rosat observation of m 67 ( belloni , verbunt & mathieu , in preparation ) , the two sources could be resolved and they are listed separately ( sources 11a and 11b ) . the optical positions are from girard et al . ( 1989 ) when available , while the coordinates for the remaining objects are from the digitized sky survey . for these , distances are accurate only to a few arcsecs . for optical identification , we adopted the results obtained by similar studies in the field ( see e.g. stocke et al qso s and emission - line galaxies , when close to the nominal error box were accepted as counterparts ; in particular , in the error boxes of the 3 extragalactic sources identified with x - ray sources , no known m 67 member exists . for the ten rosat fields with no identification , efosc images were acquired , but no stellar counterparts was found down to magnitude limits in excess of those expected for values of l@xmath4/l@xmath5 typical for coronal sources . in figure 1 , low resolution spectra of the extragalactic sources and of some of the most interesting objects in the fields with doubtful or no optical identification are shown . note that the shape of the continuum may not represent the true continuum of the objects , since in order to gain spectra of several objects simultaneously , in most cases the slit could not be aligned with the parallactic angle . thanks to the boresight correction applied by bvs , the optical and x - ray positions for most candidates agree extremely well , with differences often smaller than 10 arcsec , which is the typical nominal rosat error box ( bvs ) . therefore , we consider the 2 proposed counterparts having larger distances from the x - ray source ( b1 , b15 ) only as possible , but not likely candidates . for b19 , the uncertainty in the identification is given by the fact that , although the x - ray and the optical position of the star s364 match quite well , the high - resolution spectra obtained do not show any sign of enhanced chromospheric activity ( cfr . section 5.2 ) . this , according to the previously mentioned criteria , does not make this star a firm counterpart . 0.2 cm 0.2 cm [ cols= " > , > , < , < , < " , ] the aim of this work is to discuss the nature of the coronal sources belonging to m 67 detected in the rosat observation of bvs . for this purpose we concentrate mostly on the firmly identified sources . it is important to remember that the observations by bvs could only detect the high x - ray luminosity tail of the cluster . the longer , more recent rosat observations reveal more sources , confirming that cluster x - ray emitters might exist at lower levels than the one analyzed here ( belloni , verbunt & mathieu , in preparation ) . of the 8 known members of table 2 , six are either known binaries or show clear signs of duplicity , like photometric modulation or displaced position in the colour - magnitude diagram of the cluster . of the remaining two , s1077 is reported by bvs as a multiple system with short period , while s364 is not included in the current list of binaries . jones and smith ( 1984 ) found anomalous ddo colours for this star , but they concluded that this was probably due to due measurement uncertainties . once again , we stress that this identification is only considered as ` possible ' . _ the first conclusion is that in m 67 the strongest coronal sources are binaries_. this was expected from the results on field stars and from the age - coronal activity relationships : single , late type stars with ages comparable to the sun are not expected to emit x - ray in excess to @xmath010@xmath6 erg / s in luminosity . in figure 3 the colour - magnitude diagram of the cluster is shown , with the x - ray identifications marked . it is striking to notice that a large variety of cases exists : one blue straggler ( s1082 ) , one peculiar object ( s1063 ) , evolved ` normal ' binaries ( s1077 , s999 ) , main sequence binaries ( s1019 , s972 ) , one red straggler ( s1040 ) , and possibly a red giant ( s364 ) . among field stars such a comparison can hardly be done , due to the uncertainty in the fundamental stellar parameters . _ the second conclusion is therefore that among the m 67 sources there exists a large variety in evolutionary status and composition of binary systems . in particular , several of the strong x - ray emitters are found among objects having peculiar location in the colour magnitude diagram . _ this variety makes it difficult to understand which parameters determine the x - ray emission in these binaries . from the study of field active binaries , it emerged that the main parameter determining x - ray luminosity is stellar radius . therefore , the strongest x - ray emitters would be expected to be among the most luminous systems ( dempsey et al . 1993b ) . on the contrary , the most luminous x - ray sources in m 67 span over a range in visible luminosity of at least 4 magnitudes ( cfr . table 1 ) . although s972 , which has the lowest x - ray luminosity , is also the faintest established member of the cluster in the optical , other stars ( like s1019 or s1063 ) show the highest x - ray emission , while being relatively faint in the optical . another interesting point is the dependence of x - ray emission on the orbital ( or rotational ) period , since x - ray luminosity scales with stellar rotational velocity ( pallavicini et al . 1981 ) , which in turn is related to stellar radius and orbital period . in most cases ( and for main sequence stars ) , it is expected that short period binaries are synchronized , i.e. they have equal orbital and rotational periods . it is also expected that synchronization happens before orbit circularization ( zahn 1977 ) . all 4 stars for which the rotational modulation is known , as inferred from the photometric variability , have rotational periods of less than 10 days . for s1063 , only the orbital period is known , but this star may have a special history ( see section 5.3 ) . pending more data on the rotational period of the other objects , we can say that on the other hand , do all m 67 short period systems show strong x - ray ( or chromospheric ) activity ? in m 67 , 11 binaries have a measured orbital period shorter than 16 days ( latham et al . 1992 ) , a canonical value for the definition of rs cvn systems , but most of them were not detected in the rosat observations . three of them ( s1272 , s1284 , s1224 ) where out or at the very border of the rosat field ; one ( s999 ) was detected , but the remaining ( s1045 , s1234 , s1024 , s986 , s1009 , s1070 s1014 , s810 ) were contained in the rosat field but not detected . these systems are likely active , but with x - ray luminosities l@xmath7 30.3 ( corresponding to the sensitivity of bsv ) , as it is suggested by the likely detection of some of these stars in the deeper pointing ( s1045 , s1024 , s1234 and s1070 , belloni , verbunt & mathieu , in preparation ) . since some of the non - detected binaries , like s986 , s1009 and s810 , have periods of 10 days or shorter , the data collected up to now show that since we are dealing mostly with evolved stars , for which short periods can coexist with eccentrical orbits , eccentricity could be a relevant parameter , because the non - synchronization would allow the possibility of having a rotational period shorter than the orbital period . of the detected binaries , s1063 and s999 have highly eccentric orbits , but s1040 has an eccentricity comparable with 0 . this is surprising , because if the photometric variability detected by gilliland et al . ( 1991 ) really represents the rotational period , than this star would be circularized but not synchronized . on the other hand , s1040 had probably a very complex history , having had mass transfer in the past ( landsman et al . 1997 ) . considering this star as a special case we , could argue that non - circular binaries are favored among the strongest x - ray emitters . however , in the list of latham et al . ( 1992 ) , there are five short - period binaries with eccentricities significantly different from 0 ( s1284 , s1272 , s1234 , s1224 , s1014 ) . two of them ( s1014 and s1234 ) were in the rosat field of view but were not detected . therefore , it seems that also a high eccentricity does not represent _ per se _ a condition for strong x - ray emission . the chromospheric flux of the sun in the ca ii k line is of 3 - 5@xmath810@xmath9 erg @xmath10sec@xmath11 ( pasquini et al . 1988 ) . for the main sequence g stars in m 67 , being metallicity and age comparable with those of the sun , values comparable with this are expected . recently , dupree et al . ( in preparation ) studied the chromospheric emission in the ca ii lines for a sample of m 67 giants , using calibrations and spectra of similar quality to the ones presented here . in figure 4 the ca ii chromospheric fluxes are plotted as a function of the ( v - r ) colour , both for the stars in dupree et al . ( open squares ) and for the x - ray counterparts of table 2 ( filled squares ) . the rosat stars have a level of chromospheric activity one order of magnitude higher than the optically - selected giants in dupree et al . the only exception is s364 , which fits extremely well within normal giants of similar spectral type . for this reason , despite the reasonable agreement with the x - ray position , this star is considered as a doubtful counterpart . in order to perform an unbiased analysis of the relationship between chromospheric and coronal activity , we have to use indicators which express similar quantities , namely fluxes at the stellar surface . we computed the radii of the stars by using the same barnes - evans ( 1976 ) relationship used to calibrate the ca ii data : log@xmath12 = 0.4874 - 0.2v@xmath13 + 0.858(v - r ) . although we know that the resulting radii are probably incorrect ( for instance , because the stars are implicitly assumed to be single ) , the fact that the same relationship is used for computing the ca ii and x - ray fluxes minimizes the presence of possible systematic effects in the comparison . we assumed an absorption a(v)=0.17 and a distance of 785 parsecs ( janes 1984 ) . the resulting diameters and radii are given in tables 2 and 3 ( in units of milliarcseconds and 10@xmath14 cm respectively ) . in figure 5a , log(f@xmath4 ) is given as a function of log f@xmath15 . the relationship is rather scattered and mostly the presence of the ( doubtful ) s364 hints to the presence of a trend . since the sample contains systems whose evolutionary status is very different from each other , to further investigate this point we plot in figure 5b the ratio between the x - ray and chromospheric fluxes versus apparent magnitude . fainter ( i.e. higher gravity ) stars have much higher coronal to chromospheric flux ratios than more luminous ( i.e. lower gravity ) stars . although the number of objects is rather low , it appears that dwarfs and giants follow different trends , with dwarfs having higher f@xmath4 for a given f@xmath15 . this fact has two possible explanations : * for comparable chromospheric fluxes , higher - gravity stars are more efficient in heating their coronae than lower - gravity stars . this could indicate the presence of different coronal structures between dwarfs and evolved stars . * the assumption that the ca ii k fluxes are representative of the whole chromospheric losses may not apply when comparing stars of different luminosity . giants could for instance have a different balance in the different chromospheric lines than dwarfs . s1113 was was outside the field of the observation by bsv . because of the strong chromospheric activity observed , we would expect it to show rather strong x - ray emission , which is indeed detected in the new rosat pointing ( cfr . table 3 ) . these two stars are located below the giants branch : they are as red as single subgiants , but almost one magnitude fainter . s1063 is a known eccentric binary with a period of 18.3 days , while s1113 is a short period ( 2.82 d ) circular binary ( latham et al . 1992 , mathieu et al . in preparation ) . the two stars are classified as members in the proper motion studies of sanders ( 1977 ) and girard et al . ( 1989 ) , with a probability higher than 90@xmath16 . their peculiar position in the colour - magnitude diagram and their high level of coronal and chromospheric activity make these two object very interesting . it is not possible to simply combine two m 67 stars and obtain the magnitudes and colours of s1063 and s1113 . some mass exchange , or large mass losses in the past history of the systems , possibly still going on , look unavoidable . the high - resolution ca ii spectrum of s1113 shows neither direct evidence of duplicity , nor strong asymmetries in the ca ii core typical of strong mass losses , but not much can be derived with only one optical spectrum for this star . we stress that the conclusion that these two stars have suffered a special evolutionary history is made possible only by the fact that we they are members of a cluster ( and therefore we can firmly position them in the colour magnitude diagram ) and by the detailed optical follow - up . how many such systems exist among field binaries ? note that , in absence of detailed studies ( e.g. accurate determinations of mass , metallicity and gravity ) , similar systems in the field can not be distinguished by otherwise normal rs cvn binaries . a more detailed study in other clusters and possibly the analysis of hipparcos parallaxes of active binaries will help in understanding how common these systems are . investigation of their binarity and orbital synchronization will also be crucial to model their possible evolution with time ( see i.e. the discussion in stepien 1995 ) . one of the aims of this study is the possibility of comparing for the first time active stars in an old cluster with the field population of rs cvn . dempsey et al . ( 1993a , b ) studied the rosat detections of known rs cvn . their sample , taken from the catalogue of strassmeier et al . ( 1988 ) , contains binaries of a large variety of ages , masses and periods . we selected all the stars from dempsey et al . ( 1993a ) with known distance . we computed x - ray luminosities by converting the pspc count rates ( see also dempsey et al . 1994 ) using the same conversion factor used for the m 67 stars . the distribution in luminosity is given in figure 6 for m 67 and field rs cvn separately . although the m 67 sources do overlap well with the main body of field rs cvn s , their emission is not as high as the most active rs cvn systems . the reasons for this difference are at the moment not very clear : it could be an effect due to the old age of the cluster and/or to the evolutionary status of the sources , or to a statistical effect caused by the fact that we can only sample a few hundred stars within the cluster . identifications in clusters of different ages , as well as a detailed analysis of the strongest sources in the field , will help in understanding this open question . it is interesting to note that in the analysis of the much younger hyades , stern et al . ( 1995 ) found that binaries are the strongest x - ray emitters in this cluster . similarities with m67 exist also in that the strongest x - ray source in the hyades ( v471 tau ) is a peculiar system . finally , hyades binaries have x - ray luminosities similar or lower than those observed among the m67 sources . this could indicate that as far as the high - luminosity tail of the x - ray luminosity function is concerned , age is not the primary parameter to determine the x - ray emission in binaries , as pointed out also by ottmann et al . ( 1997 ) in their analysis of pop ii binaries . we would therefore argue that the extremely active i.e. ( l@xmath4 @xmath0 10@xmath17 erg @xmath10 sec @xmath11 ) rs cvn systems seem to be very rare and possibly limited to quite exceptional cases . metanomski et al . ( 1998 ) , in their study of hundred stars identified over a large area of the rosat all - sky survey , find results similar to ours : no such a high luminosity coronal source is indeed present in their sample . banse , k. , grosbol , p. , ponz , d. et al 1988 ` the midas image processing system ' in ` instrumentation for ground base astronomy : present and future ' , robinson , l.b . ( springer ) barnes , t.g . , evans , d. 1976 , mnras 174 , 489 basri , g. 1987 , apj , 316 , 377 belloni , t. , verbunt , f. , schmitt , j.h.m.m . 1993 , a&a 269 , 165 ( bvs ) belloni , t. , verbunt , f. 1996 , a&a 305 , 806 belloni , t. , & tagliaferri , t. , 1997 , a&a , 326 , 608 benetti , s. , savaglio , s. pasquini , l. 1997 , ` efosc operating manual ' ( eso ) dempsey , r.c . , linsky , j.l . , schmitt , j.h.m.m . , fleming , t.a . 1993b , apj , 413 , 333 dempsey , r.c . , linsky , j.l . , fleming , t.a . , schmitt , j.h.m.m . 1993a , apjs , 86 , 599 dempsey , r.c . , linsky , j.l . , fleming , t.a . , schmitt , j.h.m.m . 1994 , apjs , 94 , 829 fleming , t.a . , snowden , s.l . , pfeffermann , e. , briel , u. , greiner , j. 1996 , a&a , 316 , 147 gilliland , r. et al . 1991 , aj , 101 , 541 girard , t.m . , grundy , w.m . , lopez , c.e . , van altena , w.f . 1989 , aj 98 , 227 goranskij , v.p . , kusakin , a.v . , mironov , a.v . , moshkaljov , v.g . , pastukhova , e.n . 1992 , astron . trans . 2 , 201 guillout , f. , sterzik , m.f . , schmitt , j.h.m.m . 1998 , a&a 332 , l29 hasinger , g. , burg , r. , giacconi , r. et al . 1998 , a&a 329 , 482 janes , k.a . 1984 , in calibration of fundamental stellar quantities , iau symposium no 11 , d.s . hayes , l.e . pasinetti , a.g . davis philip eds . ( reidel ) janes , k.a . , smith , g.h . 1984 , aj 89 , 487 johnson , h.l . 1966 , ara&a 4 , 193 landsman , w. , aparicio , j. , bergeron , p. , di stefano , r. , stecher , t.p . 1997 , apj , 481 , l93 latham , d.w . , mathieu , r.d . , milone , a.e . , davis , r.j . 1992 , in ` binaries as tracers of stellar formation ' , a. duquennoy and m. mayor eds . ( cambridge un . mathys , g. 1991 , a&a 245 , 467 melnick , j. 1993 , emmi operating manual ( eso ) metanomski , a.d.f . , pasquini , l. , krautter , j. , cutispoto , g. , fleming , t.a . 1998 , a&as , in press montgomery , k.a . , marschall , l.a . , janes , k.a . 1993 aj , 106 , 181 ottmann , r. , fleming , t.a . , pasquini , l. 1997 , a&a 322 , 785 pallavicini , r. , golub , l. , rosner , r. , vaiana , g.s . , ayres , t. , linsky , j.l . 1981 , apj 248 , 279 pasquini , l. , pallavicini , r. , pakull , m. 1988 , a&a 191 , 266 pasquini , l. , pallavicini , r. , dravins , d. 1989 , a&a 213 , 261 pasquini , l. , belloni , t. , abbott , t.m.c . 1994a , a&a 290 , l17 pasquini , l. , storm , j. , dekker , h. 1994b , the messenger 77 , 5 randich , s. , schmitt , j.h.m.m . 1995 , a&a , 298 , 115 randich , s. , pasquini , l. 1996 , caspec operating manual ( eso ) sanders , w.l . 1977 , a&as 27 , 89 stepien , k. 1995 , mnras 274 , 1019 stern , r.a . , schmitt , j.h.m.m . , kahabka , p.t . , 1995 , apj 448 , 683 stocke , j.t . , morris , s.l . , 1991 , apjs , 76 , 813 strassmeier , k.g . , hall , d.s . , zeilik , m. , nelson , e. , eker , z.,fekel , f.c . , 1988 , a&as , 72 , 291 turatto , m. 1997 , ` the b&ch spectrograph user s manual ' ( eso ) vaiana , g.s . , cassinelli , j.p . 1981 , apj 245 , 163 walter , f.m . , cash , w. , charles , p.a . , bowyer , c.s . 1980 , apj 245 , 671 zahn , j.p . , 1977 , a&a 57 , 383
|
we present optical identification and high - resolution spectroscopy of rosat sources in the field of the old open cluster m 67 . for the first time it is possible to analyze coronal and chromospheric activity of active stars in a solar - age cluster , and to compare it with field stars .
rosat observed the high x - ray luminosity tail of the cluster sources . in agreement with what expected from studies of field stars ,
most of the detected x - ray sources are binaries , preferably with short periods and eccentric orbits .
in addition , several of the m 67 rosat sources have peculiar locations in the cluster colour - magnitude diagram .
this is most likely due to rather complex evolutionary histories , involving the presence of mass transfer or large mass losses .
the x - ray luminosity of the sources does not scale with the stellar parameters in an obvious way .
in particular , no relationship is found between coronal emission and stellar magnitude or binary period .
the ca ii k chromospheric flux from most of the counterparts is in excess to that of single stars in the cluster by one order of magnitude .
the x - ray luminosity of the sources in the old m 67 is one order of magnitude lower than the most active active binaries in the field , but comparable to that of the much younger binaries in the hyades .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the large - scale distribution of galaxies contains the signature of acoustic waves that propagated through the universe prior to the epoch of recombination . this signal , referred to as baryon acoustic oscillations ( bao ) , appears as a modulation in the amplitude of the galaxy power spectrum , @xmath12 , and a broad peak in the large - scale two - point correlation function , @xmath0 @xcite . the wavelength of the oscillations in @xmath12 and the location of the peak in @xmath0 can be associated with the maximum distance that these acoustic waves can travel before the decoupling of matter and radiation , that is , the sound horizon at the drag redshift , @xmath13 . as this scale can be constrained with high accuracy from observations of the cosmic microwave background ( cmb ) , the acoustic scale inferred from the clustering of galaxy samples at different redshifts can be used as a standard ruler to measure the distance - redshift relation , providing a powerful and robust probe of the expansion history of the universe @xcite . the bao signal was first detected in the clustering of the two - degree field galaxy redshift survey ( 2dfgrs , * ? ? ? * ; * ? ? ? * ) by @xcite and the luminous red galaxy ( lrg , * ? ? ? * ) sample of the sloan digital sky survey ( sdss , * ? ? ? * ) by @xcite . since then , subsequent analyses on various galaxy samples have provided bao measurements with increasing precision @xcite . using these results it is now possible to construct a hubble diagram based entirely on bao distance measurements . it has become standard practice to use this information , in combination with additional data sets , when deriving constraints on cosmological parameters . separate measurements of the acoustic scale in the directions parallel and perpendicular to the line of sight can be used to obtain constraints on the hubble parameter , @xmath14 , and the angular diameter distance , @xmath15 , through the alcock paczynski test @xcite . however , the bao signal on angle - averaged clustering measurements such as @xmath12 or @xmath0 provide estimates of the average distance @xmath16 . although most analyses have focused on angle - averaged quantities , the large volumes probed by present - day galaxy samples make it possible to extend these analyses to anisotropic clustering measurements @xcite using the full power of the bao test . the clustering of galaxies encodes additional information beyond that contained in the bao signal that can significantly improve the cosmological constraints derived from large - scale structure ( lss ) data sets . this extra information is particularly important for anisotropic clustering measurements , where the signature of the so - called redshift - space distortions ( rsd ) can be used to constrain the growth rate of cosmic structures @xcite . in this way , anisotropic clustering measurements can provide information of the expansion history of the universe and the growth of density fluctuations , which can be used to distinguish between the dark energy and modified gravity scenarios for the origin of cosmic acceleration . the most accurate bao measurements to date have been obtained from the baryon oscillation spectroscopic survey ( boss , * ? ? ? * ) , which is one of the four component surveys of sdss - iii @xcite . after applying a modified version of the reconstruction technique of @xcite , the bao signal in the galaxy clustering of boss sdss data release 9 ( dr9 , * ? ? ? * ) provided a 1.7 per cent accuracy measurement of the average distance @xmath17 at @xmath18 @xcite , as well as separate constraints on @xmath15 and @xmath14 at the same redshift with 3 and 8 per cent accuracy , respectively @xcite . these measurements have been complemented by analyses of the full shape of isotropic and anisotropic clustering measurements @xcite . besides galaxy clustering analyses , a sample of high - redshift quasars from boss has been used to detect for the first time the signature of the bao in the fluctuations of the lyman-@xmath19 forest at @xmath20 @xcite . in this paper we use information from the full - shape of the two - point correlation function and the clustering wedges statistic @xcite measured from boss data to derive constraints on cosmological parameters . we extend the analyses of @xcite based on a high - redshift galaxy sample from boss dr9 to the data corresponding to dr10 @xcite and dr11 ( internal data - release ) , including results from the low - redshift boss galaxy sample . as the statistical uncertainties characterizing different cosmological observations become smaller , it is important to explore potential systematics that can be introduced by the analysis techniques and models applied to the data . the comparison of the results obtained by applying multiple methods to the same data can be used to identify the presence of systematics errors . our analysis is part of a series of papers examining the clustering properties of the boss dr10 and dr11 galaxy samples with different methodologies . tojeiro et al . ( in preparation ) and @xcite analyse the isotropic and anisotropic bao signal in these samples and explore their cosmological implications . @xcite study the sensitivity of these bao measurements to the properties of the galaxy population being analysed . @xcite investigate the potential systematic errors affecting anisotropic bao measurements . @xcite perform a detailed analysis of the effect of the uncertainties in the covariance matrices determined from mock catalogues on the obtained constraints . these analyses are complemented by those of @xcite , @xcite and @xcite , who analyse the full shape of the monopole - quadrupole pair in configuration and fourier space . these studies attempt to condense the information of the clustering measurements into a few numbers reflecting the geometric constraints and the measurements of the growth of structures , that are then compared with the predictions from different cosmological models . we follow an alternative approach in which we perform the comparison with cosmological models at the level of the galaxy clustering measurements themselves . the consistency of the results presented here and those of our companion papers is a reassuring indication of the robustness of our results . the outline of this paper is as follows . in section [ sec : data ] we describe our galaxy sample , the procedure followed to obtain our clustering measurements and their respective covariance matrices , as well as the additional data sets included in our analysis . in section [ sec : method ] we review our model of the full shape of the correlation function and the clustering wedges and our methodology to obtain cosmological constraints . in section [ sec : results ] we present the constraints on cosmological parameters obtained from different combinations of data sets and parameter spaces . finally , section [ sec : conclusions ] contains our main conclusions . we use the lowz and cmass samples of boss corresponding to sdss dr10 @xcite and dr11 , which will become publicly available with the final data release of the survey . these galaxy samples were selected on the basis of the sdss multicolour photometric observations @xcite to cover the redshift range @xmath21 with a roughly uniform comoving number density @xmath22 ( * ? ? ? * ; * ? ? ? * padmanabhan et al . in preparation ) . up to @xmath23 and 2 per cent of lowz and cmass targets , respectively , were observed during the sdss i / ii surveys @xcite and thus already have a redshift . the remaining redshifts were measured from the spectra obtained with the double - armed boss spectrographs @xcite by applying the minimum-@xmath24 template - fitting procedure described in @xcite and @xcite . the lowz sample consists primarily of red galaxies that lie in massive haloes , with a satellite fraction of 12 per cent @xcite . the cmass sample is approximately complete down to a limiting stellar mass of @xmath25 @xcite , and has a @xmath2610 per cent satellite fraction @xcite . although this sample is dominated by early type galaxies , it contains a significant fraction of massive spirals ( @xmath2626 per cent , * ? ? ? @xcite describes the construction of catalogues for lss analyses based on these samples . we use these samples separately , restricting our analysis to the redshift ranges @xmath27 for the lowz sample , and @xmath28 for the cmass galaxies . we study the clustering properties of these galaxy samples by means of the angle - averaged correlation function , @xmath0 , and the clustering wedges statistic @xcite , @xmath29 , which corresponds to the average of the full two - dimensional correlation function @xmath30 over the interval @xmath31 , that is @xmath32 we use two wide clustering wedges , @xmath1 and @xmath2 , defined for the intervals @xmath33 and @xmath34 , respectively . the basic procedure implemented to obtain these measurements from the lowz and cmass samples is analogous to that of @xcite and @xcite . here we summarize the most important points and refer the reader to these studies for more details . we convert the observed redshifts into distances assuming a flat @xmath3cdm fiducial cosmology characterized by a matter density parameter of @xmath35 . we use the estimator of @xcite to compute the full correlation function @xmath30 of the lowz and cmass samples , with random samples following the same selection function as the original catalogues but containing 50 times more objects . the value of @xmath36 of a given pair is defined as the cosine of the angle between the separation vector , @xmath37 , and the line - of - sight direction at the midpoint of @xmath37 . we infer the correlation function @xmath0 and the clustering wedges @xmath1 and @xmath2 by averaging the full @xmath30 over the corresponding @xmath36 intervals . as discussed in @xcite , this procedure correctly accounts for the @xmath36 dependence of the random - random counts , which is ignored when the estimator of @xcite is applied to the averaged counts directly , leading to a bias in the recovered clustering measurements . when computing the pair counts , we assign a series of weights to each object in our catalogue . first , we apply a radial weight designed to minimize the variance of our measurements @xcite given by @xmath38 where @xmath39 is the expected number density of the catalogue at the given redshift and @xmath40 is a scale - independent parameter , which we set to @xmath41 . we also include angular weights to account for redshift failures and fibre collisions . for the cmass sample we apply additional weights to correct for the systematic effect introduced by the local stellar density and the seeing of the observations , as described in detail in @xcite . the left panels of figs . [ fig : measurdr10 ] and [ fig : measurdr11 ] show the resulting angle - averaged correlation function @xmath0 of the sdss - dr10 and dr11 lowz ( upper ) and cmass ( bottom ) samples , respectively , while the left panels show the corresponding clustering wedges @xmath1 ( circles ) and @xmath2 ( squares ) . the anisotropic clustering pattern generated by redshift - space distortions leads to significant differences in the amplitude and shape of the two clustering wedges , with @xmath2 showing a lower amplitude and a stronger damping of the bao peak than @xmath1 . the dashed lines correspond to the best - fitting @xmath3cdm model obtained from the combination of the lowz and cmass dr11 clustering wedges with cmb observations from the planck satellite @xcite and the cmb polarization measurements from wmap @xcite as described in section [ sec : lcdm ] , which provide an excellent description of all our measurements . when comparing our boss clustering measurements with theoretical predictions we assume a gaussian likelihood function of the form @xmath42 . the calculation of the @xmath24 value of a given model requires the knowledge of the inverse covariance matrix of our measurements , which we estimate using mock catalogues matching the selection functions of the lowz and cmass samples . these mocks were constructed from two sets of pthalos realizations @xcite , corresponding to our fiducial cosmology , as described in @xcite and manera et al . ( in preparation ) . our cmass mocks are based on 600 independent simulations with a box size of @xmath43 , while those of the lowz sample were constructed from a separate set of 500 boxes with the same volume . in the construction of these mocks , the northern galactic cap ( ngc ) and southern galactic cap ( sgc ) components of the survey were considered as being independent , and sampled separately from the same pthalos realizations . the volume of the lowz sample allowed us to obtain two separate ngc and sgc mocks per pthalos realization , leading to 1000 independent combined ngc+sgc lowz mock catalogues . the larger volume of the cmass sample makes it more difficult to construct mocks of the ngc and sgc components from the boxes without overlap . this means that the ngc and sgc cmass mocks drawn from the same box are not independent . for dr10 the overlap between the ngc and sgc mocks is approximately 75 per cent of the area covered by the sgc , while for dr11 the whole of the southern component is also covered by the ngc . to account for this overlap in our covariance matrix estimations we construct two sets of 300 independent ngc+sgc cmass mocks , drawing the matched components from different boxes .
|
we explore the cosmological implications of the angle - averaged correlation function , @xmath0 , and the clustering wedges , @xmath1 and @xmath2 , of the lowz and cmass galaxy samples from data release 10 and 11 of the sdss - iii baryon oscillation spectroscopic survey .
our results show no significant evidence for a deviation from the standard @xmath3cdm model .
the combination of the information from our clustering measurements with recent data from the cosmic microwave background is sufficient to constrain the curvature of the universe to @xmath4 , the total neutrino mass to @xmath5 ( 95% confidence level ) , the effective number of relativistic species to @xmath6 , and the dark energy equation of state to @xmath7 .
these limits are further improved by adding information from type ia supernovae and baryon acoustic oscillations from other samples . in particular , this data set combination is completely consistent with a time - independent dark energy equation of state , in which case we find @xmath8 .
we explore the constraints on the growth - rate of cosmic structures assuming @xmath9 and obtain @xmath10 , in agreement with the predictions from general relativity of @xmath11 .
cosmological parameters , large scale structure of the universe
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the general problem that this article is about is the following : consider a nonlinear equation system and an outer iteration method to solve it that consists of solving a subproblem at each step using a second , inner iteration method . now we want to answer the following question : how can we efficiently control the iteration error of the outer iteration method ? or otherwise put : how accurate do we need to solve the inner systems to obtain a certain iteration error for the outer nonlinear equation ? for the case of the outer iteration being newton s method , this problem has been successfully solved . the inner problem is a linear system and the concept of an inexact newton s method was introduced in @xcite . there , at each newton step the inner iteration is terminated when a relative tolerance criterion in the linear residual is satisfied . based on this , it is possible to give conditions on the sequence of relative tolerances to obtain linear , superlinear or quadratic convergence of the inexact newton s method . essentially , the sequence of tolerances has to converge to zero fast enough as the newton scheme progresses and then quadratic convergence is obtained . following up , a strategy that has this property and leads to a very efficient scheme was suggested in @xcite . there , the point is that the initial systems are solved quite coarsely and these schemes are part of widely used software packages , for example of sundials @xcite . note that with this knowledge , the choice of iterative solver for the inner iteration obtains a better basis : if most of the systems are solved very coarsely and thus very few iterations are needed , it is more important that the method is cheap per iteration than how fast we can reach machine accuracy . in this setting , when looking at unsymmetric linear systems and krylov subspace methods , gmres @xcite beats bicgstab @xcite , since it needs only one matrix vector product per iteration instead of two . now when looking at fixed point iterations , an iteration typically consists of evaluating a function and not of solving a system . however , two prominent and important examples where this happens are the picard iteration and fluid - structure interaction and thus we call these inexact fixed point schemes . surprisingly , the problem framed in the first paragraph has not been analyzed for these . for the picard iteration , which is a common tool in the context of the incompressible navier - stokes equations , the evaluation of the right hand side corresponds to solving a linear system . strategies for choosing a termination criterion for the inner iteration are empirically discussed for example in @xcite . in fluid - structure interaction , a standard approach are partitioned coupling schemes , where existing solvers for the subproblems are reused @xcite . commonly in the form of a dirichlet - neumann iteration , this consists of subsequently solving the fluid and the structure problem with appropriate boundary conditions and reasonable tolerances . it is common to formulate the coupling condition at the interface in the form of a fixed point equation . recently , it was suggested to use a time adaptive implicit time integration scheme for fluid structure interaction @xcite , where the time step is chosen based on an error tolerance . as is common in this setting , the tolerances for the solvers for the appearing nonlinear equation systems are chosen such that the iteration error does not interfere with the error from the time integration scheme @xcite , but nevertheless as large as possible to avoid unnecessary computations . thus , it is imperative to be able to control the iteration error . to solve our problem , we proceed in the following way . first , we will review well known results on perturbations of fixed point schemes @xcite . the general idea is then to quantify the iteration errors based on the termination criterion of the inner iteration such that the existing perturbation results can be applied . for the dirichlet - neumann iteration , we first have to extend the perturbation results to the case of a nested fixed point equation of the form @xmath0 . as it turns out , the type of termination criterion chosen in the inner solver is crucial for the answer to our problem in that when using a nonstandard relative criterion , we obtain convergence of the fixed point iteration to the exact solution independently of how accurate we solve the inner systems , leading to an efficient way of controlling the outer iteration error . on the other hand , no general statement can be made on a standard relative or absolute termination criterion , but the analysis suggests that these do not have favorable properties . all of this is confirmed by numerical results , which show that the latter criteria cause convergence to a solution that is farther away from the exact one the less accurate we solve the linear systems or otherwise put , the less error we want in the fixed point equation , the more accurate we have to solve the linear systems . consider the fixed point equation @xmath2 with @xmath3 , @xmath4 closed and where we assume that @xmath5 is lipschitz continuous with lipschitz constant @xmath6 . this implies by the banach fixed point theorem that has a unique solution @xmath7 . furthermore , we consider the perturbed fixed point iteration @xmath8 where @xmath9 is a perturbation that could originate from an iterative solver and for simplicities sake we denote the norm of @xmath9 by @xmath9 as well . we furthermore assume for simplicities sake that @xmath10 is also a self - map on @xmath4 . thus this iteration obtains a solution @xmath11 of the perturbed fixed point equation @xmath12 the question is now : how far is the solution @xmath11 of that equation away from @xmath7 ? the answer is giving by the following theorem , see for example @xcite . [ ortrhetheorem ] for the solutions @xmath11 and @xmath7 of problems and we have : @xmath13 this means that the error is of the order @xmath9 as is to be expected , but interestingly , it becomes larger , the closer the lipschitz constant of @xmath14 is to one or otherwise put , the less contractive the function is . this implies that in these cases , the error will be much larger than @xmath9 and thus a much smaller tolerance would have to be supplied to acchieve the desired error . note that in practice , we typically do not know the lipschitz constant and that in the nonlinear case , it depends on the definition of the domain @xmath4 . thus the important lipschitz constant is the local one in the solution . if we instead consider a sequence of perturbations @xmath15 , respectively a nonconstant perturbation , and thus the iteration @xmath16 the first question is when this sequence converges to @xmath7 . the answer is given by the next theorem , also from @xcite : [ perturbedtheorem ] the iteration converges to the solution of the unperturbed problem if and only if @xmath17 . a specific case is @xmath18 with @xmath19 , which we call the adaptive strategy . as an application of the above theorems , we now analyze the convergence of the picard iteration . this is often employed in the context of the incompressible navier - stokes equation and corresponds to a fixed point iteration for the equation @xmath20 where @xmath21 is an approximation of a jacobian in @xmath22 @xcite . thus , the fixed point iteration @xmath23 is implemented by solving @xmath24 for @xmath25 up to a certain tolerance using an iterative scheme . the scheme can be analyzed either as a fixed point scheme , which results in linear convergence provided that the lipschitz constant @xmath26 of @xmath27 can be bounded from above away from one or as a method of newton type where @xmath28 is an approximation of the exact jacobian and we have linear convergence as long as this approximation is good enough . when solving , either the relative termination criterion @xmath29 the relative termination criterion @xmath30 or the absolute criterion @xmath31 are used , where @xmath32 and @xmath33 are relative and adaptive tolerances . to analyze the consequences of choosing one of these using theorems 1 and 2 , we need to quantify the perturbation in the form . thus , we define @xmath34 to obtain @xmath35 and we can write @xmath36 in the case of the relative termination criterion , we can estimate the norm of the right hand side in by @xmath37 we furthermore have @xmath38 thus , @xmath39 all in all , we obtain with the condition number @xmath40 @xmath41 with the additional and reasonable assumption that @xmath42 is bounded , this is a perturbation of the form and @xmath15 converges to zero independent of the choice of @xmath32 ! thus by theorem [ perturbedtheorem ] this iteration converges to the exact solution independently of how accurate we solve the linear equation systems . we now formulate this as a theorem . [ picardtheorem ] let @xmath43 and the function @xmath28 be given that maps the closed set @xmath44 onto quadratic regular matrices . assume that the function @xmath45 is lipschitz continuous with lipschitz constant @xmath6 and correspondingly has a unique fixpoint @xmath7 . furthermore assume that @xmath46 is bounded on @xmath4 and that the inexact fixedpoint iteration defined by converges to a limit @xmath11 . then @xmath47 , independent of the choice of @xmath32 . in case of the relative criterion , the estimate @xmath48 for the norm of the left hand side in holds which is bounded away from zero provided that @xmath49 is . thus , it is not clear if this iteration satisfies theorem [ perturbedtheorem ] , but in the general case , the iteration will not converge to @xmath7 . similarly if we use the absolute termination criterion , we obtain @xmath50 which is also bounded away from zero if @xmath49 is . numerical results that confirm theorem [ picardtheorem ] and demonstrate that the other two iterations behave like being of the form can be found in section [ picardresultssection ] . we would like to point out that the criterion is sometimes suggested in the literature on the picard iteration , e.g. @xcite , but that an absolute termination criterion is suggested in @xcite . there it is suggested to just `` gain one digit '' , meaning to use a tolerance of 0.1 . now consider two functions @xmath51 and @xmath52 with @xmath53 closed and the fixed point equation @xmath54 again with solution @xmath7 . we now consider an iteration where both the evaluation of @xmath55 and of @xmath56 are perturbed , namely @xmath56 is perturbed by @xmath57 and @xmath55 by @xmath15 : @xmath58 again , assume that this iteration is well defined and that this sequence has the limit @xmath11 . then , we obtain the following theorem . [ fsitheorem ] let @xmath55 and @xmath56 be lipschitz continuous with lipschitz constants @xmath59 and @xmath60 , respectively . assume that @xmath61 . then we have , if @xmath62 for all @xmath63 , that @xmath64 in the case @xmath65 and @xmath66 , we obtain @xmath67 finally , @xmath47 if and only if both @xmath57 and @xmath15 converge to zero . proof : the proof is technically identical to the one of theorem [ ortrhetheorem ] . we have due to the lipschitz continuity @xmath68 and thus in the limit @xmath69 , @xmath70 for a constant perturbation overall , e.g. @xmath62 for all @xmath63 , we obtain in the limit @xmath71 which proves the inequality . if we have constant but separate perturbations @xmath9 and @xmath72 of @xmath56 and @xmath55 , we obtain from @xmath73 in the general case , due to positivity , the right hand side of is zero if and only if both @xmath15 and @xmath57 are such that for @xmath74 or @xmath75 , @xmath76 by an identical proof to theorem 2 , this is the case if and only if both @xmath15 and @xmath57 converge to zero . note that this implies that the sequence @xmath15 perturbing the inner function @xmath55 is less important by a factor of the lipschitz constant @xmath60 of the outer function . thus , a possible strategy is to define @xmath77 meaning that we solve the fluid part less accurate by a factor of @xmath60 . unfortunately , @xmath60 has to be known for this . as an application of the theory from section 2.3 , we consider a problem that is a basic building block in fluid structure interaction , namely the transmission problem , where the laplace equation with right hand side @xmath78 on a domain @xmath4 is cut into two domains @xmath79 using transmission conditions at the interface @xmath80 : @xmath81 we now employ a standard dirichlet - neumann iteration to solve it . using any linear discretization , this corresponds to alternately solving the problems @xmath82 and @xmath83 were problem corresponds to a discretization of the transmission problem on @xmath84 only with dirichlet data on @xmath85 given by @xmath86 on the coupling interface and problem corresponds to a discretization of on @xmath87 only with neumann data on @xmath85 given by the discrete normal derivative of @xmath88 on @xmath85 . it can be shown that convergence of the approximate solutions on the whole domain is equivalent to the convergence of the solution on the interface only @xcite . by considering - as one iteration , we obtain a fixed point formulation @xmath89 where @xmath90 is @xmath91 on the interface , @xmath92 and @xmath93 . hereby @xmath94 is the matrix that computes the discrete normal derivatives in @xmath84 on @xmath85 and @xmath95 is the discrete trace operator with respect to @xmath85 . otherwise put , @xmath95 is the projection of the space that @xmath91 is in onto the space of discrete unknowns on @xmath85 . in practice , the linear equation systems are solved iteratively , typically using the conjugate gradient method ( cg ) up to a relative tolerance of @xmath96 . thus , we obtain a perturbed nested fixed point iteration of the form and the question is now again if we can quantify this perturbation . we have @xmath97 and @xmath98 for the iteration we obtain @xmath99 again , the second factor is what is tested in the termination criterion of cg . in the case of the relative criterion , here stated as @xmath100 we obtain @xmath101 @xmath102 now the point is that since the whole iteration is linear , we can write down a linear mapping that maps @xmath103 onto @xmath104 for arbitrary @xmath63 . let this have lipschitz constant @xmath105 , then we have @xmath106 thus the perturbation has limit zero if @xmath107 . this is the case if and only if the sequence @xmath108 is convergent , which is in fact the case provided that @xmath78 is sufficiently harmless , as can be seen from the literature on domain decomposition methods , e.g. ( * ? ? ? analagously to the picard iteration , if we choose the absolute termination criterion or the relative one based on the right hand side , we obtain a bound of the form @xmath109 respectively @xmath110 again , we can not make a statement on the limit of @xmath15 . in the second case , meaning the iteration with neumann data , we obtain @xmath111 and analogous arguments produce the same results for @xmath57 . thus by theorem [ fsitheorem ] , we have that when using the relative criterion we obtain convergence to the exact solution for any @xmath32 . it is important to note that under the assumptions , all sequences considered , wether perturbed or not , are convergent and therefore , the fixed point iteration will terminate when using the standard criterion @xmath112 however , as just shown , the perturbed iteration converges to an approximation of the unperturbed fixed point and thus , the algorithm can terminate when we are in fact not @xmath113-close to the solution . a further difference between the different iterations that should be stressed is that the iterations perturbed by a constant are fixed point iterations , wheras the schemes with a variable perturbation are in fact , not . thus , the convergence speed , which is otherwise linear with constant @xmath26 , is not clear and numerical evidence suggests that it is in fact slower than for the other iteration . thus , we could argue to employ the schemes with constant perturbation , measure the lipschitz constant numerically after a few iterations and then adjust the tolerance based on theorem [ ortrhetheorem ] or [ fsitheorem ] . unfortunately , it is not clear what the @xmath9 from these theorems is , respectively , it is based on quantities that are hard to measure like @xmath114 . thus , we can not guarantee a certain iteration error in this way , to do this we must employ the nonstandard relative termination criterion . finally , it is important to note that this analysis is mostly relevant to the time independent case . otherwise , when considering this inside an implicit time integration scheme , additional requirements on the solutions appear , namely that the solutions in the subdomains have a certain accuracy . for all numerical experiments , the fixed point iteration is terminated when the norm @xmath115 is smaller than @xmath116 . furthermore , with the exception of the results on the picard iteration , all computations were performed in matlab , where matlab 2012a was used for all computations with the exception of the results in section 3.2.3 , where matlab 2013a was employed . .[problem2xepsminusxstar]@xmath117 ( left ) and @xmath118 ( right ) for different values of @xmath9 and @xmath26 for the solution of the scalar nonlinear equation [ cols="^,^,^,^",options="header " , ] the results are depicted in table [ fsiproblemefficiency ] . for the computation with a - divergence was observed . otherwise , the schemes roughly obey the desired behavior that the error is proportional to tol , although all are above the desired accuracy . otherwise , the scheme corresponding to is about a factor of five faster than that corresponding to and up to a factor of two faster than that corresponding to while providing more accurate results . we considered perturbed fixed point iterations where the perturbation results from inexact solves of equation systems by iterative solvers . thereby , we extended a perturbation result for fixed point equations to the case of a nested fixed point equation . applying these results to the picard- and the dirichlet - neumann iteration for steady states , we showed that these converge to the exact solution indepently of the tolerance in the subsolver , if a specific relative termination criterion is employed . this justifies extremely coarse solves in the inner solvers and suggests the use of gmres as krylov subspace solver for unsymmetric systems . if an absolute or standard relative criterion is used , the theory indicates that we will not converge to the exact solution . numerical results demonstrate this behavior . thus , to obtain a certain accuracy in the fixed point solution when using a standard relative or absolute criterion , we have to solve the inner systems more accurate the tighter the tolerance , whereas for the nonstandard relative criterion we can solve the inner systems very coarsely independent of desired accuracy . numerical results show that this is the most efficient way to treat these systems . part of this work was funded by the german research foundation ( dfg ) as part of the collaborative research area sfb trr 30 , project c2 . furthermore , i d like to thank gunar matthies for performing the numerical experiments on the picard iteration in section 3.1.2 . , _ cfd - based nonlinear computational aeroelasticity _ , in encyclopedia of computational mechanics , e. stein , r. de borst , and t. j. r. hughes , eds . 3 : fluids , john wiley & sons , 2004 , ch . 13 , pp . 459480 .
|
we analyze inexact fixed point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed point iteration .
important applications are the picard iteration and partitioned fluid structure interaction .
we prove that the iteration converges irrespective of how accurate the inner systems are solved , provided that a specific relative termination criterion is employed , whereas standard relative and absolute criteria do not have this property . for the analysis ,
the iteration is modelled as a perturbed fixed point iteration and existing analysis is extended to the nested case @xmath0 .
= 0.9 _ @xmath1 numerical analysis , centre for the mathematical sciences , lund university , box 118 , 22100 lund , sweden + institute of mathematics , university of kassel , heinrich - plett - str .
40 , 34132 kassel , germany + department of mathematics / computer science , university of osnabrck , albrechtstr .
28a , 49076 osnabrck , germany + email : philipp.birkenna.lu..de_ _ keywords : fixed point iteration , picard iteration , transmission problem , dirichlet - neumann iteration , termination criteria _
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in nature and technology , a broad variety of active media where pattern selection occurs is governed by kuramoto sivashinsky type equations . in the presence of an imposed advectional transport @xmath0 in the @xmath1-direction the modified kuramoto sivashinsky equation reads @xmath2 this equation describes two - dimensional large - scale natural thermal convection in a horizontal fluid layer heated from below @xcite and is still valid for a turbulent fluid @xcite , a binary mixture at small lewis number @xcite , a porous layer saturated with a fluid @xcite , etc . in these fluid dynamical systems , except for the turbulent one @xcite , the plates bounding the layer should be nearly thermally insulating ( in comparison to the fluid ) for a large - scale convection to arise . in the problems mentioned , equation governs evolution of temperature perturbations @xmath3 which are nearly uniform along the vertical coordinate @xmath4 and determine fluid currents . the origin of such a frequent occurrence of equation is its general validity , which may be argued as follows . basic laws in physics are conservation ones . this often results in final governing equations having the form @xmath5+\nabla\!\cdot\![\mbox{flux of quantity}]=0 $ ] . such conservation laws lead to kuramoto sivashinsky type equations . while the original kuramoto sivashinsky equation has a quadratic nonlinear term ( cf @xcite ) , this term should be replaced by a cubic one for the systems with the sign inversion symmetry of the fields , which is widespread in nature , or for description of a spatiotemporal modulation of an oscillatory mode . thus the governing equation takes the form . on these grounds , we state that equation describes pattern formation in a broad variety of physical systems . we restrict this paper to the case of convection in a porous medium @xcite for the sake of definiteness ; nonetheless , most of our results may be extended in a straightforward manner to the other physical systems mentioned . is already dimensionless and below we introduce all parameters and variables in appropriate dimensionless forms . in the large - scale ( or long - wavelength ) approximation , which we use , the characteristic horizontal scales are assumed to be large against the layer height @xmath6 . in equation , @xmath7 represents the local supercriticality : @xmath8 is the sum of relative deviations of the heating intensity and of the macroscopic properties of the porous matrix ( porosity , permeability , heat diffusivity , etc . ) from the critical values for the spatially homogeneous case @xcite . for positive spatially uniform @xmath9 , convection sets up , while for negative @xmath9 , all the temperature perturbations decay . in a porous medium @xcite , the macroscopic fluid velocity field is @xmath10 where @xmath11 is the stream function amplitude , and the reference frame is such that @xmath12 and @xmath13 are the lower and upper boundaries of the layer , respectively [ ( b ) ] . though the temperature perturbations obey equation for diverse convective systems , function @xmath14 , which determines the relation between the flow pattern and the temperature perturbation , is specific to each case . notice that @xmath0 is not presented in expression owing to its smallness in comparison to the excited convective currents @xmath15 . the impact of a weak imposed advective flow on the evolution of temperature perturbations is caused by its symmetry properties : the gross advective flux through the vertical cross - section is @xmath0 , while the convective flow @xmath15 possesses zero gross flux and , therefore , yields a less effective heat transfer along the layer @xcite . although equation is valid for a large - scale inhomogeneity @xmath7 , which means @xmath16 , one can set such a hierarchy of small parameters , namely @xmath17 , that a frozen random inhomogeneity may be represented by white gaussian noise @xmath18 : @xmath19 where @xmath20 is the disorder intensity and @xmath21 is the mean supercriticality ( i.e. departure from the instability threshold of the disorder - free system ) . numerical simulation reveals only time - independent solutions to establish in with @xmath22 and such @xmath7 @xcite ; for a small non - zero @xmath0 , stable oscillatory regimes are of low probability by continuity . in the stationary case for @xmath22 the linearized form of equation , i.e. , @xmath23 is a stationary schrdinger equation for @xmath24 with @xmath21 instead of the state energy and @xmath25 instead of the potential . therefore , similarly to the case of the schrdinger equation ( see @xcite ) , all the solutions @xmath26 to the stationary linearized equation are spatially localized for arbitrary @xmath21 ; asymptotically , @xmath27 where @xmath28 is the localization exponent . such a localization can be readily seen for the solution to the nonlinear problem in ( a ) for @xmath29 , @xmath22 , which is a solitary vortex . for @xmath30 the solitary patterns are still exponentially localized [ ( a ) , @xmath31 . however , their localization properties change drastically because instead of the second - order linear ode with respect to @xmath24 , equation , one finds a third - order equation : @xmath32 the two symmetric modes @xmath33 and trivial solution @xmath34 ( @xmath35 ) turn into three modes @xmath36 , @xmath37 , of equation with @xmath38 , @xmath39 ( see sample spectrum of @xmath40 in ; cf @xcite for details ) . specifically , @xmath41-mode is the successor of @xmath42 , @xmath43-mode is the one of the trivial homogeneous mode , and @xmath44-mode is that of @xmath45 . thus , the upstream flank of the localized pattern is now composed by two modes decaying in the distance from the pattern : @xmath46 where functions @xmath47 and @xmath48 neither grow nor decay over large distances . the @xmath41-mode , which disappears for @xmath22 , i.e. @xmath49 , decays slowly for a small finite @xmath0 , prevails over the @xmath43-mode decaying rapidly , and , thus , determines the upstream localization properties of the pattern . the upstream localization length @xmath50 can become remarkably large leading to upstream delocalization of patterns , which can be seen in . one should keep in mind , that consideration of solitary patterns makes sense where such patterns can be distinguished , i.e. , are sparse enough in space . this is the case of negative @xmath51 . in , for a sample realization of @xmath18 , one can see that localized patterns can be discriminated for @xmath52 , @xmath53 and the localization properties are very well pronounced for @xmath29 . here we would like to emphasize the fact of existence of convective currents below the instability threshold of the disorder - free system . these currents considerably and nontrivially affect transport of a pollutant ( or other passive scalar ) , especially when its molecular diffusivity is small in comparison to the thermal one , which is quite typical in nature ( for instance , at standard conditions the molecular diffusivity of nacl in water is @xmath54 against the heat diffusivity of water which is @xmath55 ) . transport of a nearly indiffusive passive scalar , quantified by the effective ( or eddy ) diffusivity coefficient , is the object of our research , as a `` substance '' which is essentially influenced by these localized currents and , thus , provides an opportunity to observe manifestation of disorder - induced phenomena discussed in @xcite . in @xcite we studied the problem for the case of no advection ( @xmath22 ) and calculated ( both numerically and analytically ) the enhancement of the effective diffusivity by disorder - induced currents ; this enhancement is especially strong for low molecular diffusivity deep below the instability threshold of the disorder - free system ( see ) . in this paper we address the role of an imposed advection in this problem . the interest to advection is provoked by its dramatic influence on localization properties , i.e. , the upstream delocalization of convective currents that is described above . we expect this delocalization to result in a giant increase of the effective diffusivity for a nearly indiffusive pollutant and , particularly , in the lowering of the mean supercriticality ( @xmath21 ) value at which the transition from sets of localized convective currents to an almost everywhere intense ` global ' flow occurs . in this section we describe the transport of a passive pollutant by a steady convective flow ; `` passive '' means that the flow is not influenced by the pollutant . the assumption of passiveness is practically relevant because ( biologically / chemichally ) significant concentrations of a pollutant can be very small and mechanically negligible . the flux @xmath56 of the pollutant concentration @xmath57 is @xmath58 where the first term describes the convective transport , the second one represents the molecular diffusion , and @xmath59 is the molecular diffusivity . the establishing time - independent distributions of the pollutant obey @xmath60 [ with account for ] yields a distribution of @xmath57 which is uniform along @xmath4 and is determined by @xmath61 where @xmath62 is the constant pollutant flux along the layer . detailed derivation of equation for @xmath22 can be found in @xcite where it was performed in the spirit of the standard multiscale method ( interested readers can consult , _ e.g. _ , @xcite ) . remarkably , advection velocity @xmath0 is not presented in the last equation : its direct contribution to convective currents transferring the pollutant is small in comparison to the one of excited convective currents . instead , it influences the heat transfer and , consequently , excited flows , drastically changing properties of the field @xmath26 . notice that , for the other convective systems which we mentioned in , the result differs only in the factor ahead of @xmath63 . thus we come to introducing the effective diffusivity for the system under consideration ( general ideas on the effective diffusivity in systems with irregular currents can be found , e.g. , in @xcite ) . let us consider the domain @xmath64 $ ] with the imposed concentration difference @xmath65 at the ends . then the establishing pollutant flux @xmath62 is defined by the integral [ cf ] @xmath66 for a lengthy domain the specific realization of @xmath18 becomes insignificant : @xmath67 hence , @xmath68 which means that @xmath69 can be treated as an effective diffusivity . the effective diffusivity @xmath70 turns into @xmath59 for vanishing convective flow . for small @xmath59 the regions of the layer where the flow is damped , @xmath71 , make large contribution to the mean value appearing in and diminish @xmath69 , thus , leading to the locking of the spreading of the pollutant . the disorder strength @xmath20 can be excluded from equations by the appropriate rescaling of parameters and fields . as a consequence , the results on the effective diffusivity can be comprehensively presented in terms of @xmath72 , @xmath73 , @xmath74 , and @xmath75 : @xmath76 provides calculated dependencies of effective diffusivity @xmath73 on @xmath74 for moderate and small values of molecular diffusivity @xmath72 . concerning these dependencies the following is worth noticing : * ( a ) * for small @xmath72 a quite sharp transition of effective diffusivity @xmath73 between moderate values and ones comparable with @xmath72 occurs near @xmath77 ( note logarithmic scale of the vertical axis ) , suggesting the transition from an almost everywhere intense ` global ' flow to a set of localized currents to take place . * ( b ) * in the presence of a weak imposed advection , @xmath78 , the transition to ` global ' flow occurs at the value of @xmath74 which is considerably lower than that without advection . * ( c ) * below the instability threshold of the disorder - free system , where only sparse localized currents are excited , the effective diffusion can be significantly enhanced by these currents . * ( d ) * the disorder - induced enhancement of the effective diffusivity is especially drastic in the presence of an imposed advection ; e.g. , for @xmath79 , @xmath80 , the effective diffusivity is increased by one order of magnitude compared to the molecular diffusivity without advection ( @xmath81 ) and by two orders of magnitude for @xmath78 . the effective diffusivity can be analytically evaluated for a small molecular diffusivity ( @xmath82 ) and sparse domains of excitation of convective currents ( the spacial density of the excitation domains @xmath83 ) . in @xcite it was evaluated for the case of no advection , @xmath84 where one can use the asymptotic expressions for the density of the excitation domains @xmath85 , @xmath86 and @xmath87 which are valid for @xmath88 . the latter expression is known from the classical theory of al ( e.g. , see @xcite ) . in the following we advance the evaluation procedure realized in @xcite in order to account for the asymmetry between up- and downstream localization exponents . now we calculate the average @xmath89 due to ergodicity , this average over @xmath1 for a given realization of @xmath18 coincides with the average over realizations of @xmath18 at a certain point @xmath90 . we set the origin of the @xmath1-axis at @xmath90 and find @xmath91 . when the two nearest to the origin excitation domains are distant and localized near @xmath92 and @xmath93 ( see ) , @xmath94 where @xmath95 and @xmath96 characterize the amplitude of temperature perturbation modes excited around @xmath97 and @xmath98 , respectively . for small @xmath72 and density @xmath85 , the contribution of the excitation domains to @xmath99 is negligible against that of the extensive regions where flow is weak , but @xmath100 is still larger than @xmath72 . therefore , one may be not very subtle with `` cores '' of excitation domains and may utilize expression even for small @xmath101 : @xmath102 where @xmath103 [ @xmath104 is the density of the probability to observe the nearest right [ left ] excitation domain at @xmath105 [ @xmath106 . for probability distribution @xmath107 , one finds @xmath108 , i.e. , @xmath109 . hence , @xmath110 , and probability density @xmath111 . as regards averaging over @xmath112 , it is important that the multiplication of @xmath112 by factor @xmath113 is effectively equivalent to the shift of the excitation domain by @xmath114 , which is insignificant for @xmath115 in the limit case that we consider . hence , one can assume @xmath116 ( the topological difference between different combinations of signs of @xmath112 is not to be neglected ) and rewrite equation as @xmath117 . \nonumber\end{aligned}\ ] ] for @xmath118 and @xmath43-mode dominating over @xmath41-mode ( that is the case in ) , the last formula yields @xmath119 here we assume that advection is weak and suppresses thermal convection in a negligible fraction of the excitation centers and the asymptotic expression is still valid . noticeable difference between equation for @xmath81 , i.e. , @xmath120 , and equation is actually insignificant up to our approximations , because moderate number @xmath121 risen to small power @xmath122 , @xmath123 which is the ratio of these equations , approximately equals 1 . is simpler than that we consider here and admits analytical evaluation of integrals with a fewer number of approximations . ] for instance , in , the curves given by analytic expressions and with @xmath120 are visually undistinguishable . for small finite @xmath75 , smallness of @xmath43 in equation gives rise to a significant enhancement of effective diffusivity @xmath73 , which is in agreement with the results of numerical simulation presented in . shows that for @xmath78 the effective diffusivity in the presence of advection is always stronger than without it ; the larger the difference between the effective and the molecular diffusivity the stronger advectional enhancement of the effective diffusivity is . for instance , for @xmath124 , @xmath80 the effective diffusivity in the presence of advection @xmath78 is by factor 10 larger than without advection , and this factor grows as @xmath72 decreases . noteworthy , for @xmath78 expression provides slightly overestimated value of the effective diffusivity while for @xmath81 the analytical estimation is accurate . the inaccuracy appears because in our analytical theory we ignore three factors : ( a ) decrease of the spatial density of the excitation centers owing to advectional suppression ( washing - out ) of weak excitation centers ; ( b ) for small @xmath75 the rapidly decaying upstream @xmath41-mode is significant because of the smallness of the slowly decaying @xmath43-mode ; and ( c ) as the advection strengthens , currents in some excitation domains disappear via a hopf bifurcation @xcite and thus there is non - zero probability to observe oscillatory flows for small @xmath75 even though there is no stable time - dependent solutions for @xmath81 . unfortunately , inaccuracies caused by these three assumptions can not be minimized simultaneously : the first and third assumptions require @xmath125 , while the second one needs @xmath75 to be small but finite . oscillatory localized patterns discovered in the dynamic system for non - zero @xmath75 ( ( a ) ; see @xcite for details ) are statistically improbable and rare when @xmath75 is small . notably , their relative contribution to the effective diffusivity is much larger than their fraction among the excited localized patterns . in ( b ) one can see the soaring of the effective diffusivity along a finite region as a localized pattern turns oscillatory ( @xmath126 ) before disappearing ( @xmath127 ) . ( a ) reveals the origin of this soaring : the oscillatory pattern is not so well localized as the time - independent one . indeed , the localization properties of the oscillatory pattern of frequency @xmath128 are determined by the following linearization of equation : @xmath129 in contrast to , this is already a 4th - order differential equation , which yields four localization exponents . the newly appeared 4th mode possesses @xmath130 , i.e. , decays slowly , and contributes to the downstream flank of localized patterns ( evidence of these facts is beyond the scope of this paper and will be presented elsewhere ) . as well as the @xmath43-mode results in upstream delocalization of time - independent patterns , the new mode leads to downstream delocalization of oscillatory patterns , which appear to be weakly localized both up- and downstream , as one sees this in ( a ) . nevertheless , owing to the smallness of the fraction of the oscillatory patterns among all the localized patterns at small @xmath75 , their contribution to the effective diffusivity over large domains is still negligible . this is additionally confirmed by the accuracy of our analytical theory disregarding oscillatory currents [ equation ] . meanwhile , the analytical theory accounting for the oscillatory patterns should involve the distribution of frequencies of excited patterns , which are to be determined only from the nonlinear problem : this is not an analytically solvable problem . we have studied the transport of a pollutant in a horizontal fluid layer by spatially localized two - dimensional thermoconvective currents appearing under frozen parametric disorder in the presence of an imposed longitudinal advection . though we have considered the specific physical system , a horizontal porous layer saturated with a fluid and confined between two nearly thermally insulating plates , our results can be in a straightforward manner extended to a broad variety of fluid dynamical systems ( like ones studied in @xcite ) . we have calculated numerically the dependence of the effective diffusivity on the molecular one and the mean supercriticality for a non - zero advection strength ( see ) . the results reveal that advectional delocalization of convective currents greatly assists transfer of a nearly indiffusive pollutant ( @xmath131 ) below the instability threshold of the disorder - free system : the effective diffusivity can become by several orders of magnitude larger in comparison to that without advection . the analytical theory focusing on advectional delocalization of localized current patterns yields results which are in a fair agreement with the results of numerical simulation . this correspondence confirms our treatment of importance of disorder - induced patterns and their localization properties in active / dissipavite media , which provoked works @xcite .
|
frozen parametric disorder can lead to appearance of sets of localized convective currents in an otherwise stable ( quiescent ) fluid layer heated from below .
these currents significantly influence the transport of an admixture ( or any other passive scalar ) along the layer .
when the molecular diffusivity of the admixture is small in comparison to the thermal one , which is quite typical in nature , disorder can enhance the effective ( eddy ) diffusivity by several orders of magnitude in comparison to the molecular diffusivity .
in this paper we study the effect of an imposed longitudinal advection on delocalization of convective currents , both numerically and analytically ; and report subsequent drastic boost of the effective diffusivity for weak advection .
_ special issue _ : article preparation , iop journals + disorder in operation conditions of a dynamic system is known to be able to play not only trivial destructive role , distorting the system behavior , but also constructive one , inducing certain degree of order and leading to various non - trivial effects : anderson localization @xcite , stochastic @xcite and coherence resonances @xcite , noise - induced synchronization @xcite , etc .
one of the most distinguished and fair effects is the anderson localization ( al ) , which is the localization of states in spatially extended linear systems subject to a frozen parametric disorder ( random spacial inhomogeneity of parameters ) .
al was first discovered and discussed for quantum systems @xcite .
later on , investigations were extended to diverse branches of semiclassical and classical physics : wave optics @xcite , acoustics @xcite , etc .
the phenomenon was comprehensively studied and well understood mathematically for the schrdinger equation and related mathematical models @xcite .
the role of nonlinearity in these models was addressed in literature as well ( e.g. , @xcite ) .
while extensively studied for conservative media , the localization phenomenon did not receive comparable attention for active / dissipative ones , like in problems of thermal convection or reaction - diffusion .
the main reason is that the physical interpretation of formal solutions to the schrdinger equation is essentially different from that of governing equations for active / dissipative media ; therefore , the theory of al may be extended to the latter only under certain strong restrictions ( see @xcite for reference ) .
nevertheless , effects similar to al can be observed in fluid dynamical systems @xcite . in @xcite we addressed the problem where localized thermoconvective currents excited in a horizontal porous layer under frozen parametric disorder ( spacial inhomogeneity of the macroscopic permeability , the heat diffusivity , etc . )
drastically influence the process of transport of a passive scalar ( e.g. , a pollutant ) along the layer .
below the threshold of instability of the disorder - free system , the effective diffusivity quantifying this transport has been found to be faithfully determined by localization properties of patterns .
meanwhile , in @xcite these properties have been revealed to be greatly affected by a weak imposed longitudinal advection .
hence , one can expect a weak advection to lead to a significant enhancement of the effective diffusivity of a nearly indiffusive pollutant .
treatment of this effect is the subject of the present paper .
the paper is organized as follows . in
we formulate the specific physical problem that we deal with and introduce the relevant mathematical model ; in particular , we discuss disorder - induced excitation of localized currents below the instability threshold of the disorder - free system and advectional delocalization of these patterns . presents the results of a numerical simulation and calculation of the effective diffusivity . in
we develop an analytical theory for the effective diffusivity in the presence of an imposed advection . ends the paper with conclusions .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
supermassive black holes ( smbh ) , with masses between @xmath4 and @xmath5 , are found at the center of nearly every galaxy where there have been sensitive searches . while we suspect that the seeds for these smbh might all have a common origin , their formation mechanism is not well understood . many galaxies at redshifts @xmath6 contain quasars , i.e. , smbh in the midst of luminous accretion . the soltan argument @xcite suggests black hole masses are largely accounted for via growth due to luminous accretion . there are far fewer quasars at low redshift @xcite , implying that at some point , smbh cease their luminous accretion . the quasar _ turnoff _ mechanism is not well understood ( thacker et al . finally , the black hole mass - stellar velocity ( @xmath7 ) relation @xcite suggests that smbh are in some way co - evolving with their host galaxies , but the @xmath7 relation merely describes an end state and is not a theoretical explanation . as we know of no rapid process by which smbh can lose mass , the evolutionary tracks for smbh consist of stages at progressively higher masses . these stages must , in order , involve ( 1 ) a formation mechanism , ( 2 ) a period of luminous growth ( the ` quasar phase ' ) , perhaps along with a period of nonluminous growth ( ` turnoff ' ) , and ( 3 ) a period in which smbh lie at the centers of galaxies without substantial growth , as we observe them today . much current theoretical research concerns the origin and turnoff phases of quasar evolution , while the ` quasar phase ' appears to be relatively well understood . in this paper we find a new feature of smbh evolution during their quasar phase . we examine the evolution of the quasar locus in mass - luminosity space as a function of redshift . the @xmath8 locus is traditionally shown with all quasars on the same plot ( as in figure [ fig : allml ] ) . the large size of the sloan digital sky survey ( sdss ) dr5 quasar catalogue @xcite allows a subdivision into several redshift bins , each containing thousands of quasars . the eddington limit produces an absolute upper bound on the luminosity of quasars which is proportional to the black hole mass , @xmath9 @xcite . while strictly applicable only for a spherical accretion flow of ionized gas , models for more realistic accretion configurations with rotation are still limited by a luminosity of this order ( except under special circumstances , e.g. begelman 2002 ) . using a smaller data set ( @xmath10 ) than sdss , kollmeier et al . ( 2006 ) appeared to confirm the applicability of @xmath11 to quasars , using an @xmath8 locus ( similar to figure [ fig : allml ] ) to show that the most luminous quasars reach but do not exceed @xmath12 . for the lowest black - hole masses at every redshift , we confirm the conclusion of @xcite , but we show that the quasars with the highest smbh mass at every redshift fall well short of @xmath11 . in [ sec : methods ] , we review the methods used to estimate masses and bolometric luminosities . while we have added nothing original to this methodology , the remainder of our results are entirely dependent upon its accuracy . in [ sec : lm ] , we subdivide the sdss dr5 quasar catalogue by redshift and show that the quasar mass - luminosity distribution does not match what we should expect given our current theoretical understanding of quasar accretion . in particular , we show that instead a sub - eddington boundary ( seb ) is present in each redshift bin . we also use our limited statistics to make a first estimate for how the seb evolves with redshift . in [ sec : tests ] , we consider several alternative explanations for the discrepancies between the observed quasar locus and the eddington luminosity , focusing on potential sources of measurement uncertainty or bias . in [ sec : discussion ] , we comment on the potential implications of our new results . the remainder of this work relies upon an ability to accurately estimate the bolometric luminosities and central black hole masses of quasars at cosmological redshifts . the primary results in this work are drawn from the shen et al . ( 2008 ) virial mass catalogue for sdss dr5 @xcite quasars . the luminosity determination is fairly straightforward and uses the relatively settled techniques discussed in richards et al . the mass determination , on the other hand , uses relatively new techniques , and has a larger uncertainty . therefore , we review here the basic assumptions in virial mass estimation . potential sources of error or bias in the shen et al . virial masses are discussed in more detail in [ sec : tests ] the determination of black hole masses from spectral emission lines relies upon two basic assumptions : ( 1 ) that the orbital velocity of gas in the broad - line region ( blr ) is dominated by the virial velocity due to the central black hole and ( 2 ) that there is a scaling relationship between luminosity and radius . as a result of ( 1 ) , we can calculate the mass @xmath13 of the black hole from emission lines in the blr as @xmath14 where @xmath15 is the radius and @xmath16 the velocity of gas emitting the blr spectral lines . marconi et al . ( 2009 ) suggest corrections to the virial approximation for radiation pressure might be needed at high luminosity , particularly when using civ lines to determine @xmath16 . this first scaling relationship is based upon black hole masses determined using reverberation mapping , which uses the time delay between variability in the continuum and emission lines to determine @xmath15 ( cf . peterson & horne 2004 ) . virial mass estimates also require assumption ( 2 ) , an empirical scaling relationship very close to the @xmath17 that we would expect for a black body @xcite , although thermal processes in the accretion disc are likely to be substantially more complex . this second assumption transforms ( [ eq : virial ] ) into a scaling relation of the form @xmath18 \nonumber,\end{aligned}\ ] ] where fwhm is the full width at half - maximum of the corresponding line profile , @xmath19 is the luminosity per unit wavelength at rest - frame wavelength @xmath20 , and @xmath21 and @xmath22 are constants . vestergaard & peterson ( 2006 ) determined these constants in black hole mass scaling relations based on h@xmath0 and civ emission lines ( vp06 ) , while mclure & jarvis ( 2002 ) and mclure & dunlop ( 2004 ) developed a mass relation ( md04 ) by scaling mgii - based estimates against h@xmath0-based estimates . the scaling relations used in this work are summarized in table [ table : vmesummary ] . multiple scaling relations are required because sdss spectra only cover the range 39009100 , so that none of these three blr emission lines are accessible over the entire redshift range of the sdss catalogue . the additive constants @xmath23 are determined by calibrating virial mass scaling relationships against other methods for black hole mass estimation as part of a ` black hole mass ladder'@xcite . direct estimates based upon local stellar and gas kinematics ( cf . ferrarese & ford 2005 provide our best estimates for nearby black holes . reverberation mapping is calibrated against these estimates , and the virial mass scaling relations are in turn calibrated against reverberation mapping . these calibrations use very few quasars compared to the 62,185 in the shen et al . ( 2008 ) sample : only 28 quasars are used to calibrate the h@xmath0 and 27 to calibrate the civ scaling relations@xcite . .summary of the virial mass estimates used [ cols="^,^,^,^,^,^",options="header " , ] [ table : tests ] a large , unknown systematic error in virial @xmath24 measurements is the most plausible explanation by comparison for how the seb might be produced without additional underlying physics , but requires that virial mass estimates are @xmath25 dex incorrect , reverberation mapping is @xmath25 dex incorrect , and that quasars at low redshift lie within a spread of just 1.3 decades of central black hole mass . it would appear likely that the explanation for these new limits lies in new theory rather than observational errors . quasar catalogues such as the sloan digital sky survey are now large enough to yield useful statistical conclusions after being subdivided . in this work , we have explored just a few of the many ways in which the catalogue might be divided . it is immediately apparent that when the quasar distribution is considered at any given redshift , the most massive central black holes do not accrete at their eddington luminosity , but rather all fall well short of @xmath26 . while there are many measurements that contribute to these quasar distributions , each with its own set of assumptions , a close examination reveals no evidence that any these are responsible for massive quasars remaining sub - eddington . rather , it appears that this ` sub - eddington boundary ' is a new physical limit for quasars . thus , quasar accretion must be more complicated than had been previously thought . the sub - eddington boundary helps to recast the problem in a two - dimensional form . this form emphasizes the luminosity as being a function of the mass and accretion rate , fundamental properties of the quasar , in a non - trivial way . quasar luminosity functions and quasar mass functions are both one - dimensional projections of the mass - luminosity plane . the full two - dimensional quasar distribution contains complexities difficult to discern from either of these projections . the sub - eddington boundary presents a theoretical puzzle . not only must an improved theory explain the sub - eddington boundary , but it must also explain its detailed shape and explain the evolution of that shape with redshift . while the slope of the sub - eddington boundary may not vary greatly with redshift , the mass scale at which the boundary becomes relevant is larger at higher redshift . the results in this paper can be used to develop a series of tests which should be capable of discriminating between models . given the complications introduced by the sub - eddington boundary , the case could be made that we really understand surprisingly little about the supermassive black holes that appear to be at the center of nearly every galaxy . not only is their seeding mechanism unknown , but as shown in this paper , the growth mechanism is quite poorly understood , contrary to expectations . the eddington limit is relevant at low black hole masses , but is only part of the story . the sub - eddington boundary developed in this work is the latest addition to a growing collection of puzzles regarding every phase of the evolution of galactic nuclei and surrounding regions . the authors would like to thank mihail amarie , forrest collman , doug finkbeiner , margaret geller , lars hernquist , gillian knapp , avi loeb , ramesh narayan , jerry ostriker , and michael strauss for valuable comments . this work was supported in part by chandra grant number go7 - 8136a ( chandra x - ray center ) .
|
we use 62,185 quasars from the sloan digital sky survey dr5 sample to explore the relationship between black hole mass and luminosity .
black hole masses were estimated based on the widths of their h@xmath0 , mgii , and civ lines and adjacent continuum luminosities using standard virial mass estimate scaling laws .
we find that , over the range @xmath1 , the most luminous low - mass quasars are at their eddington luminosity , but the most luminous high - mass quasars in each redshift bin fall short of their eddington luminosities , with the shortfall of order ten or more at @xmath2 .
we examine several potential sources of measurement uncertainty or bias and show that none of them can account for this effect .
we also show the statistical uncertainty in virial mass estimation to have an upper bound of @xmath3 dex , smaller than the 0.4 dex previously reported .
we also examine the highest - mass quasars in every redshift bin in an effort to learn more about quasars that are about to cease their luminous accretion .
we conclude that the quasar mass - luminosity locus contains a number of new puzzles that must be explained theoretically .
[ firstpage ] black hole physics galaxies : evolution galaxies : nuclei quasars : general accretion , accretion discs
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the framework of the semiclassical theory of waves [ 1 - 5 ] , i.e. , short wavelength asymptotics , the uniform ( global ) description of the wavefield is complicated by the formation of caustic singularities [ 1 , 2 , 6 - 8 ] . although a complete and deep understanding of the wavefield structure near caustic regions is obtained on the basis of catastrophe theory @xcite and the unfolding of the corresponding singularities can be treated by means of symplectic techniques @xcite , the application of such methods to realistic cases , e.g. , to waves in magnetically confined plasmas @xcite , appears rather difficult . therefore , with specific regard to physical applications , several asymptotic methods have been developed which yield numerically tractable equations , though being limited concerning the global properties of the asymptotic solutions . such asymptotic techniques can be classified into two different families , depending on whether the relevant wave equation is described in the phase space , _ microlocal techniques _ , or directly in the configuration space where the wavefield is defined , _ quasi - optical methods_. this work aims to give a detailed comparative analysis of two such techniques , namely , the wigner - weyl kinetic formalism [ 11 - 13 ] and the complex geometrical optics ( cgo ) method [ 14 - 17 ] which can be considered as benchmarks for microlocal and quasi - optical methods , respectively . specifically , in sec.2 , the wigner - weyl formalism and the complex geometrical optics method are reviewed and compared . in particular , it is pointed out that , within the wigner - weyl formalism , physically meaningful solutions should have a specific form , referred to as momentum distribution , which is characterized in sec.3 . on the basis of the mathematical properties of momentum distributions , our main result is obtained in sec.4 . in particular , it is shown that , in correspondence to appropriate boundary conditions , there exists a specific asymptotic solution of the wave kinetic equation relevant to the wigner - weyl formalism that can be written in terms of the corresponding solution of the complex geometrical optics equations . this allows us to relate the two considered methods as well as to determine the specific class of boundary conditions for which they are equivalent . in sec.5 , this general result is illustrated by means of an analytically tractable example , i.e. , the propagation of a gaussian beam of electromagnetic waves in an isotropic `` lens - like '' medium . in conclusion , a summary of the main results is given in sec.6 . in order to set up the framework , let us discuss the relevant boundary value problem for a generic _ scalar _ pseudodifferential wave equation , together with the required mathematical definitions . specifically , we will consider the case of a scalar ( real or complex ) wavefield @xmath0 propagating in the @xmath1-dimensional linear space @xmath2 with @xmath3 a generic set of cartesian coordinates and denote by @xmath4 the corresponding coordinates in the dual space @xmath5 . time - dependent wavefields are included as one of the coordinates can play the role of time , e.g. , @xmath6 , @xmath7 being a reference speed , and the corresponding dual coordinate is related to frequency , e.g. , @xmath8 . to some extent , the results for a scalar wave equation are valid also for a multi - component wave equation as the latter can be reduced to a set of _ independent _ scalar equations far from mode conversion regions @xcite . thereafter , the wigner - weyl formalism will be formulated entirely in the space @xmath9 , with coordinates @xmath10 , which is viewed as the trivial cotangent bundle @xcite over the configuration space @xmath11 where the wavefield is defined . no explicit reference to the propagation direction is made , differently from the classical derivations @xcite . as for the cgo method , it has been originally developed for solving the second - order partial differential equation relevant to the propagation of electromagnetic wave beams in stationary spatially nondispersive media . hence , we will need to discuss its application to generic pseudodifferential equations . in particular , it is shown that the cgo method yields an approximation of the wavefield directly in the configuration space , provided that the wave equation satisfies an appropriate condition . first , let us define the class of wave equations undergone to solution . a pseudodifferential wave equation is an equation of the form @xmath12 which admits propagating wave solutions . the operator @xmath13 is a pseudodifferential operator [ 3,18 - 20 ] acting on the wavefield as a fourier integral operator @xcite characterized by the bilinear phase function @xmath14 . here , @xmath15 belongs to a particular class of smooth functions , referred to as _ symbols _ , which , roughly speaking , behave like a polynomial in @xmath16 for @xmath17 large enough . specifically , a smooth function @xmath18 with @xmath19 and @xmath20 is a symbol of order @xmath21 if for every multi - indices @xmath22 , @xmath23 there is a constant @xmath24 such that @xmath25 . in virtue of the symbol estimate ( [ 2n ] ) the integral in ( [ 1n ] ) makes sense for @xmath26 , the space of tempered distribution @xcite . moreover , the scale length @xmath27 characterizes the variations of symbols with respect to the spatial coordinate @xmath28 and it can be eliminated by the rescaling @xmath29 and @xmath30 . it is worth noting that any linear differential operator with smooth and bounded coefficients is a pseudodifferential operator @xcite . boundary conditions of cauchy type are given on an @xmath31-dimensional hypersurface @xmath32 : for simplicity , one can assume @xmath32 to be the hyperplane @xmath33 where the wavefield @xmath34 , @xmath35 , is assigned together with as many derivatives @xmath36 as appropriate . we are interested in semiclassical solutions for which only the covectors @xmath16 with @xmath37 are significant , in the integral in ( [ 1n ] ) . as a consequence , the wavefield should be a highly oscillating function on the ( large ) scale length @xmath27 and it should correspond to a specific set of highly oscillating boundary conditions of the form @xmath38 the amplitude @xmath39 being slowly varying , that is , @xmath40 . in the wigner - weyl formalism , the weyl - symbol map @xmath41 is applied in order to represent the wave equation ( [ 1n ] ) in @xmath42 which is naturally endowed with a phase - space structure . the weyl - symbol map transforms an operator @xmath43 , @xmath44 being the space of schwartz s functions @xcite , into a tempered distribution by acting on the schwartz kernel @xmath45 of the operator @xmath46 according to @xcite @xmath47 it is worth noting that for the pseudodifferential operator in ( [ 1n ] ) the schwartz kernel is @xmath48 thus , the image of @xmath13 under the weyl - symbol map amounts to the formal series of decreasing order symbols @xmath49 where taylor expansion has been used and @xmath50 . series of that kind admit always an asymptotic resummation @xcite to a symbol of order @xmath51 , @xmath52 which is referred to as _ weyl symbol _ of @xmath13 . in ( [ weyl ] ) the @xmath53 denotes the asymptotic equivalence of symbols @xcite . on the other hand , one can consider the correlation operator @xcite @xmath54 whose schwartz kernel is given by the tensor product @xmath55 , then the weyl - symbol map yields the _ wigner function _ @xmath56 the wave equation ( [ 1n ] ) can be written in the equivalent form @xmath57 and , on applying the weyl - symbol map , one gets @xcite @xmath58 in the semiclassical limit one has @xmath59 so that , on assuming the same ordering for the wigner function @xcite , the foregoing equation separates into [ 1.1 ] @xmath60 where @xmath61 and @xmath62 are the real and imaginary parts of the weyl symbol ; in particular , one has @xmath63 which is the condition for weak absorption and/or instabilities @xcite . equation ( [ 1.1a ] ) is a constraint to eq.([1.1b ] ) which , on the other hand , has the form of a kinetic equation in the @xmath64-@xmath16 phase space , @xmath65 being the corresponding poisson brackets . we will refer to the whole system ( [ 1.1 ] ) as the _ wave kinetic equation_. in general , a solution of the wave kinetic equation is a tempered distribution , however , one usually restricts the class of solutions to semiclassical measures . this allows us to make sense of the integrals of the form @xcite @xmath66 which expresses the expectation value of a physical quantity represented by the pseudodifferential operator @xmath46 with weyl symbol @xmath67 . in the following we will assume further regularity with respect to @xmath64 so that the expectation values of physical quantities can be defined _ locally _ , that is , [ physics ] @xmath68 makes sense as a smooth function in @xmath69 . in particular , the wavefield intensity amounts to @xmath70 ) . on the hyperplane @xmath71 the wavefield @xmath34 has been assigned and one can compute the corresponding wigner function @xmath72 , @xmath73 being the correlation operator associated to @xmath74 and @xmath75 the coordinates dual to @xmath76 ; then a solution @xmath77 should match @xmath78 in some appropriate sense . specifically , one should impose that the local value @xmath79 of any physical quantity evaluated on @xmath32 is the same whether it is evaluated by @xmath77 or by @xmath78 . within this formulation , the weyl symbol @xmath67 should be restricted to @xmath80 where @xmath81 is defined . one can note that the suitable embeddings @xmath82 of @xmath80 into @xmath83 such that @xmath82 lies over @xmath32 , i.e. , @xmath84 with @xmath85 the canonical projection @xcite , are of the form @xmath86 with @xmath87 a generic smooth function ; correspondingly , the restriction of a symbol is readily defined as @xmath88 . then the boundary value conditions read [ 6n ] @xmath89 which is equivalent to @xmath90 the function @xmath91 can not be arbitrary as ( [ 6nb ] ) should satisfy the constraint ( [ 1.1a ] ) which reads @xmath92 hence , the appropriate functions @xmath91 are obtained on solving the so - called _ local dispersion relation _ evaluated on @xmath32 . since @xmath32 has been assumed to be noncharacteristic , i.e. , @xmath93 on @xmath82 , the function @xmath91 is well defined and smooth at least locally , in view of the implicit function theorem . on the other hand , it is not unique since ( [ 7n ] ) may have multiple solutions , each one corresponding to a specific branch of the dispersion relation . in virtue of the superposition principle for linear wave equations , the total wavefield is a linear superposition of the contributions from each branch of the dispersion relation , the coefficients being determined by the cauchy boundary values of the normal derivatives @xmath94 . therefore , one has a specific cauchy boundary value problem for the wigner function of each branch and the sum over all branches yields the total wigner function . since one has @xmath95 where the indices @xmath96 and @xmath97 run over all branches , and , on noting that the average @xmath98 over short scale oscillations cancels out the mixed terms , @xmath99 for @xmath100 , whereas @xmath101 , one gets @xmath102 that is , the projection of the total wigner function yields the _ averaged wavefield intensity _ and , thus , it does not account for , e.g. , the formation of short - scale diffraction patterns . let us now turn to the complex geometrical optics ( cgo ) method and , in particular , let us discuss its application to pseudodifferential wave equations . this is based on approximating the solution of ( [ 1n ] ) by a smooth wave function of the form @xmath103 where , according to the semiclassical limit , @xmath104 and @xmath105 . in addition to the standard oscillating exponential @xmath106 , the wave object ( [ 1.3 ] ) exhibits a novel scale length @xmath107 which accounts for intermediate - scales variations of the amplitude profile @xmath108 with @xmath109 . in general , such an intermediate scalelength @xmath110 can be determined by both ( strong ) absorption @xcite and diffraction @xcite ; however , in this paper , it is assumed that the medium is weakly nondissipative [ cf . comments after equations ( [ 1.1 ] ) ] so that only diffraction effects are significant . the total short- and intermediate - scale variations of the wavefield are accounted for by the _ complex eikonal function _ @xmath111 . the relevant equations for the three unknown functions @xmath112 , @xmath113 and @xmath114 are determined on substituting the ansatz ( [ 1.3 ] ) into the wave equation ( [ 1n ] ) . for the specific case for which @xmath13 is a differential operator this is straightforward . on the other hand , for the general case , one should deal with the nonlocal response of the operator @xcite . with this aim it is convenient writing ( [ 1n ] ) in the configuration space in terms of the schwartz kernel , namely , @xmath115 where the exact kernel has been replaced by the @xmath116 where @xmath117 is the weyl symbol . actually , one could make use of other symbol maps @xcite , yielding asymptotically equivalent results ; here the weyl - symbol maps has been chosen for direct comparison with the wigner - weyl approach . let us further assume that the kernel @xmath118 amounts to a distribution smoothly dependent on @xmath119 and with compact support in @xmath120 . in virtue of the paley - wiener - schwartz theorem @xcite , this is equivalent to assume that the corresponding weyl symbol @xmath121 extends to an _ entire _ function of the complex - valued dual vector @xmath122 , smoothly dependent on @xmath123 . from a physical standpoint , the foregoing assumption implies that nonlocal effects have a finite range : the response @xmath124 of the operator @xmath13 depends only on the value of the wavefield @xmath125 in a compact set . within this condition , one can substitute the complex eikonal ansatz ( [ 1.3 ] ) into ( [ 1n ] ) and expand in taylor series with respect to @xmath126 . as a result one has @xcite @xmath127 where @xmath128 and , for the paley - wiener - schwartz theorem , the estimate @xmath129 is still valid for complex - extended symbols . to leading orders in @xmath130 , and for a weakly dissipative media , i.e. , @xmath63 , one gets [ 11n ] @xmath131 \right ] u(x),\end{aligned}\ ] ] @xmath132 and @xmath133 being the real and imaginary parts of @xmath117 extended in the complex @xmath134 space and , in general , are complex valued . it is worth noting that the foregoing equations can be formally obtained from the standard geometrical optics equations , e.g. , in the form given by littlejohn and flynn @xcite , by replacing @xmath135 with @xmath136 . the cgo equations ( [ 11n ] ) have been dealt with both by means of the characteristics method in the complex domain @xcite and on expanding the equations with respect to @xmath137 @xcite . in particular , on referring to the latter approach , in the weak - diffraction regime @xmath138 , terms up to order @xmath139 should be considered in the cgo equation for the complex eikonal @xmath140 , which , after separating the real and imaginary parts , amounts to [ 1.5 ] @xmath141 equations ( [ 1.5a ] ) and ( [ 1.5b ] ) constitute a set of coupled first - order partial differential equations for @xmath142 and @xmath143 with @xmath144 and @xmath145 . as for the complex amplitude @xmath146 , only the lowest order approximation with respect to @xmath147 is significant , so that the real amplitude @xmath148 is decoupled from the phase @xmath149 $ ] ( not considered hereafter ) and determined by means of the transport equation @xmath150 = 2d '' \big(x , k(x)\big)\ |u(x)|^{2}.\ ] ] this is formally the same equation as the geometrical optics transport equation @xcite , but diffraction effects are accounted for through the wavevector - field @xmath135 which differs from that obtained in the geometrical optics . the approximated form ( [ 1.5 ] ) of the cgo equations is the one used in physical applications . moreover , in the zero - diffraction regime ( @xmath151 ) , one has @xmath152 , thus terms up to first order only should be considered , with the result that equations ( [ 1.5a ] ) and ( [ 1.5b ] ) are decoupled and the whole set of cgo equations ( [ 1.5 ] ) reduces to the standard geometrical optics equations , @xmath113 being effectively zero . equations ( [ 1.5a ] ) and ( [ 1.5b ] ) are usually solved by computing the characteristic curves @xcite of ( [ 1.5a ] ) with ( [ 1.5b ] ) regarded as a constraint with the result that the characteristics curves thus obtained resemble the geometrical optics rays @xcite . therefore , the appropriate boundary conditions should be enough to determine the initial values of the complex vector @xmath153 evaluated on the boundary surface @xmath32 . such conditions are obtained from the cauchy data ( [ 1nbound ] ) on writing @xmath154 for some functions @xmath155 and @xmath156 such that @xmath157 and @xmath158 . from ( [ 12n ] ) one readily gets the value of the component of the complex vector @xmath159 tangent to @xmath32 , namely , @xmath160 and @xmath161 . the remaining normal component is obtained on imposing that the cgo equations ( [ 1.5 ] ) are satisfied on @xmath32 ; this yields two equation for @xmath162 and @xmath163 , _ viz . _ , [ 13n ] @xmath164 where @xmath165 and @xmath166 are obtained in terms of the first- and second - order @xmath16 derivatives of @xmath167 and evaluated at @xmath168 . equation ( [ 13na ] ) is an @xmath169 perturbation of the local dispersion relation ( [ 7n ] ) , hence , it can be solved by @xmath170 and , in correspondence of ( [ 14n ] ) , eq.([13nb ] ) yields @xmath171 . as in the wigner - weyl formalism , if multiple solutions are found , one should write the wavefield as a sum of contributions from each branch of the local dispersion relation . from the foregoing discussion , one should note that the cgo method yields the solution directly in the configuration space , but one should deal with the set of partial differential equations ( [ 1.5 ] ) , the numerical solution of which can be rather cumbersome . although the characteristics technique can be used for eq.([1.5a ] ) , the constraint ( [ 1.5b ] ) should be solved in parallel , thus increasing the computational complexity of the problem . as for the global properties of the cgo solution , to our knowledge no general result is still available , though numerical solutions @xcite show that the cgo solution is regular even near focal points where the standard geometrical optics solution exhibits a caustic singularity . in contrast , the wigner - weyl formalism appears better suited for numerical solutions . in particular , the wave kinetic equation can be solved along the corresponding hamiltonian orbits in the phase space so that it is reduced to a set of ordinary differential equations that require limited computational efforts and the solution thus obtained has a global validity in the phase space since the hamiltonian orbits do not cross each other . in this respect , the constraint ( [ 1.1a ] ) does not constitute a limitation as @xmath167 is a constant of motion . moreover , there is no limitation on the nonlocal response of pseudodifferential operators to which the wigner - weyl formalism applies . on the other hand , the solution in the phase space should be projected into the configuration space and , thus , an integral with respect to the momentum @xmath16 should be carried out numerically . notwithstanding these differences , the wigner - weyl kinetic formalism and the complex geometrical optics method share a number of features , e.g. , the solution of the local dispersion relation ( [ 14n ] ) relevant to the cgo method is obtained , to the lowest significant order in @xmath147 , on evaluating the corresponding solution ( [ 7n ] ) for @xmath172 . in the following sections , it will be proved that one can project the wave kinetic equation from the phase space into the configuration space in such a way that the cgo eq.([1.5 ] ) are recovered . as discussed in sec.2 , the solutions of the wave kinetic equation are usually sought in the space @xmath173 of tempered distributions @xcite , or in the space of semiclassical measures @xcite . the first formulation is the more general , whereas the second follows from the physical requirement that expectation values ( [ star ] ) are well defined . moreover , we have pointed out that , for general wave propagation problems , physics requires a stronger condition on the wigner function , namely , the expectation values of physical quantities should be locally defined according to ( [ starstar ] ) . for instance , if one deals with a time - dependent wavefield for which @xmath174 and @xmath175 , the integral @xmath176 yields the wave action density @xmath177 in the space - time @xcite . in this section , a mathematical characterization of such novel solutions is given and the corresponding differential calculus is put forward . first , let us note that for any schwartz function @xmath178 and for any tempered distribution @xmath179 one can define a tempered distribution @xmath180 over the configuration space only , given by @xmath181 where , in general , angle brackets and the integral are alternative ways to denote the action of a distribution on the corresponding test function . the distribution @xmath182 is smooth with respect to @xmath69 if and only if @xmath183 amounts to a smooth function @xmath184 . in this case the map @xmath185 for @xmath186 defines at every point location @xmath123 a tempered distribution @xmath187 with @xmath188 this is a consequence of the completeness of @xmath189 along with the identity @xmath190 where @xmath191 is a compact - supported function that approximates the dirac s @xmath192-function for @xmath193 . let us now consider a symbol @xmath194 , that is , @xmath39 fulfills the symbol estimate ( [ 2n ] ) for every order @xmath51 . it follows that @xmath195 and for any @xmath196 which is smooth with respect to @xmath64 one can define @xmath197 and this is a smooth function on @xmath11 as required in ( [ starstar ] ) . the definition ( [ 1 m ] ) should be extended for any symbol @xmath198 with arbitrary order . with this aim , the class of physically admissible solutions [ in the sense of ( [ starstar ] ) ] is restricted . in particular , it is appropriate considering the tempered distributions @xmath199 that satisfy the following conditions : 1 . @xmath182 is smooth with respect to @xmath64 , 2 . the restriction @xmath200 amounts to a distribution with compact support , i.e. , @xmath201 , where @xmath202 is continuously embedded in @xmath189 in the weak topology . such a distribution @xmath182 will be called _ momentum distribution _ since for every @xmath64 its restriction @xmath200 represents the distribution of momentum @xmath16 over @xmath64 . for short let us write @xmath203 for the space of momentum distributions . within this formulation for every @xmath204 and for every @xmath198 eq.([1 m ] ) is well posed and defines a smooth function on @xmath11 . let us note that the foregoing definition of the space @xmath205 is not the optimal one as functions rapidly decreasing in @xmath16 are also admissible momentum distributions . however , in the semiclassical limit these functions can be ignored and only compact - supported distributions are significant . let us now address the derivatives of a momentum distribution @xmath204 . first , the derivatives with respect to the momentum @xmath16 are defined throughout every order . specifically , since @xmath206 @xmath207 is smooth with respect to @xmath64 ; moreover , for every @xmath186 , @xmath208 hence , @xmath209 is compactly supported , and , thus , @xmath210 . in terms of the integral notation the latter result evaluated for symbols reads @xmath211 which is the `` integration - by - parts '' formula . on the other hand , the derivatives with respect to @xmath64 should be dealt with more carefully . for simplicity , let us consider first - order derivative @xmath212 for @xmath203 . one has that @xmath213 in virtue of ( [ 1mprimo ] ) , so that @xmath212 is smooth with respect to @xmath64 . furthermore , @xmath214 hence @xmath215 is compactly supported and @xmath216 . the explicit formula for the derivative is obtained on noting that for every symbol @xmath198 , @xmath217 and @xmath218 which in the integral notation takes the form @xmath219 the same result would be obtained from the definition @xmath220 through straightforward but longer calculations . higher - order derivatives can be defined by recurrence , but they are not explicitly needed in the following . searching for solutions of the wave kinetic equations ( [ 1.1 ] ) in the space of momentum distributions leads to the weak formulation [ 2.1bar ] @xmath221a(x , k)\ d^{n}k=0,\end{gathered}\ ] ] with @xmath222 . furthermore , we are interested in semiclassical solutions for which only large - enough momenta are significant . on recalling that @xmath223 , we will search for solution of ( [ 2.1bar ] ) in the form @xmath224 where @xmath142 and @xmath225 are smooth functions to be determined ; in particular , @xmath142 defines a lagrangian manifold @xmath226 in the @xmath64-@xmath16 phase space . moreover , @xmath227 is normalized , i.e. , @xmath228 , and such that @xmath229 for any multi - index @xmath22 . the integrals in ( [ 2.3 ] ) are well posed since @xmath230 are symbols of order @xmath231 , and the corresponding quantities @xmath232 express the statistical moments of the distribution @xmath182 , i.e. , they expresses how important are the _ deviations _ @xmath233 of the momentum from the lagrangian manifold @xmath226 . in particular , @xmath234 in view of the normalization condition . in the semiclassical limit it is assumed that the scale length @xmath235 characterizing the range of the momentum deviations is large enough as compared to @xmath236 , namely , @xmath237 . in appendix , it is proved that , within the weak formulation ( [ 2.1bar ] ) , the momentum distribution @xmath182 , satisfying the foregoing conditions , can be represented by the asymptotic series , cf . eq.([a.5 ] ) , @xmath238 controlled by the small parameter @xmath239 . it is worth noting that @xmath240 is thus represented by a distribution which is point supported on the lagrangian manifold @xmath241 and completely determined by its statistical moments @xmath242 . in correspondence of the asymptotic expansion ( [ 2.5 ] ) , eq.([2.1abar ] ) reduces to @xmath243 as shown in details in appendix , cf . , in particular , eq.([a.6 ] ) . formally , eq.([2.6 ] ) constitutes an infinite set of algebraic equations for the statistical moments @xmath244 characterizing the momentum distribution , where each equation , labelled by @xmath23 , is expressed as an asymptotic series in @xmath245 ; the function @xmath142 is determined by imposing that the system ( [ 2.6 ] ) admits nontrivial solutions . equations ( [ 2.6 ] ) are valid for a general momentum distribution @xmath182 which satisfies ( [ 2.1abar ] ) . in particular , on setting @xmath246 for @xmath247 , eq.([2.5 ] ) reduces to @xmath248 which is the geometrical - optics - like solution obtained by bornatici and kravtsov @xcite and by sparber , markowich and mauser @xcite , whereas in ( [ 2.6 ] ) the only nontrivial equation reduces to the geometrical optics eikonal equation @xcite for @xmath142 , namely , @xmath249 . on the basis of the asymptotic expansion ( [ 2.5 ] ) for a momentum distribution , one can prove the main result of this paper , that is , relating the wave kinetic equation to the cgo equations for suitable boundary conditions . first let us consider the specific momentum distribution for which @xmath250 that is , odd - order moments have been set to zero , whereas even - order moments have been related to a single vector field @xmath145 , with @xmath113 an unknown smooth function . correspondingly , the momentum distribution ( [ 2.5 ] ) takes the form @xmath251^{2n } \delta\big(k-\partial_{x}s(x)\big),\end{aligned}\ ] ] the second identity being obtained by means of the multinomial formula @xmath252 with @xmath253 ( no sum over @xmath254 ) . let us note that the momentum distribution ( [ 2.8bar ] ) is symmetric with respect to the lagrangian manifold @xmath241 , as even - order moments only appear ; in particular , the second order moment @xmath255 for @xmath256 is negative , so that such a distribution can not be interpreted as a probability measure . the momentum distribution ( [ 2.8bar ] ) should be multiplied by @xmath225 to get the whole wigner function ( [ 2.1 ] ) . let us consider the case for which @xmath257 ) and @xmath258 is ordered according to @xmath259 with @xmath260 . as a consequence , @xmath235 is the shortest scale length characterizing the wavefield intensity @xmath225 , hence it can be identified with the scale length @xmath110 defined after eq.([1.3 ] ) , namely , @xmath261 and @xmath262 . then one has the following : first , let us prove the statement @xmath263 . in view of the ansatz ( [ 2.7 ] ) , all the equations obtained from ( [ 2.6 ] ) with @xmath23 such that @xmath264 is an even integer reduce to the same equation which reads [ 2.10 ] @xmath265^{2n } d'\big(x , k(x)\big ) = 0,\ ] ] and , analogously , all the equations obtained from ( [ 2.6 ] ) with @xmath23 such that @xmath264 is an odd integer reduce to @xmath266^{2n+1 } d'\big(x , k(x)\big)=0,\ ] ] where the second identity in both equations ( [ 2.10 ] ) follows on using the multinomial formula ( [ 2.9bar ] ) . equations ( [ 2.10 ] ) constitute a set of two coupled equations for the real functions @xmath142 and @xmath143 , and , to lowest significant orders in @xmath147 , they are the same as the cgo equations ( [ 1.5a ] ) and ( [ 1.5b ] ) ; this completes the proof of @xmath263 . as for @xmath267 , on account of the differential calculus for momentum distributions put forward in sec.3 , the term connected with the poisson brackets in eq.([2.1bbar ] ) should be written in the form , cf . eq.([5 m ] ) , @xmath268 using the specific momentum distribution ( [ 2.8bar ] ) for which @xmath269 yields @xmath270 - \left[\frac{\partial d'\big(x , k(x)\big)}{\partial k_i}\frac{\partial a\big(x , k(x)\big)}{\partial x^i}\right . \\ \nonumber & \qquad\qquad \left . -\frac{\partial d'\big(x , k(x)\big)}{\partial x^i } \frac{\partial a\big(x , k(x)\big)}{\partial k_i}\right ] |\psi(x)|^2 + o(\epsilon^2)\\ \label{17n } & = \frac{\partial}{\partial x^i } \left[\frac{\partial d'\big(x , k(x)\big)}{\partial k_i } |\psi(x)|^2 \right ] a\big(x , k(x)\big ) + o(\epsilon).\end{aligned}\ ] ] the last identity follows on noting that taking the derivative of ( [ 2.10a ] ) with respect to @xmath271 yields @xmath272 with @xmath273 . equation ( [ 17n ] ) implies that @xmath274 \delta\big(k-\partial_{x}s(x)\big ) + o(\epsilon)\ ] ] in the weak sense . hence , from the wave kinetic equation , to lowest order in @xmath147 , one gets the transport equation @xmath275 = 2 d''\big(x , k(x)\big)\ |\psi(x)|^{2},\ ] ] which reduces to the cgo transport eq.([1.5c ] ) for @xmath276 , cf . equation ( [ 2.10bar ] ) , on noting that , to lowest significant order , @xmath277 = e^{-2\phi(x)}\ \frac{\partial}{\partial x^{i}}\left[\frac{\partial d'\big(x , k(x)\big)}{\partial k_{i}}\ |u(x)|^{2}\right],\ ] ] in view of eq.([2.10b ] ) . this concludes the proof . the foregoing result shows that there exists a specific form of the wigner function for which the wave kinetic equation is reduced to the cgo equations . in order to compare the wavefield intensities predicted by the wave kinetic description with that obtained on solving the cgo equations , one should complete the foregoing argument by discussing cauchy boundary conditions . with reference to ( [ 6n ] ) and ( [ 7n ] ) one has the following : _ let @xmath77 be the weak solution of the wave kinetic equation ( [ 1.1 ] ) corresponding to the cauchy boundary conditions @xmath278 for some smooth functions @xmath279 , @xmath280 and @xmath281 with @xmath282 satisfying the cgo ordering defined after ( [ 12n ] ) , and let @xmath142 , @xmath143 and @xmath258 be solution of the cgo equations ( [ 1.5 ] ) with cauchy boundary conditions given by the same function @xmath283 , @xmath156 and @xmath284 . then @xmath77 can be approximated according to @xmath285 in the weak sense of sec.3 . _ first , let us note that the cauchy data ( [ 20n ] ) is a particular case of ( [ 6nb ] ) which corresponds to @xmath286 in particular , the wigner function corresponding to the complex - eikonal wave object ( [ 12n ] ) can be written in the form ( [ 22n ] ) . in order to prove the foregoing statement , we will make use of the previous result of this section . specifically , we have proved that the wigner function @xmath287 @xmath182 being the momentum distribution ( [ 2.8bar ] ) , solves asymptotically the wave kinetic equation in the weak sense within an @xmath288 accuracy . moreover , @xmath289 in view of ( [ 16bar ] ) . as for the boundary condition ( [ 20n ] ) one gets @xmath290 where @xmath291 . according to ( [ 14n ] ) , @xmath292 , and , thus , @xmath293 in the weak sense , so that @xmath294 matches the boundary conditions ( [ 20n ] ) . since the solution of the wave kinetic equation along with the cauchy boundary condition ( [ 20n ] ) is unique and since @xmath295 is an @xmath288 solution , it follows that @xmath296 which concludes the proof of ( [ 21n ] ) . this implies that , whenever the solutions of both the wave kinetic equation and the cgo equations exist , thus , in particular , the cauchy boundary conditions are of the form ( [ 22n ] ) , the wigner - weyl formalism and the complex geometrical optics method are equivalent within an @xmath288 accuracy . in particular , the wavefield intensity predicted by the wigner - weyl kinetic formalism is the same as that predicted by the cgo method , namely , @xmath297 ) . in the next section , an analytically tractable case is considered as an example . specifically , the solution of the wave kinetic equation relevant to the paraxial propagation of a gaussian wave beam in a `` lens - like '' medium is obtained and shown to be the same as the corresponding cgo solution . let us address the case of a monochromatic @xmath298 beam of electromagnetic waves propagating in a _ loss less _ `` lens - like '' medium @xcite with _ real _ refractive index @xmath299^{\frac{1}{2}}$ ] . it is assumed that the wavefield is localized near the axis @xmath300 of the medium , that is , @xmath301 ; moreover , the wave electric field is written in the form @xmath302 , i.e. , it is polarized along the @xmath303 axis and propagates in the @xmath64-@xmath28 plane . the relevant wave equation for the wavefield real amplitude @xmath304 is thus the helmholtz equation . the corresponding weyl symbol is real valued and given by @xmath305,\ ] ] thus , the dispersion relation @xmath306 yields two branches , to be referred to as the progressive and the regressive waves . as for the cauchy boundary conditions , let us assume that the wavefield is purely gaussian at @xmath307 , i.e. , @xmath308 $ ] , @xmath309 being the initial width , and the propagation occurs along the @xmath28 axis , so that one should solve the dispersion relation @xmath306 for @xmath310 . on assuming that each branch of the dispersion relation carries half of the wavefield intensity , one can consider the progressive wave only which reads @xmath311 where @xmath312 is the wavevector at @xmath300 and the paraxial approximation @xmath313 has been exploited as relevant to the weak - diffraction regime @xcite . it is convenient noting that the dispersion relation corresponding to the second form of ( [ 3.1 ] ) can be written as @xmath314 which is formally analogous to the dispersion relation relevant to a quantum harmonic oscillator @xcite with unit mass and @xmath315 , @xmath316 , @xmath317 being the characteristic frequency of the oscillator , @xmath318 and @xmath319 . in particular , the frequency @xmath320 corresponds to the shifted wavevector @xmath321 along the propagation direction @xmath28 . the shift occurs because of the oscillations of the wavefield along the propagation direction @xmath28 . this analogy allows to make use of the well - known solution of the wave kinetic equation for the quantum harmonic oscillator @xcite to describe the paraxial propagation of a gaussian beam in the lens - like " medium . more specifically , the solution of the wave kinetic equation for the harmonic oscillator corresponding to an initially gaussian wave packet @xmath322 $ ] is @xcite [ 3.2pre ] @xmath323\ w_{0}^{2 } , \end{gathered}\ ] ] where @xmath324 is the width of the wave packet as a function of time and @xmath325 , @xmath51 being the mass of the oscillator and @xmath326 the initial displacement of the gaussian from the centre of the elastic force acting on the oscillator . correspondingly , the solution for the wave electric field intensity in the `` lens - like '' medium , with the considered launching conditions , is [ 3.2 ] @xmath327w_{0}^{2 } = \left[1+\big(\big(\frac{l}{z_{r}}\big)^{2}-1\big)\sin^{2}(z / l)\right ] w_{0}^{2},\end{gathered}\ ] ] with @xmath328 the rayleigh range in the medium . in eq.([3.2b ] ) , it has been explicitly indicated that the solution obtained from the wave kinetic equation amounts to the averaged intensity @xmath329 , rather than to the exact value @xmath330 since two branches of the dispersion relation exist each one carrying half of the wavefield intensity , cf . comments after eqs.([7nprime ] ) . the intensity ( [ 3.2a ] ) and the beam width ( [ 3.2b ] ) are the same as the corresponding quantities obtained from the cgo solution @xcite . as a consequence the intensity profile ( [ 3.2 ] ) accounts for diffraction effects as shown in fig.1 . one can conclude that , according to results of sec.4 , the _ kinetic formalism can be used to describe the effects of diffraction on the propagation of wave beams _ , and , for the case under consideration , it yields the same result as the cgo method . nevertheless , the detailed structure of the wavefield , i.e. , the oscillations along the propagation direction @xmath28 , the effects of the curvature of phase fronts and the gouy shift , which are available from the cgo solution @xcite , _ can not be resolved by means of the wave kinetic equation _ , which instead gives the averaged intensity distribution . within the framework of semiclassical wave propagation , two specific asymptotic techniques have been considered , namely , the wigner - weyl kinetic formalism and the complex geometrical optics ( cgo ) method . a detailed comparative analysis of these techniques has been given in sec.2 , for the case of scalar pseudodifferential wave equations , with cauchy boundary conditions . in particular , in the wigner - weyl formalism , the wavefield is represented in the phase space by the wigner function which is a solution to the wave kinetic equation . in the most general case the wigner function amounts to a tempered distribution . however , physical considerations lead to the definition of a novel class of weak solutions which have been characterized in sec.3 . such specific weak solutions are referred to as _ momentum distributions _ since , for every point location @xmath64 in the configuration space , they give the distribution of momentum @xmath16 over @xmath64 , in the @xmath64-@xmath16 phase space . on the other hand , the cgo method yields an asymptotics solution of a pseudodifferential wave equation directly in the configuration space , in terms of three smoothly varying functions representing , the phase , the wavefield envelope and the amplitude , respectively . in sec.4 , on the basis of the mathematical framework developed in sec.3 , we have proved that , whenever both the solutions of the wave kinetic equation and of the cgo equations exist , thus , in particular , the cauchy data are of the form ( [ 22n ] ) , the former can be approximated by a momentum distribution , cf . , equation ( [ 21n ] ) , written in terms of the three smoothly varying functions that solve the cgo equations ( [ 1.5 ] ) . as a consequence , the two considered techniques are asymptotically equivalent and , in particular , to lowest significant order , the wavefield intensity predicted by the wigner - weyl formalism is the same as that predicted by the cgo method , cf . eq.([intensity ] ) . in addition , one can conclude that the wigner - weyl kinetic formalism properly describes the wavefield near focal points . this is also shown by comparing the solution of the kinetic equation to that of the cgo equations for a specific case , namely , the propagation of electromagnetic gaussian wave beams in an isotropic `` lens - like '' medium , cf . in particular , the relevant solution of the wave kinetic equation has been obtained on the basis of the analogy between the `` lens - like '' medium and the quantum harmonic oscillator . the author is grateful to m. bornatici for many useful discussions , suggestions and for carefully reading the manuscript . for the revised version of this paper , valuable discussions with c. dappiaggi and a. orlandi are gratefully acknowledged . this work was supported by the italian ministry of university scientific research and technology ( murst ) and the italian institute for the physics of matter ( infm ) . in this appendix the asymptotic series expansion ( [ 2.5 ] ) of the momentum distribution is proved and the corresponding equations ( [ 2.6 ] ) are derived from the weak form ( [ 2.1abar ] ) of the dispersion relationship ( [ 1.1a ] ) . since both @xmath331 and @xmath67 are symbols , they are , in particular , smooth functions , and one can apply the taylor s formula @xmath332 @xmath333 being an @xmath1-dimensional multi - index and @xmath334 the remainder of order @xmath335 relevant to the expansions of @xmath167 and @xmath39 , respectively . more specifically , on making use of ( [ a.2 ] ) to evaluate the left - hand side of ( [ 2.1abar ] ) , one gets @xmath336 where @xmath337 are the statistical moments of the momentum distribution @xmath338 , c.f . eq.([2.3 ] ) . in virtue of the symbol estimate ( [ 2n ] ) , symbols are such that , e.g. , @xmath339 in the semiclassical limit @xmath340 uniformly in @xmath64 , hence , the asymptotic series expansion ( [ a.3 ] ) is controlled by the ( small ) parameter @xmath341 . moreover , on noting that @xmath342 eq.([a.3 ] ) takes the form @xmath343 a(x , k)\ d^{n}k\end{gathered}\ ] ] and , in view of the arbitrariness of @xmath67 , one gets @xmath344 in the weak sense . it is worth noting that the derivation of ( [ a.4 ] ) does not depend on the explicit form of the symbol @xmath331 , thus , on setting @xmath345 , eq.([a.4 ] ) reduces to @xmath346 which is just the general asymptotic expansion ( [ 2.5 ] ) of the momentum distribution . going back to eq.([2.1abar ] ) , its solution is obtained on setting the expansion ( [ a.4 ] ) to zero and exploiting the linear independence of the derivatives of the dirac s @xmath192-function , thus yielding a set of equations for the statistical moments , namely , @xmath347 which is just eq.([2.6 ] ) . it is worth noting that eq.([a.6 ] ) can be also obtained on substituting ( [ a.5 ] ) into ( [ 2.1abar ] ) and exploiting the leibniz s formula @xmath348 which expresses the derivative of a product to any orders . v. guillemin and s. stenberg , _ geometric asymptotics _ ( amer . math survey , amer . soc . , new york 1976 ) . j. j. duistermaat , comm . pure appl . vol.xxvii , 207 ( 1974 ) . a. martinez , _ an introduction to semiclassical and microlocal analysis _ ( springer - verlag , new york , 2002 ) . a. kravtsov and yu . i. orlov , _ geometrical optics of inhomogeneous media _ ( springer - verlag , berlin , 1990 ) . r. g. littlejohn and w. g. flynn , phys . a * 44 * , 5239 ( 1991 ) . a. kravtsov and yu . i. orlov , _ caustics , catastrophes and wave fields _ ( springer - verlag , berlin , 1999 ) . j. ehlers and e. t. newman , _ j. math . phys . _ * 41 * , 3344 ( 2000 ) , and references therein . j. f. nye , _ natural focusing and fine structure of light _ ( institute of physics publishing , bristol , 1999 ) and references therein . v. guillemin and s. stenberg , _ symplectic techniques in physics _ ( cambridge university press , cambridge , 1984 ) . t. stix _ plasma waves _ ( american institute of physics , new york , 1992 ) . s. w. mcdonald and a. n. kaufman , phys . a * 32 * , 1708 ( 1985 ) . r. g. littlejohn , phys . rep . * 138 * , 193 ( 1986 ) . s. w. mcdonald , phys . * 158 * , 337 ( 1988 ) . s. choudhary and l. b. felsen , proc . of the ieee * 62 * , 1530 ( 1974 ) . a. kravtsov , g. w. forbes and a. a. asatryan , in progress in optics vol.xxxix ( 1999 ) , pp.1 - 61 . a. bravo - ortega and a. h. glasser , phys . fluids b * 3 * , 529 ( 1991 ) . e. mazzucato , phys . fluids b * 1 * , 1855 ( 1989 ) . m. e. taylor , _ partial differential equations ii : qualitative studies of linear equations _ , springer , new york ( 1996 ) . l. hrmander , _ lectures on nonlinear hyperbolic differential equations _ ( springer - verlag , berlin heidelberg , 1997 ) , section 8.4 . j. j. duistermaat , _ fourier integral operators _ , birkhuser , boston ( 1996 ) . l. hrmander , _ the analysis of linear partial differential operators i : distribution theory and fourier analysis _ , springer , berlin ( 2003 ) . o. maj and m. bornatici , _ proc . of the 30@xmath349 eps conf . on controlled fusion and plasma physics _ eca * 27a * , p-2.1 ( 2003 ) . m. bornatici and yu . a. kravtsov , plasma phys . controlled fusion * 42 * , 255 ( 2000 ) . c. sparber , p. a. markowich and n. j. mauser , asymptotic analysis * 33 * , 153 ( 2003 ) . m. bornatici and o. maj , plasma phys . controlled fusion * 45 * , 707 ( 2003 ) .
|
the relationship between two different asymptotic techniques , namely , the wigner - weyl kinetic formalism and the complex geometrical optics method , is addressed within the framework of semiclassical theory of wave propagation . more specifically , in correspondence to appropriate boundary conditions , the solution of the wave kinetic equation , relevant to the wigner - weyl formalism , is obtained in terms of the corresponding solution of the complex geometrical optics equations . in particular , this implies that the two considered techniques yield the same wavefield intensity . such a result is also discussed on the basis of the analytical solution of the wave kinetic equation specific to gaussian beams of electromagnetic waves propagating in a `` lens - like '' medium for which the complex geometrical optics solution is already available .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the elucidation of dynamo action is a problem of central importance in nonlinear dynamics because it has implications for a variety of physical systems . dynamo instabilities , which amplify weak magnetic fields in a turbulent conducting fluid , are believed to be the principal mechanism for the generation of magnetic fields in celestial bodies and in the interstellar medium @xcite , and in liquid - metal systems @xcite studied in laboratories . in these situations the kinematic viscosity @xmath3 and the magnetic diffusivity @xmath4 can differ by several orders of magnitude , so the magnetic prandtl number @xmath5 can either be very small or very large ; e.g. , @xmath6 at the base of the sun s convection zone , @xmath7 in the liquid - sodium system , and @xmath8 in the interstellar medium . this prandtl number is related to the reynolds number @xmath9 and the magnetic reynolds number @xmath10 that characterize the conducting fluid ; here @xmath11 and @xmath12 are typical length and velocity scales in the flow ; clearly @xmath13 . two dissipative scales play an important role here ; they are the kolmogorov scale @xmath14 [ @xmath15 at the level of kolmogorov 1941 ( k41 ) phenomenology @xcite ] and the magnetic - resistive scale @xmath16 [ @xmath17 in k41 ] . for large prandtl numbers , i.e. , @xmath18 , @xmath19 so the magnetic field grows predominantly in the dissipation range of the fluid till it is strong enough to affect the dynamics of the fluid through the lorentz force . this behavior is a characteristic of a small - scale turbulent dynamo , in which dynamo action is driven by a smooth , dissipative - scale velocity field . in the initial stage of growth , called the kinematic stage of the dynamo , the magnetic field is not large enough to act back on the velocity field . dynamo action can be obtained for values of @xmath1 that are large enough to overcome joule dissipation ; and the dynamo - threshold value @xmath20 decreases as @xmath0 increases @xcite . @xmath21 in liquid - metal flows @xcite so they lie in the small - prandtl - number region , @xmath22 , for which the growth of the magnetic energy occurs initially in the _ inertial scales _ of fluid turbulence , because @xmath23 ; here the velocity field is not smooth and the local strain rate is not uniform : at the k41 level the turnover velocity of an eddy of size @xmath24 is @xmath25 , so the rate of shearing @xmath26 . direct numerical simulations ( dns ) are playing an increasingly important role in developing an understanding of such dynamo action . most dns studies of mhd turbulence @xcite have been restricted , because of computational constraints , either to low resolutions or to the case @xmath27 ; small - scale dynamos , with @xmath28 have also been studied via dns @xcite . however , given the large range spanned by @xmath0 in the physical settings mentioned above , some recent dns studies of the mhd equations have started to explore the @xmath0 dependence of dynamo action ; the range of @xmath29 covered by such pure dns studies @xcite is quite modest ( @xmath30 ) . to explore the dynamo boundary in the @xmath2 plane over a large range of @xmath0 , one recent study @xcite has used a combination of numerical methods , some of which require small - scale models like large - eddy simulations ( les ) or lagrangian - averaged mhd ( lamhd ) , and others , like a pseudospectral dns , in which the only approximations are the finite number of collocation points and the finite step used in time marching ; yet another dns study @xcite has introduced hyperviscosity of order @xmath31 to study the low @xmath0 regime ; by using this combination of methods these studies has been able to cover the range @xmath32 and to obtain the boundary between dynamo and no - dynamo regions but with fairly large error bars . we have carried out extensive , high - resolution , numerical studies that have been designed to explore in detail the boundary between the dynamo and no - dynamo regimes in the @xmath2 plane in a shell model for three - dimensional mhd @xcite . this shell model allows us to explore a much larger range of @xmath0 than is possible if we use the mhd equations . although our study uses a simple shell model , it has the virtue that it can explore the boundary between dynamo and no - dynamo regions in great detail without resorting to the modelling of small spatial scales . shell - model studies of dynamo action have also been attempted in refs . @xcite but these have concentrated on aspects of the dynamo problem that are different from those we consider here . our study suggests that it is natural to think of the boundary between dynamo and no - dynamo regimes in the @xmath2 plane as a first - order phase boundary that is the locus of first - order , nonequilibrium phase transitions from one nonequilibrium statistical steady state ( ness ) to another . the first ness is a turbulent , but statistically steady , conducting fluid in which the magnetic energy is negligibly small compared to the kinetic energy ; the second ness is also a statistically steady turbulent state but one in which the magnetic energy is comparable to the kinetic energy . indeed , the ratio of the magnetic and fluid energies @xmath33 turns out to be a convenient order parameter for this nonequilibrium phase transition since it vanishes in the no - dynamo phase and assumes a finite , nonzero value in the dynamo state . the other , intriguing result of our study is that the boundary between these phases is very intricate and might well have a fractal character ; this provides an appealing explanation for the large error bars in earlier attempts to determine this boundary @xcite . the analogy with first - order transitions that we have outlined above is not superficial . as in any first - order transition we find that our order parameter shows hysteretic behavior @xcite as we scan through the dynamo boundary by changing the forcing term at a nonzero rate . we also find some evidence of nucleation - type phenomena : the closer we are to the dynamo boundary , the longer it takes for a significant magnetic field to nucleate and thus lead to dynamo action . we compare our results with earlier studies such as ref . @xcite , which have suggested that dynamo action occurs because of a subcritical bifurcation . the remaining part of this paper is organised as follows : in sec . [ models ] we describe the shell model for mhd @xcite and the numerical method we employ . sec . [ results ] is devoted to our results and sec . [ conclusion ] contains a concluding discussion . to study dynamo action it is natural to use the equations of magnetohydrodynamics(mhd ) . in three dimensions the mhd equations are @xmath34 where @xmath3 and @xmath4 are the kinematic viscosity and the magnetic diffusivity , respectively , the effective pressure @xmath35 , and @xmath36 is the pressure . for low - mach - number flows , to which we restrict ourselves , we use the incompressibility condition @xmath37 ; and @xmath38 . as we have mentioned above , a dns of the mhd equations poses a significant computational challenge , even on the most powerful computers available today , if we want to cover a large part of the @xmath2 plane and to locate the dynamo boundary accurately . therefore one study @xcite has used a combination of les , lamhd , and dns to obtain this boundary . we employ a complementary strategy : we use a simple shell model for mhd @xcite that allows us to carry out very extensive numerical simulations to probe the nature of the dynamo boundary without using les or lamhd . shell models comprise a set of ordinary differential equations with nonlinear coupling terms that mimic the advection terms in and respect the shell - model analogs of the conservation laws of the parent hydrodynamic equations in the inviscid , unforced limit @xcite . for the case of mhd each shell @xmath39 is characterized by a complex velocity @xmath40 and a complex magnetic field @xmath41 in a logarithmically discretized fourier space with wave vectors @xmath42 ; furthermore , there is a direct coupling only between velocities and magnetic fields in nearest and next - nearest neighbor shells . the mhd shell model equations @xcite are @xmath43^\ast+f_n^u , \\ \frac{db_n}{dt}= & -&\eta k_n^2b_n + i[d_n(u_{n+1}b_{n+2}-b_{n+1}u_{n+2})\nonumber \\ & + & e_n(u_{n-1}b_{n+1}-b_{n-1}u_{n+1})\nonumber \label{shell2}\\ & + & f_n(u_{n-2}b_{n-1}-b_{n-2}u_{n-1})]^\ast+f_n^b , \end{aligned}\ ] ] where @xmath44 denotes complex conjugation , @xmath45 , with @xmath46 the total number of shells , the wave numbers @xmath47 , with @xmath48 , and @xmath49 and @xmath50 the forcing terms in the equations for @xmath40 and @xmath41 , respectively . in our studies of dynamo action , we set @xmath51 . the parameters @xmath52 , are obtained by demanding that these equations conserve all the shell - model analogs of the invariants of 3dmhd , in the inviscid , unforced case , and reduce to the well - known gledzer - ohkitani - yamada ( goy ) shell model @xcite for fluid turbulence if @xmath53 , @xmath54 . in particular , to ensure the conservation of shell - model analogs of the total energy @xmath55 , cross helicity @xmath56 , and magnetic helicity @xmath57 , in the unforced and inviscid case , and to obtain the goy - model limit for the fluid , we choose @xmath58 the only adjustable parameters are the forcing terms and @xmath3 and @xmath4 . the ratio @xmath59 yields the magnetic prandtl number @xmath0 . grashof numbers yield nondimensionalized forces @xcite but , for easy comparison with earlier studies @xcite , we use the fluid and magnetic integral - scale fluid and magnetic reynolds numbers whose shell - model analogs are , respectively , @xmath60 , where @xmath61 and @xmath62 , @xmath63 and @xmath64 . we use the following boundary conditions : @xmath65 we set @xmath66 and use a fifth - order , adams - bashforth scheme for solving the shell - model equations , i.e. , for an equation of the type @xmath67 we use @xmath68,\end{aligned}\ ] ] where @xmath69 is the time step . we have found that this numerical scheme works well for the integration of eqs.([shell1 ] ) and ( [ shell2 ] ) so long as @xmath70 and @xmath71 . in all our calculations we use @xmath72 . characteristic time scales include the time scale for diffusion @xmath73 and the large - eddy - turnover time @xmath74 , where @xmath75 is the box - size length scale and @xmath76 is the root - mean - square velocity . the initial conditions we use are as follows : we first obtain a statistically steady state for the goy - shell - model equations , which are obtained from eq.([shell1 ] ) by setting all @xmath77 ; the forcing terms are chosen to be @xmath78 , with @xmath79 in all our runs , except ones in which we study hysteretic behavior , and @xmath51 . we choose the goy - model shell velocities at time @xmath80 to be @xmath81 , with @xmath82 a random phase distributed uniformly on the interval @xmath83 . to make sure we have a statistically steady state we evolve the shell velocities @xmath40 till @xmath84 . this yields the shell - model energy spectrum @xmath85 that has the k41 form @xmath86 if we ignore intermittency corrections . we now introduce a small seed magnetic which is such that @xmath87 and then follow the temporal evolution of @xmath40 and @xmath41 that is given by eqs.([shell1 ] ) and ( [ shell2 ] ) . versus time @xmath88 , with @xmath89 the magnetic - diffusion time , in the dynamo region ( blue , dashed curve ) , near the dynamo boundary ( green , full line ) , and in the no - dynamo regime ( red , full line in the inset).,title="fig : " ] + 50 30 50 30 [ fig : energy ] + 50 30 plane : red circles indicate dynamo action ; green stars are used if no dynamo occurs . the boundary between the two regions shows an intricate , interleaved pattern of fine , dynamo and no - dynamo regimes ( see inset for a detailed view ) . we have drawn two black , dashed lines ; the region above the upper one of these lines is predominantly in the dynamo regime ; the area below the lower one of these lines is principally in the no - dynamo regime . , title="fig : " ] 50 30 50 30 50 30 given the numerical scheme that we have described in the previous section , we obtain the time series for @xmath40 and @xmath41 from the mhd - shell - model equations . an analysis of these time series shows two types of nonequilibrium statistical steady states ( ness ) . we refer to the first as the no - dynamo state and to the second as the dynamo state . these states have been found in several earlier studies such as refs . our main goal is to explore in detail the phase boundary between these two states . this can be done most easily by the introduction of a dynamo order parameter a natural candidate for which is the ratio @xmath33 , where the fluid and magnetic energies are , respectively , @xmath90 in fig . [ fig : tseries ] : the blue , dashed curve shows the evolution of @xmath33 in the dynamo regime ; note that here the dynamo order parameter rises rapidly , fluctuates significantly for @xmath91 , and finally reaches a statistical steady state with equipartition , i.e. , @xmath92 . the red , full curve in the inset of fig . [ fig : tseries ] shows how @xmath33 vanishes rapidly in the no - dynamo state . the behavior of the dynamo order parameter is more complicated than these two simple possibilities in the vicinity of the phase boundary between dynamo and no - dynamo states as shown by the green , full curve in fig . [ fig : tseries ] ; @xmath33 rises much more slowly from zero than in the dynamo regime and then it fluctuates significantly for a long time ; the difficulty of pinpointing the dynamo boundary is a consequence of these fluctuations . the time series for the dynamo order parameter are obtained from those for @xmath93 and @xmath94 ; representative plots for these are shown , via red and blue - dashed curves , in figs . [ fig : energy](a ) , ( b ) , ( c ) , ( d ) , ( e ) , and ( f ) for a very large range of magnetic prandtl numbers , namely , @xmath95 and @xmath96 , respectively . the values of @xmath3 are ( a ) @xmath97 , ( b ) @xmath97 , ( c ) @xmath96 , ( d ) @xmath98 , ( e ) @xmath99 , and ( f ) @xmath98 ; and the corresponding values of the diffusion time scale @xmath100 and @xmath101 , respectively . clearly dynamo action occurs in figs . [ fig : energy](a)-(d ) but not figs . [ fig : energy](e ) and ( f ) . by obtaining many such plots we can identify the dynamo boundary in the @xmath102 plane as we discuss later . in the dynamo regime the shell - model kinetic and magnetic energy spectra defined , respectively , by @xmath103 and @xmath104(a ) , ( b ) , and ( c ) for @xmath105 and @xmath106 , respectively , we show the evolution of @xmath107 with time : the curves with red stars , green diamonds , blue hexagons , cyan circles , and magenta triangles , are obtained , respectively , for @xmath108 and @xmath109 ; the analogs of these plots for @xmath110 are given in figs . [ fig : spectra](d ) , ( e ) , and ( f ) . note that the initial growth of @xmath107 occurs principally at large values of @xmath42 if @xmath0 is large , i.e. , we have a small - scale dynamo ; this growth of @xmath107 moves to low values of @xmath42 as @xmath0 decreases ; earlier studies @xcite have observed similar trends but not over the large range of @xmath0 we cover . as @xmath107 grows , the velocity spectra are also affected but much less than their magnetic counterparts as can be seen by comparing figs . [ fig : spectra](d ) , ( e ) , and ( f ) with figs . [ fig : spectra](a ) , ( b ) , and ( c ) , respectively . in all these plots the curves with black squares indicate @xmath110 from the initial steady state for the goy shell model ; and the black lines with no symbols show the k41 @xmath111 spectrum for comparison . from this line we see that the ness that is obtained , once dynamo action has occurred , is such that both velocity and magnetic - field energy spectra display a substantial inertial range with k41 scaling ; these inertial ranges are not large enough , at least near the dynamo boundary in our runs , for a reliable estimation of multiscaling corrections to the @xmath112 exponent . if @xmath113 then the scaling ranges in velocity and magnetic - field spectra are comparable ; as @xmath0 decreases ( increases ) , the scaling range for the magnetic spectrum decreases ( increases ) relative to its counterpart in the velocity spectrum ; these trends are clearly visible in the representative plots in fig . [ fig : spectra ] . we return now to the identification of the dynamo boundary . a close scrutiny of the plots in fig . [ fig : energy ] shows that the initial growth of @xmath94 is not monotonic . it is important , therefore , to set a threshold value of the magnetic energy @xmath114 : for a given pair of values for @xmath0 and @xmath1 , if @xmath115 for @xmath116 , where @xmath117 is the time at which the threshold value is crossed , we conclude that dynamo action occurs ; if not , then there is no dynamo formation . by examining the growth of @xmath118 we can , therefore , map out the dynamo boundary in the @xmath119 plane . the crossing time @xmath117 depends on @xmath0 and @xmath120 . note that , if @xmath121 , the length of time for which we integrate eqs.([shell1 ] ) and ( [ shell2 ] ) , we would conclude , _ incorrectly _ , that no dynamo action occurs for this of values of @xmath0 and @xmath1 . in other words the dynamo boundary depends on @xmath122 ; we have checked this explicitly in several cases . an important questions arises now : is there a well - defined dynamo boundary in the @xmath2 plane as @xmath123 ? earlier studies @xcite have began to answer this question . they find that , if @xmath124 , then a well - defined dynamo boundary is obtained . however , since they work with the mhd equations the error bars on this boundary are large and the range of values of @xmath0 and @xmath1 rather limited . the simplicity of our model allows us to carry out a systematic study of the dynamo boundary . we find that , at least in our shell model for mhd , we can obtain an asymptotic dynamo boundary ( see fig . [ fig : stability ] ) if we choose @xmath125 , i.e. , we conclude that dynamo action has occurred if @xmath118 exceeds @xmath126 ; furthermore , if @xmath118 falls below @xmath127 we say that dynamo action will never be achieved . we continue the temporal evolution of eqs.([shell1 ] ) and ( [ shell2 ] ) till one of these criteria is satisfied . for all values of @xmath0 and @xmath120 that we have used we find that this @xmath122 , the run time required to decide whether or not dynamo action occurs , is several orders of magnitude lower than @xmath89 . we have also checked for several representative pairs of values for @xmath0 and @xmath1 that runs of length @xmath124 do not change our conclusions about such dynamo action . the dynamo boundary that we obtain is shown in the stability diagram of fig . [ fig : stability ] . red circles indicate parameter values at which we obtain dynamo action whereas green stars are used for values at which no dynamo occurs . the most important result that follows from this stability diagram is that the boundary between dynamo and no - dynamo regimes is very complicated . it seems to be of fractal - type , with an intricate pattern of fine , dynamo regions interleaved with no - dynamo regimes . this is especially apparent in the inset of fig . [ fig : stability ] , which shows a detailed view of the stability diagram in the vicinity of the dynamo boundary . earlier studies seem to have missed this fractal - type of boundary because they have not been able to examine the transition in as much detail as we have for our shell model . however , fractal - type boundaries between different dynamical regimes have been suggested in other extended dynamical systems ; recent examples include the transition to turbulence in pipe flow @xcite and different forms of spiral - wave dynamics in mathematical models for cardiac tissue @xcite . in fig . [ fig : stability ] we have drawn two black , dashed lines ; the region above the upper one of these lines is predominantly in the dynamo regime ; the area below the lower one of these lines is predominantly in the no - dynamo regime . these two lines give an approximate indication of the error bars we might expect in the determination of the dynamo boundary in a study that can not scan through points in the @xmath128 plane as finely as we have . from fig . [ fig : tseries ] we see that the order parameter @xmath33 jumps from a very small value in the no dynamo region to a value @xmath129 in the dynamo state . it is natural , therefore , to think of the dynamo boundary as a nonequilibrium , first - order boundary . in an equilibrium , first - order transition the order parameter shows hysteretic behavior if we scan through a first - order boundary by , say , changing , at a finite rate , the field that is conjugate to the order parameter @xcite . it is natural to ask if we see such hysteretic behavior at the dynamo boundary . indeed , we do , as we show in fig . [ fig : hysterisis ] where we cross the dynamo boundary by changing the amplitude @xmath130 of the forcing term in eq.([shell1 ] ) . figure [ fig : hysterisis ] shows representative plots of the dynamo order parameter @xmath33 versus @xmath130 ; these illustrate the hysteretic behavior that occurs when @xmath130 is cycled at a finite , nonzero rate across the dynamo boundary ; here @xmath131 and @xmath132 . as @xmath130 increases , @xmath33 follows the blue , full line : it increases and then saturates ; fluctuations are superimposed on these mean trends . if we now decrease @xmath130 , then @xmath33 follows the red dotted line , and not the blue one , i.e. , we have a hysteresis loop . the faster the rate at which we change @xmath130 the wider is the hysteresis loop as is known from studies of hysteresis in spin systems @xcite . here we increase @xmath130 in steps of @xmath133 from an initial value of @xmath133 ; we keep @xmath130 constant for a time period @xmath134 in fig . [ fig : hysterisis](a ) and @xmath135 in fig . [ fig : hysterisis](b ) ; the red , dotted - line segments of the hysteresis loops are obtained by decreasing @xmath130 at the same rates as for the blue , full - line segments ; the loop in the former case is narrower than in the latter . given the analogy with first - order transitions that we have outlined above , it is natural to ask if nucleation - type phenomena @xcite are also associated with dynamo formation . it would be interesting to check this in a dns of the mhd equations . at the level of our shell model , the best we can do is to try to see if , for a given @xmath0 , when we obtain a dynamo , the time required for dynamo action @xmath117 diverges as we approach the dynamo boundary . our data are consistent with an increase of @xmath117 as we approach this boundary from the dynamo side as shown by the representative plots in fig . [ fig : time ] . however , it is hard to fit a precise form to the behavior of @xmath117 near the dynamo boundary partly because of the complicated nature of this boundary which makes it difficult to estimate the position @xmath20 reliably ( the plot in fig . [ fig : time ] is motivated by the form of eq . ( 27 ) in ref . we have presented a detailed study of dynamo action in a shell model of turbulence @xcite . our study has been designed to explore the nature of the boundary between dynamo and no - dynamo regimes in the @xmath119 plane over a much wider range of @xmath0 than has been attempted in earlier numerical studies . the dynamo boundary emerges as a first - order nonequilibrium phase boundary between one turbulent , nonequilibrium statistical steady state ( ness ) and another @xcite . this point of view is implicit in earlier work , e.g. , in studies of the kazantsev dynamo @xcite or in studies that view dynamo generation as a subcritical bifurcation @xcite . one of these studies @xcite has remarked that when dynamo action ... is obtained in a fully turbulent system , where fluctuations are of the same order of magnitude as the mean flow ... the traditional concept of amplitude equation may be ill - defined and one may have to generalize the notion of subcritical transition `` for turbulent flows ... '' . we believe that the natural generalization is the nonequilibrium , first - order transition we suggest above . we have explored the explicit consequences of such a view in far greater detail than has been attempted hitherto . in particular , the ratio @xmath33 is a convenient order parameter for this nonequilibrium phase transition ; it shows hysteresis across the dynamo boundary like order parameters at any first - order transition ; and nucleation - type phenomena also seem to be associated with dynamo formation . last , and perhaps most interesting , we find that the dynamo boundary seems to have a fractal character ; this provides a natural explanation for the large error bars in earlier attempts to determine this boundary @xcite . furthermore , this fractal - type boundary might well be the root cause of magnetic - field reversals discussed , e.g. , in refs . @xcite . it is important to check , of course , that our shell - model results carry over to the mhd equations . this requires large - scale dns that might well be beyond present - day computing capabilities if we want to explore issues like the possible fractal nature of the dynamo boundary . however , analogs of the hysteretic behavior we mention above have been obtained in dns studies of the mhd equations @xcite ; hysteresis has also been seen in a numerical simulation that includes turbulent convection @xcite . in some of these studies hysteretic behavior is obtained by changing the viscosity of the magnetic prandtl number . we have obtained hysteresis by changing the forcing ; this change of forcing might be easier to effect in experiments than a change of the viscosity or magnetic diffusivity . to the best of our knowledge , earlier studies have not noted the increase in the dynamo - formation time @xmath117 as the dynamo boundary is approached from the dynamo side . we have suggested that this is akin to the increase in the time required to form a critical nucleus as we approach a first - order boundary @xcite . it would be interesting to see if such an increase of @xmath117 can be obtained in dns studies of dynamo formation with the mhd equations . it is worth noting here that some dns studies @xcite have suggested that simulation times comparable to the diffusion time scale @xmath89 are required to confirm dynamo formation ; by contrast our shell - model study yields dynamo action on a much shorter time @xmath117 , which increases as we approach the dynamo boundary . perhaps the large simulation times required for dynamo action in full mhd simulation might have arisen because these simulations have been carried out in the vicinity of the dynamo boundary . to settle completely whether the dynamo boundary is of fractal - type , very long simulations might be required to make sure that the apparent fractal nature is not an artifact of long - lived metastable states to make sure that our calculations do not suffer from such an artifact , we have carried out very long runs for representative points in the region of the dynamo boundary in fig . [ fig : stability ] ; we have found that these long runs do not change our results . furthermore , it is useful to check whether , instead of one dynamo boundary , there is a sequence of transitions , with more and more complicated temporal behaviors for the order parameter , as has been seen in the turbulence - induced melting of a nonequilibrium vortex crystal @xcite . we have not found any conclusive evidence for this but , in the vicinity of the dynamo boundary , the order parameter can oscillate for fairly long times ( see , e.g. , the green full curve in fig . [ fig : tseries ] ) . to decide conclusively whether these oscillations characterize a new nonequilibrium oscillating state , different from the simple dynamo and no - dynamo nesss we have mentioned , requires extensive numerical studies that lie beyond the scope of this paper . in equilibrium statistical mechanics different ensembles are equivalent ; in particular , we may determine a first - order phase boundary by using either the canonical or the grand - canonical ensemble . however , such an equivalence of ensembles does not apply to transitions between different nonequilibrium statistical steady states ( nesss ) ; examples may be found in driven diffusive systems @xcite or in the turbulence - induced melting of a nonequilibrium vortex crystal @xcite . given that the dynamo boundary separates two turbulent nesss , we might expect that this boundary might depend on precisely how the system is forced . evidence for this exists already : for example , the dynamo boundary depends on whether a stochastic external force is used @xcite or whether a taylor - green force is used @xcite ; furthermore , this boundary is different if the fluid is helical @xcite , as in most astrophysical dynamos . we thank r. karan , s.s . ray , and s. ramaswamy for discussions , serc(iisc ) for computational resources and dst , ugc and csir india for support . one of us is a member of the international collaboration for turbulence research ( ictr ) . et al . _ , new j. phys . * 4 * , 84 ( 2002 ) ; phys . lett . * 92 * , 054502 ( 2004 ) ; new j. phys . * 9 * , 300 ( 2007 ) ; a.a . schekochihin , s.c . cowley , and s.f . taylor , astrophys . j. * 612 * , 276 ( 2004 ) . f. ptrlis and s. fauve , europhys . lett . * 22 * , 273 ( 2001 ) ; * 76 * , 602 ( 2006 ) ; s. fauve and f. ptrlis , in _ peyresq lectures on nonlinear phenomena _ , edited by j .- a . sepulchre ( world scientific , singapore , 2003 ) , vol . 2 , pp . 164 ; s. fauve , f. ptrlis , c.r . physique * 8 * , 87 ( 2007 ) . minnini , d.c . montgomery , and a. pouquet , phys . fluids * 17 * , 035112 ( 2005 ) ; p.d . _ , astrophys . j. * 626 * , 853 ( 2005 ) ; p. mininni , a. alexakis , and a. pouquet , phys . e * 72 * , 046302 ( 2005 ) ; p.d . mininni and a. pouquet , phys . lett . * 99 * , 254502 ( 2007 ) . j. leorat , p. lallemand , j.l . guermond , and f. plunian , in _ dynamo and dynamics : a mathematical challenge _ , edited by p. chossat _ ( kluwer academic , dordrecht , 2001 ) , p. 25 - 33 ; r. stepanov and f. plunian , j. turbulence , * 7 * , n 39 ( 2006 ) . shajahan , s. sinha , and r. pandit , phys . e * 75 * , 011929 ( 2007 ) ; t.k . shajahan , a.r . nayak , and r. pandit , plos one * 4*(3 ) , e4738 ( 2009 ) . oxtoby , j. phys . condens . matter * 4 * , 7627 ( 1992 ) . we use the expression first - order transition as in equilibrium statistical mechanics since the order parameter ( the first derivative of a free energy ) jumps at the transition . since there is no free energy for the nesss we consider , one can , just as well , refer to this as a discontinuos transition .
|
we carry out systematic and high - resolution studies of dynamo action in a shell model for magnetohydrodynamic ( mhd ) turbulence over wide ranges of the magnetic prandtl number @xmath0 and the magnetic reynolds number @xmath1 .
our study suggests that it is natural to think of dynamo onset as a nonequilibrium , first - order phase transition between two different turbulent , but statistically steady , states .
the ratio of the magnetic and kinetic energies is a convenient order parameter for this transition . by using this order parameter
, we obtain the stability diagram ( or nonequilibrium phase diagram ) for dynamo formation in our mhd shell model in the @xmath2 plane .
the dynamo boundary , which separates dynamo and no - dynamo regions , appears to have a fractal character .
we obtain hysteretic behavior of the order parameter across this boundary and suggestions of nucleation - type phenomena .
# 1 # 2 # 3 ( # 1,#2)(0,0 ) ( 0,0)(#1,#2)#3
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
with the advent of the x - ray free - electron lasers @xcite there are new possibilities to explore matter on atomic and single molecule levels . on these length scales , of the order of a few ngstrm , quantum effects play an important role in the dynamics of the electrons . quantum effects have been measured experimentally both in the degenerate electron gas in metals and in warm dense matters @xcite . it has also been found that quantum mechanical effects must be taken into account in intense laser - solid density plasma interaction experiments @xcite . the interaction of large amplitude electromagnetic waves with the plasma can lead to various parametric instabilities @xcite . at laser intensities around @xmath0 and above , the nonlinearity associated with relativistic electron mass increase in short laser pulses plays a significant role . furthermore , the relativistic ponderomotive force @xcite of intense laser pulses produces density modifications . thus , in a classical plasma , nonlinear effects associated with relativistic electron mass increase and relativistic ponderomotive force very important , since they provide the possibility of the modulational instability @xcite followed by a compression and localization of intense electromagnetic waves . in addition to the modulational instability , there are relativistic raman forward and backward scattering instabilities @xcite and the two - plasmon decay @xcite instability that lead to strong collisionless heating of the plasma in the relativistic regime . the parametric instabilities of intense electromagnetic waves in magnetized plasmas have also been investigated @xcite . however , for intense electromagnetic waves interacting with the plasma in the x - ray and @xmath1-ray regimes , both relativistic and quantum effects must me taken into account on equal footing . accordingly , in this paper , we present a simple nonlinear model , based on the klein - gordon ( kg ) equation coupled with the maxwell equations that are capable of treating both the relativistic and quantum effects . our work has applications in laboratories @xcite , in quantum free electron laser systems @xcite , as well as in astrophysical settings @xcite where white dwarf cores @xcite and neutron stars @xcite are strong sources of x - rays and @xmath1-rays . the manuscript is organized as follows . in sec . ii , we present our mathematical model for the coupled kg and maxwell equations , exhibiting nonlinear interactions between relativistic electrons and electromagnetic fields . linear properties of the electrostatic and electromagnetic waves are discussed in sec . section iv shows hoe our governing equations lead to the wave equation that reveals the phenomena of relativistic self - focusing and relativistic self - induced transparency of electromagnetic waves . section v is concerned with the theoretical and numerical investigations of the relativistic parametric instabilities in the quantum regime . section vi deals with relativistic optical solitary waves . the nonlinear dynamics of interacting intense localized electromagnetic pulses , as well as the new phenomena of the formation of nonlinear bernstein - greene - krushkal ( bgk)-like modes and associated electron acceleration are described in sec . section viii contains a brief summary and conclusions . historically , the klein - gordon equation ( kge ) for an electron is obtained from the relativistic relation between the energy @xmath2 and the momentum @xmath3 , viz . @xmath4 where @xmath5 is the speed of light in vacuum and @xmath6 the electron mass . by the substitution @xmath7 and @xmath8 in ( 1 ) , where @xmath9 is the planck constant divided by @xmath10 , we obtain the kge for a free electron as @xmath11 where @xmath12 is the electron wave function . the free - particle kge fulfills the continuity equation @xmath13 where @xmath14 and @xmath15 we have multiplied the right - hand sides of eqs . ( [ eq4 ] ) and ( [ eq5 ] ) by the electron charge @xmath16 , so that @xmath17 can be interpreted as the electric charge density and @xmath18 as the electric current density . since @xmath17 is neither positive or negative definite , it can not be interpreted as a probability density , however , it can be interpreted as a charge density which need not has a definite sign . we now wish to use the charge and current densities as sources for the self - consistent electromagnetic scalar and vector potentials @xmath19 and @xmath20 for a quantum plasma . we , therefore , let @xmath12 represent an ensemble of the electrons . introducing the electromagnetic potentials into the kge , we make the usual substitutions @xmath21 and @xmath22 , obtaining @xmath23 where we have defined the energy and momentum operators as @xmath24 and @xmath25 respectively . the electric charge and current densities are now obtained as @xmath26 , \label{rho_e}\ ] ] and @xmath27 , \label{j_e}\ ] ] respectively . we note that the charge and current densities obey the continuity equation @xmath28 the self - consistent vector and scalar potentials are obtained from the electromagnetic wave equations @xmath29 and @xmath30 where @xmath31 is the magnetic vacuum permeability and @xmath32 is the electric permittivity in vacuum , and @xmath33 is the neutralizing positive charge density due to the ions . for immobile , singly charged ions , one can assume that @xmath34 , where @xmath35 is the equilibrium ion number density . using the coulomb gauge @xmath36 , we obtain from eqs . ( [ vector_pot ] ) and ( [ scalar_pot ] ) @xmath37 and @xmath38 respectively . taking the divergences of both sides of eq . ( [ wave2 ] ) , we have @xmath39 so that eq . ( [ wave2 ] ) can be written as @xmath40 equations ( [ kg ] ) , ( [ poisson ] ) and ( [ wave3 ] ) are our desired system that describes intense laser - plasma interactions in the quantum regime . the non - relativistic limit is obtained from eq . ( [ kg ] ) by substituting @xmath41 , and by using the condition @xmath42 , together with the normalization of @xmath43 such that @xmath44 is the electron number density at the equilibrium . in this limit , ( [ kg ] ) , yields the schrdinger equation @xmath45 here , and in what follows , we have used a simplified model and neglected the electron degeneracy pressure . the latter is important in dense matters where the electron degeneracy pressure appears due to the pauli exclusion principle . for a non - relativistic plasma , the quantum statistical pressure has been introduced in a nonlinear schrdinger model @xcite , but this has to be investigated for relativistic quantum plasmas . in the absence of the electromagnetic field ( viz . @xmath46 ) , we still have electrostatic waves due to the charge separation between the electrons and ions . at short wavelengths , the quantum effects become important and give rise to dispersive effects in the electrostatic wave . at these wavelengths , there is an interplay between collective electron oscillations and free electron motion . when the wavelength is comparable to the compton wavelength , the electrons become relativistic , and there are relativistic corrections to the dispersion relation for the electrostatic wave . in the derivation of the dispersion relation for relativistic electrons , it is convenient to first make the transformation @xmath47 , where the wave function @xmath48 obeys the wave equation @xmath49 and the electron charge density is @xmath50 we next linearize the system ( 19 ) by setting @xmath51 and @xmath52 , where @xmath53 + complex conjugate , @xmath54 , and where @xmath55 . separating different fourier modes , we obtain from ( 19 ) the dispersion relation for the electrostatic oscillations as @xmath56 , where @xmath2 is the dielectric constant and the electron susceptibility is @xmath57 } { \hbar^2(\omega^2-c^2 k^2)^2 - 4 m_e^2 c^4\omega^2 } , \label{chi_e}\ ] ] where @xmath58 is the electron plasma frequency . we note that in the classical limit @xmath59 , we have @xmath60 , while in the non - relativistic limit @xmath61 , we have @xmath62 . after some reordering of terms , the dispersion relation can be written as @xmath63 in the classical limit @xmath59 we obtain the langmuir oscillations @xmath64 , while in the limit @xmath61 , we retain the non - relativistic result @xmath65 on the other hand , in the limit @xmath66 , eq . ( [ es_disp ] ) yields two possibilities , one of which is the langmuir oscillations at the plasma frequency , @xmath64 and the other one is oscillations with the frequency @xmath67 . the latter corresponds to a negative energy state , which can be interpreted as positronic state . we note that there is a non - dimensional quantum parameter @xmath68 in eq . ( [ es_disp ] ) that determines the relative importance of the quantum effect . typical values are @xmath69 for the electron number density @xmath70 in solid density laser - plasma experiments and @xmath71 may be representable of modern laser - compressed matter experiments with @xmath72 . this corresponds to @xmath73 and @xmath74 for @xmath69 , and @xmath75 and @xmath76 for @xmath71 , where @xmath77 is the electron skin depth . the non - relativistic result ( [ es_disp2 ] ) is valid for electrostatic waves with wave numbers in the range @xmath78 . for @xmath79 , the quantum corrections to @xmath80 are different from ( [ es_disp2 ] ) and turns the wave frequency slightly lower than @xmath81 . however , this effect is negligible for small values of @xmath82 , and may be smaller than the degeneracy electron pressure effect , which is neglected here . on the other hand , the limit @xmath83 corresponds to relativistic particles with @xmath84 , where @xmath85 is the reduced compton wavelength . for @xmath86 , we obtain the relativistic free particle dispersion relation @xmath87 where the upper sign ( - ) corresponds to the motion of a free electron and the lower sign ( + ) can be interpreted as the motion of a free positron . vs. @xmath88 ) for the electrostatic oscillations for @xmath69 ( solid curves ) , @xmath71 ( dashed curves ) , where @xmath89 . for @xmath83 , the particle motion turns from weakly relativistic to ultra - relativistic.,width=321 ] in fig . 1 , we have plotted the solutions of the dispersion relation ( [ es_disp ] ) for @xmath69 and @xmath71 . both the electron plasma oscillations and the positronic states are shown . the electron plasma oscillations have a cutoff frequency @xmath90 when @xmath91 , while the positronic states have a cutoff frequency @xmath92 , corresponding to @xmath93 at @xmath66 in fig . 1 . for the electron plasma oscillations , the increase in the wave frequency due to the quantum effect becomes noticeable approximately where @xmath94 , or @xmath95 where @xmath96 is the bohr radius . this corresponds to a wavelength of @xmath97 for @xmath69 and @xmath98 for @xmath71 . the positronic states are associated with , for example , the zitterbewegung effect @xcite , in which the interference between the positive and negative energy states are predicted to give oscillations on compton wavelength scales in space . the zitterbewegung effect is still debated and has not yet been observed in experiments . it is well - known @xcite that a large amplitude electromagnetic wave propagating in a classical plasma changes the dispersive properties of the plasma due to the relativistic mass increase of the electrons . we show here that the same effect occurs in our klein - gordon - maxwell system . we consider for simplicity the propagation of a right - hand circularly polarized electromagnetic ( cpem ) wave of the form @xmath99 $ ] , where @xmath100 is the wave frequency and @xmath101 the wavenumber . due to the circular polarization , the oscillatory parts in the nonlinear term proportional to @xmath102 in the klein - gordon equation vanish . assuming that @xmath12 depends only on time and not on space , and that @xmath103 , we obtain from eq . ( [ kg ] ) @xmath104 where @xmath105 can be interpreted as the relativistic gamma factor due to the electron mass increase in the cpem wave field . equation ( [ kg_time ] ) has the solution @xmath106 where the constant @xmath107 is determined by assuming the constant density @xmath108 in eq . ( [ rho_e ] ) . inserting ( [ psi ] ) into ( [ rho_e ] ) with @xmath108 , we obtain @xmath109 on the other hand , inserting ( [ psi ] ) into ( [ j_e ] ) we have @xmath110 which can be inserted into ( [ wave2 ] ) to obtain @xmath111 equation ( [ plane_wave ] ) admits the nonlinear dispersion relation @xmath112 which predicts a relativistic downshift of the cpem wave frequency due to the relativistic electron mass increase in the cpem wave field . since the effective plasma frequency is decreased by a factor @xmath113 , the model predicts the well - known self - induced transparency where the cpem wave can propagate at frequencies below the electron plasma frequency . this is identical to the case of classical plasmas @xcite . we now consider the instability of an intense cpem wave in the quantum regime . in the presence of intense electromagnetic waves , we have the relativistic down - shift in the wave frequency given in ( [ omega0 ] ) , as well as the possibility of exciting electrostatic oscillations via the parametric instabilities . as an example , we will here consider stimulated raman scattering instability , in which an intense electromagnetic wave decays into a daughter em wave and an electron plasma wave . the two - plasmon decay instability , in which the cpem wave decays into two electrostatic waves , will be treated elsewhere . it is convenient to first introduce the transformation @xmath114 , where @xmath105 and @xmath115 is the amplitude of the em carrier wave @xmath116 . the wavefunction @xmath48 obeys the modified klein - gordon equation @xmath117 and the electron charge density is given by @xmath118 now , we linearize our system by introducing @xmath119 ( where @xmath120 is assumed to be constant ) , @xmath121 , and @xmath122 . using @xmath34 into eq . ( [ poisson ] ) , we note that the equilibrium quasi - neutrality requires that @xmath120 is normalized such that @xmath123 . using that @xmath116 fulfills the plane wave equation ( [ plane_wave ] ) , the linearized kge ( [ kg5 ] ) , poisson s equation ( [ poisson ] ) and the em wave equation ( [ wave3 ] ) then become @xmath124 @xmath125 and @xmath126\ } , \end{split}\ ] ] respectively . we note that the term proportional to @xmath127 in eq . ( [ kg_linear ] ) gives rise to the two - plasmon decay , which we , however , do not consider here . we now introduce the fourier representations @xmath128 , @xmath129 + c.c . , @xmath130 + c.c . , and @xmath131 $ ] + c.c . , where we introduced @xmath132 and @xmath133 , and c.c . stands for complex conjugate . in one of the steps , we take the scalar product of both sides of the em wave equation by @xmath134 and use the fact that @xmath135 = ( { \bf k}_\pm\times\widehat{\bf a}_0)\cdot ( \widehat{\bf a}_0^\ast \times{\bf k}_\pm)=-|{\bf k}_{\pm}\times \widehat { \bf a}_0|^2 $ ] . separating different fourier modes and eliminating the fourier coefficients , we find the nonlinear dispersion relation @xmath136 } \bigg[\frac{c^2 e^2|{\bf k}_+\times \widehat{\bf a}_0|^2}{k_{+}^2 d_a(\omega_{+},{\bf k}_{+ } ) } + \frac{c^2 e^2|{\bf k}_{-}\times \widehat{\bf a}_0|^2}{k_{-}^2 d_a(\omega_{-},{\bf k}_{-})}\bigg ] , \end{split } \label{nonlin_disp2}\ ] ] where the electromagnetic sidebands are governed by @xmath137 . the electric susceptibility in the presence of the laser field is given by @xmath138 } { \gamma_a[\hbar^2(\omega^2-c^2k^2)^2 - 4\gamma_a^2 m_e^2 c^4 \omega^2]}.\ ] ] after reordering of terms , the nonlinear dispersion relation ( [ nonlin_disp2 ] ) can be written as @xmath139=0 , \end{split } \label{nonlin_disp}\ ] ] where the electron plasma oscillations in the presence of the laser field are represented by @xmath140 we note that @xmath141 gives the dispersion relation for pure electrostatic oscillations in the presence of a large amplitude electromagnetic wave . in the classical limit @xmath142 , we have @xmath143 , and the nonlinear dispersion relation takes the form @xmath144 , \end{split}\ ] ] which can be written in a more familiar form as @xmath145=0 , \end{split}\ ] ] with @xmath146 . these results can be compared with , for example , the dispersion relations obtained in refs . @xcite for the relativistic case and in @xcite for the non - relativistic case . to proceed with the numerical evaluation of the nonlinear dispersion relation , we choose a coordinate system such that the cpem takes the form @xmath147 and @xmath148 , and , without loss of generality , we choose @xmath149 then , we have @xmath150 , @xmath151 , @xmath152|\widehat{a}_0|^2 $ ] , and @xmath153 . we also use that the carrier wave @xmath116 obeys the nonlinear dispersion relation @xmath154 . vs. @xmath155 and @xmath156 ) for stimulated raman scattering in the presence of a large amplitude cpem wave , for the amplitudes @xmath157 , @xmath158 , and @xmath159 ( left to right panels ) where @xmath160 for @xmath69 ( top panels ) and @xmath71 ( bottom panels ) . , width=321 ] we now assume that the wave frequency is complex valued , @xmath161 , where @xmath162 is the real frequency and @xmath163 the growth rate , and solve numerically the dispersion relation ( [ nonlin_disp ] ) for @xmath164 . in fig . 2 , we have plotted the growth rate for stimulated raman scattering instability as a function of the wavenumbers @xmath155 and @xmath156 , for a few values of @xmath165 and @xmath166 . for all cases in fig . 2 , we used @xmath167 , which corresponds to a wavelength of @xmath168 for @xmath69 and to @xmath169 for @xmath71 , which is in the x - ray regime . we observe that for @xmath69 , there is a broad spectrum of unstable waves , in particular for @xmath158 and @xmath170 . for @xmath71 , we observe a reduction in the spectrum of unstable waves and the growth rate ( relative to the electron plasma frequency ) is slightly reduced . this is due to the fact that the wavelength of the unstable electrostatic oscillation approaches the critical wavelength , where quantum dispersive effects become important compared to the plasma frequency oscillations . for @xmath69 , this wavelength is @xmath171 , corresponding to a critical wavenumber @xmath172 , and for @xmath71 , we have @xmath173 , corresponding to @xmath174 . hence , for @xmath71 we have @xmath175 , which leads to the reduction of the growth rate due to the quantum dispersion effect . furthermore , it should be mentioned that we do not find raman - type instabilities involving the positronic states in fig . this is consistent from the point of view of the conservation of charges , since the production of positrons would violate the conservation of electric charges . vs. @xmath155 and @xmath156 ) for the modulational instability in the presence of a large amplitude cpem wave , for the amplitudes @xmath157 ( left panel ) , @xmath158 ( middle panel ) and @xmath159 ( right panel ) where @xmath160 . we used @xmath71 for all cases . , width=321 ] in addition to stimulated raman scattering instabilities , we have also the modulational instability that dominates for the pump frequencies @xmath176 , and the corresponding wavenumbers @xmath177 . the modulational instability usually occurs for small modulation wavenumbers , and saturates nonlinearly by the formation of relatively small localized structures / solitary waves . in the past , such nonlinear structures have been studied for classical plasmas in 1d @xcite and 3d @xcite . we have investigated the modulational instability for the cpem dipole field with @xmath178 , and have plotted the results in fig . 3 for different amplitudes @xmath157 , @xmath158 and @xmath159 . we find that the growth rate is relatively insensitive to the quantum parameter @xmath82 . we have used @xmath71 in fig . 3 , but @xmath179 gives almost identical results . this is understandable since the modulational instability takes place on relatively large scales and the quantum effect is thus negligible . however , we will investigate the quantum effect on the relatively small scale nonlinear structures below . here we illustrate the existence of large amplitude localized cpem wave excitations at the quantum scale in our system . we restrict our investigation to one - space dimension , which has also been studied for classical plasmas @xcite . far away from the local excitation , one can assume that the dynamics of the plasma is non - relativistic . to shorten the algebraic steps , it is convenient first to introduce a new wave - function @xmath180 and the potential @xmath181 via the transformations @xmath182 and @xmath183 , and which satisfy the kge @xmath184 where @xmath185 . in this gauge , the wave function @xmath43 is non - oscillatory in time , and the new potential takes the value @xmath186 , far away from the solitary wave where the plasma is at rest . the electron charge density is expressed as @xmath187 . \label{e - charge}\ ] ] we now study quasi - steady state structures propagating with a constant speed @xmath188 , so that @xmath189 and @xmath190 , where @xmath191 and @xmath192 . the cpem wave vector potential is of the form @xmath193 $ ] . it is convenient to introduce the eikonal representation @xmath194 $ ] , where @xmath195 and @xmath196 are real - valued . then , the kge ( [ kg3 ] ) takes the form @xmath197 -i \hbar e v_0 \frac{d\phi } { d\xi } p \\ & -2i\hbar e \phi v_0 \left(\frac{dp}{d\xi}+ i p\frac{d\theta}{d\xi}\right ) + ( e^2\phi^2-m_e^2 c^4\gamma_a^2)p=0 . \end{split } \label{kg4}\ ] ] setting the imaginary part of eq . ( [ kg4 ] ) to zero , we obtain @xmath198 where @xmath199 the solution of eq . ( [ pu ] ) is @xmath200 constant . using the boundary conditions @xmath201 , @xmath103 ( hence @xmath202 ) and @xmath203 at @xmath204 , we have @xmath205 . hence , we obtain @xmath206 where we have denoted @xmath207 the electron charge density ( [ e - charge ] ) now takes the form @xmath208 which , by using eq . ( [ theta ] ) , can be written as @xmath209 and hence poisson s equation ( [ poisson ] ) , with @xmath34 , becomes @xmath210 on the other hand , by setting the real part of eq . ( [ kg4 ] ) to zero , we have @xmath211 + 2\hbar e\phi v_0 p\frac{d\theta}{d\xi}+ ( e^2 \phi^2-m_e^2 c^4\gamma_a^2)p=0 , \end{split}\ ] ] which , by using eq . ( [ theta ] ) , can be written as @xmath212p=0 . \label{p}\ ] ] finally , inserting the ansatz @xmath193 $ ] for the vector potential , together with the current @xmath213 into eq . ( [ wave3 ] ) , we obtain the em wave equation @xmath214a=0 , \label{a}\ ] ] where we used @xmath215 , and where @xmath216 is a nonlinear eigenvalue of the system that determines the wave frequency @xmath100 . the coupled system ( [ poisson2 ] ) , ( [ p ] ) and ( [ a ] ) describes the profile of electromagnetic solitary waves in a quantum plasma . it has the conserved quantity @xmath217 , where @xmath218 the conserved quantity @xmath219 can be used to check that the numerical scheme used to solve the nonlinear system ( [ poisson2 ] ) , ( [ p ] ) and ( [ a ] ) produces correct results . ( left panels ) and @xmath220 ( right panels ) for standing solitary waves ( @xmath221 ) , and for @xmath222 . , width=321 ] , @xmath223 , and @xmath222 . , width=170 ] in fig . [ fig : comparison3 ] , we have compared the present model with our previous results in ref . @xcite where we used a simplified model to describe the nonlinear interaction interaction of intense cpem wave with a quantum plasma . we used the same parameters as in fig . 2 , of ref . @xcite to produce the profiles of the cpem wave potential , the electron charge density and the electrostatic potential . we observe that the present results are almost identical to our previous work @xcite . for our sets of parameters , the quantum effect on the profiles of the solitary waves are small , and there is only a slight difference in the profiles of the electron density for the two values @xmath220 and @xmath71 . for standing solitary waves , such as the ones in fig . [ fig : comparison3 ] , the solutions are localized with exponentially decaying tails . by linearizing the system ( [ poisson2 ] ) , ( [ p ] ) and ( [ a ] ) one can show that far away from the soliton , @xmath224 decays as @xmath225 , while @xmath195 and @xmath19 are proportional to @xmath226 , where @xmath88 is found from the dispersion relation @xmath227 for @xmath221 ( and @xmath228 ) , we see immediately that there exist only complex - valued @xmath88 , which means that the quasi - stationary wave solutions decay exponentially far away from the solitary wave . however , in the classical limit @xmath59 , we instead have the plasma wake oscillations given by real valued @xmath229 . hence , an electromagnetic pulse will create an oscillatory wake that extends far away from the em pulse . in one - space dimension , there also exist special classes of propagating localized em envelope solutions @xcite . in addition to the wake oscillations , we also have quantum oscillations in quantum plasmas . in fig . 5 , we show an example of a slowly moving envelope soliton , where small - scale oscillations in the charge density are clearly visible . we note that the cold fluid results can be retained in the classical limit @xmath59 . then , eq . ( [ p ] ) can be written as @xmath230 by setting @xmath231 , where @xmath232 is the electron number density , in eq . ( [ rho_e2 ] ) and solving for @xmath195 , we obtain @xmath233\frac{m_e c^2}{e\phi},\ ] ] which can be inserted into ( [ p2 ] ) to obtain @xmath234/\bigg [ \frac{n_e^2}{n_0 ^ 2}-\frac{v_0 ^ 2}{c^2}\bigg(\frac{n_e}{n_0}-1\bigg)^2 \bigg]^{1/2 } = \frac{e\phi}{\gamma_a m_e c^2 } , \label{fluid}\ ] ] which relates @xmath232 to @xmath181 and @xmath235 at a given speed @xmath188 . the relation ( [ fluid ] ) can also be obtained from the cold electron fluid model @xcite and hence confirms the classical limit of the quantum model used here . if , furthermore , we assume standing waves such that @xmath221 , then we have from ( [ fluid ] ) @xmath236 solving for @xmath181 and inserting the result into poisson s equation ( [ poisson2 ] ) , we have @xmath237 where @xmath77 is the electron skin depth . finally , solving for @xmath238 and inserting the result into eq . ( [ a ] ) , we obtain @xmath239 where we have used @xmath221 and , therefore , @xmath178 . equation ( [ a2 ] ) is identical to the model of marburger and tooper @xcite for the nonlinear optical standing wave in a classical cold fluid electron plasma . the relativistic mass increase is reflected by the ratio @xmath240 in the right - hand side of eq . ( [ a2 ] ) . the nonlinear electron density fluctuations , which are reflected in the term proportional to @xmath241 in the right - hand side of eq . ( [ a2 ] ) can often be neglected in the weakly relativistic case @xcite . we note that our previous model @xcite can be recovered in the weakly relativistic limit , in the following manner . assuming that , to first order , we have a balance between the ponderomotive and electrostatic pressure so that @xmath242 and that @xmath243 , and @xmath244 . accordingly , we have @xmath245 poisson s equation @xmath246 and the cpem wave equation @xmath247 we now make a simple change of variables @xmath248 . then , we have , by neglecting terms containing @xmath249 , the model @xcite @xmath250 @xmath251 and @xmath252 where the relativistic mass increase appears explicitly in the cpem wave equation . , and initially a dipole field @xmath157 with @xmath178.,width=321 ] in order to study the dynamics of the nonlinear interaction between intense cpem waves and a quantum plasma , we have carried out numerical simulations of the klein - gordon - maxwell system of equations . we have here restricted our study to one - space dimension , along the @xmath253 direction in space , and written our governing equations in the form @xmath254 @xmath255 and @xmath256 we used a periodic simulation box in space , of length @xmath257 and used of the order @xmath258 grid points to resolve the solution in space . it is important to resolve the relatively long electron skin depth scale as well as the shorter length scale associated with accelerated electrons with the momentum @xmath259 and the associated wavelength @xmath260 . since we need at least two grid - points per wavelength to represent the solution , the required grid size is @xmath261 , which can be written @xmath262 . for example , to represent the wave function of relativistic electrons with the momentum @xmath263 , we need a spatial grid with @xmath264 to represent the wave function for @xmath71 . the solution was advanced in time with the standard 4th - order runge - kutta scheme , using a timestep of order @xmath265 . . top to bottom panels show a ) the electromagnetic vector potential of the laser pulse ( the arrows show the propagation directions of the pulses ) , b ) the electron charge density , c ) the electrostatic potential , and d ) the distribution of electrons in phase space in a 10-logarithmic color scale . parameters are @xmath71 , amplitude @xmath157 and wavenumber @xmath266 . the laser pulses excite large amplitude oscillatory potential wakes behind them , as they penetrate the plasma slab.,width=321 ] at time @xmath267 . groups of electrons are accelerated to ultra - relativistic speeds by the large amplitude electrostatic wake field.,width=321 ] we first study the growth and nonlinear saturation of the modulational instability , which is relevant for dense matters where the plasma is overdense or close to overdense . as initial conditions , we used a cpem pump wave of the form @xmath268 $ ] with @xmath178 and @xmath269 . a small amplitude noise ( random numbers ) was added to @xmath224 in order to seed any instability in the system . as initial conditions for the wave function , we used @xmath270 and @xmath271 , corresponding to a pure electronic state at equilibrium . the initially homogeneous electron density was set to @xmath272 , corresponding to @xmath71 [ cf . ( [ h ] ) ] . in this situation , the electromagnetic wave is unstable due to the modulational instability , which has instability for small wavenumbers , as shown in fig . 3 . in the nonlinear stage , the em waves self - focus into localized wavepackets similar to the ones in fig . 4 . figure [ fig : modulational ] depicts the late stage of the modulational instability . the collapse of the cpem wave packet leads to relativistically strong ponderomotive potentials that accelerate the electrons to relativistic speeds . the relativistic electrons are associated with small - scale spatial oscillations in the wave function , where the wavelength is comparable to or even smaller than the compton wavelength . we see in fig . [ fig : modulational ] that the cpem wave envelope has been focused into localized wavepackets , associated with depletions in the electron density and positive localized electrostatic potentials . in order to study the distribution of electrons both in space and momentum space , we have performed a fourier transform of the wavefunction @xmath12 using a moving window technique ( using a hann window ) in space . the width of the window has been tuned so that it provides a good resolution both in space and in momentum space . the resulting spatial spectrogram gives a representation of the distribution of the electrons both in space and in momentum space ; see fig . [ fig : modulational](d ) , where the color indicates the density of electrons in phase space . in fig . 6(d ) the horizontal axis shows the spatial dependence and the vertical axis shows the momentum dependence via the relation @xmath259 between the momentum @xmath273 and the wavenumber @xmath274 . in this figure , it is clear that in the collapse stage of the solitary waves , bunches of electrons are accelerated to relativistic speeds and form self - trapped , bernstein - greene - kruskal ( bgk)-like modes that propagate away from the collapsed electromagnetic wavepackets . next , we investigate a scenario of the short em pulse propagation and the wake - field generation in a quantum plasma . this concept is traditionally used for the electron acceleration in classical plasmas @xcite . the numerical results are displayed in figs . [ fig : colliding1 ] and [ fig : colliding2 ] . here , two atto - second pulses are injected from each side of a plasma slab and are allowed to collide at the center of the slab . as initial conditions , we used a cpem pump wave of the form @xmath275 $ ] with @xmath266 and envelopes of the form @xmath276 propagating into the plasma slab . the plasma slab is initially centered between @xmath277 with equal electrons with the number densities @xmath35 , where the electron wave function was set to @xmath278 and @xmath279 . after a time @xmath280 , we see in fig . [ fig : colliding1](b ) and ( c ) that the large amplitude cpem pulses excite plasma wake oscillations associated with large - amplitude positive potentials , and with an approximate wavelength of @xmath281 , corresponding to a leading wavenumber of @xmath282 . the positive potentials of the plasma wake oscillations are starting to capture populations of the electrons at edges of the plasma slab , at @xmath283 . a high - frequency diffraction pattern is formed in the electron density , as faster electrons overtake slower electrons . later , at @xmath267 , the two laser pulses have collided and passed through each other . the trapped electrons have been further accelerated to ultra - relativistic speeds , as seen in fig . [ fig : colliding2](d ) at @xmath284 . , where the fastest electrons have reached a momentum of @xmath285 in this paper , we have developed a relativistic model for the interaction between intense electromagnetic waves and a quantum plasma . our nonlinear model is based on the coupled klein - gordon and maxwell equations for the relativistic electron momentum and the electromagnetic fields . in our fully relativistic model , the electron current and charge densities are calculated self - consistently from the kge , and they enter as sources for the nonlinear em and electrostatic waves in the maxwell equation . the kg - maxwell system of equations has been used to derive the linear dispersion relation for the electrostatic and electromagnetic waves , as well as for investigating stimulated raman scattering and modulational instabilities in the presence of relativistically intense cpem waves . in the linear regime , the general dispersion relation for the electrostatic waves exhibits the quantum effect associated with the overlapping wave function . at long - wave - lengths , we have the dispersive langmuir waves with frequencies close to the electron plasma frequency , while at shorter - wavelengths , we have the oscillation frequency of free electrons . at wavelengths comparable to or larger than the compton wavelength , the electron motion is fully relativistic . in the nonlinear regime , we have demonstrated the existence of fully relativistic stimulated raman scattering and modulational instabilities . while the raman amplification is of much interest for generating a coherent electromagnetic radiation , the modulational instability gives rise to the localization and collapse of the cpem waves into localized solitary em wavepackets . indeed , numerical simulations of the coupled kg - maxwell equations reveal the collapse and acceleration of the electrons in the nonlinear stage of the modulational instability , as well as the possibility of wake - field acceleration of the electrons to relativistic speeds by short laser pulses at nanometer scales . in conclusion , we stress that the present investigation of nonlinear effects dealing with intense em wave interactions with quantum plasmas is relevant for the compression of x - ray free - electron laser pulses to attosecond duration @xcite , as well as to the understanding of localized intense x - ray and @xmath1-ray bursts that emanate from compact astrophysical objects @xcite . 99 e. hand , nature ( london ) * 461 * , 708 ( 2009 ) . s. h. glenzer _ et al . _ , lett . * 98 * , 065002 ( 2007 ) ; p. neumayer _ _ , _ ibid . _ * 105 * , 075003 ( 2010 ) ; s. h. glenzer and r. redmer , rev . phys . * 81 * , 1625 ( 2009 ) . a. v. andreev , jetp lett . * 72 * , 238 ( 2000 ) . g. mourou _ et al . _ , rev . phys . * 78 * , 309 ( 2006 ) . m. marklund and p. k. shukla , rev . phys . * 78 * , 591 ( 2006 ) . j. f. drake , p. k. kaw , y. c. lee , g. schmidt , c. s. liu , and m. n. rosenbluth , phys . fluids * 17 * , 778 ( 1974 ) . r. p. sharma and p. k. shukla , phys . fluids * 26 * , 87 ( 1983 ) . g. murtaza and p. k. shukla , j. plasma phys , * 31 * , 423 ( 1984 ) . et al . _ , phys . 138 * , 1 ( 1986 ) . c. j. mckinstrie and r. bingham , phys . fluids b * 1 * , 230 ( 1989 ) . l. n. tsintsadze , sov . j. plasma phys . * 17 * , 872 ( 1991 ) . c. j. mckinstrie and r. bingham , phys . fluids b * 4 * , 2626 ( 1992 ) . a. s. sakharov and v. i. kirsanov , phys . e * 49 * , 3274 ( 1994 ) . s. gurin , g. laval , p. mora , j. c. adam , a. hron , and a. bendib , phys . plasmas * 2 * , 2807 ( 1995 ) . j. c. adam , a. hron , g. laval , and p. mora , phys . lett . * 84 * , 3598 ( 2000 ) . b. quesnel , p. mora , j. c. adam , a. hron , and g. laval , phys . plasmas * 4 * , 3358 ( 1997 ) . l. stenflo , phys . * 14 * , 320 ( 1976 ) . l. stenflo , phys . * 21 * , 831 ( 1980 ) . l. stenflo and h. wilhelmsson , phys . rev . a * 24 * , 1115 ( 1981 ) . v. m. malkin , n. j. fisch , and j. s. wurtele , phys . e * 75 * , 026404 ( 2007 ) . a. serbeto , j. t. mendona , k. h. tui _ et al . _ , phys . plasmas * 15 * , 013110 ( 2008 ) . a. serbeto , l. f. monteiro , k. h. tsui , and j. t. mendona , plasma phys . fusion * 51 * 124024 ( 2009 ) . n. piovella , m. m. cola , l. volpe , a. schiavi , r. bonifacio , phys . lett . * 100 * , 044801 ( 2008 ) . g. chabrier _ _ , j. phys . : condens . matter * 14 * , 9133 ( 2002 ) ; j. phys . a : math . gen . * 39 * , 4411 ( 2006 ) . m. j. coe _ _ , nature ( london ) * 272 * , 37 ( 1978 ) ; d. k. galloway and j. l. sokoloski , astrophys . j. * 613 * , l61 ( 2004 ) . k. hurley _ et al . _ , nature ( london ) * 434 * , 1098 ( 2005 ) ; a. k. harding and d. lai , rep . . phys . * 69 * 2631 ( 2006 ) . g. manfredi and f. haas , phys . b * 64 * , 075316 ( 2001 ) ; p. k. shukla and b. eliasson , phys . rev . lett . * 96 * , 245001 ( 2006 ) ; phys . usp . * 53 * , 51 ( 2010 ) . m. g. fuda and e. furlani , am . * 50 * , 545 ( 1982 ) . r. gerritsma _ et al . _ , nature ( london ) * 463 * , 68 ( 2010 ) . a. i. akhiezer and r. v. polovin , sov . jetp * 3 * , 696 ( 1956 ) [ zh . eksp . . fiz . * 30 * , 915 ( 1956 ) ] ; p. kaw and j. dawson , phys . fluids * 13 * , 472 ( 1970 ) ; c. max and f. perkins , phys . * 27 * , 1342 ( 1971 ) . j. h. marburger and r. f. tooper , phys . * 35 * , 1001 ( 1975 ) . j. i. gersten and n. tzoar , phys . * 35 * , 934 ( 1975 ) . p. k. shukla and b. eliasson , phys . lett . * 99 * , 096401 ( 2007 ) . p. k. kaw , a. sen , and t. katsouleas , phys . lett . * 68 * , 3172 ( 1992 ) . v. saxena , a. das , a. sen , and p. kaw , phys . plasmas * 13 * , 032309 ( 2006 ) . k. shukla , n. n. rao , m. y. yu , and n. l. tsintsadze , phys . rep . * 138 * , 1 ( 1986 ) . t. tajima and j. m. dawson , phys . * 43 * , 267 ( 1979 ) . r. bingham _ et al . _ , plasma phys . fusion * 46 * , r1 ( 2004 ) . _ , science * 292 * , 1689 ( 2001 ) . r. m. g. m. trines , f. fiuza , r. bingham _ et al . _ , nature phys . * 6 * , 1793 doi:10.1038/nphys1793 ( 2010 ) .
|
we investigate the nonlinear interaction between a relativistically strong laser beam and a plasma in the quantum regime .
the collective behavior of the electrons is modeled by a klein - gordon equation , which is nonlinearly coupled with the electromagnetic wave through the maxwell and poisson equations .
this allows us to study the nonlinear interaction between arbitrarily large amplitude electromagnetic waves and a quantum plasma .
we have used our system of nonlinear equations to study theoretically the parametric instabilities involving stimulated raman scattering and modulational instabilities .
a model for quasi - steady state propagating electromagnetic wavepackets is also derived , and which shows the possibility of localized solitary structures in the quantum plasma .
numerical simulations demonstrate the collapse and acceleration of the electrons in the nonlinear stage of the modulational instability , as well as the possibility of wake - field acceleration of the electrons to relativistic speeds by short laser pulses at nanometer length scales .
the study has importance for the nonlinear interaction between a super - intense x - ray laser light and a solid - density plasma , where quantum effects are important .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the rvs is a slitless spectrograph whose spectral domain is 847 - 874 nm and resolving power @xmath2 . the expected accuracy is 1 km s@xmath1 for f0 to k0 stars brighter than v@xmath3 , and for k1 to k4 stars brighter than v@xmath4 . the main scientific objectives of rvs are the chemistry and dynamics of the milky way , the detection and characterisation of multiple systems and variable stars ( for more details , see * ? ? ? those objectives will be achieved from a spectroscopic survey of : * radial velocities ( @xmath5 objects , v @xmath6 ) * rotational velocities ( @xmath7 objects , v @xmath8 ) * atmospheric parameters ( @xmath7 objects , v @xmath8 ) * abundances ( @xmath9 objects , v @xmath10 ) each star will be observed @xmath11 times on average by rvs over the 5 years of the mission . because the rvs has no calibration module on board , the zero point of its radial velocities has to be determined from reference sources . ground - based observations of a large sample of well - known , stable reference stars as well as of asteroids are thus critical for the calibration of the rvs . a sample of 1420 candidate standard stars has been established @xcite and has to be validated by high spectral resolution observations . two measurements per candidate star are being made before gaia is launched ( or one , depending on already available archived data ) . another measurement will occur during the mission . the measurements will allow us to check the temporal stability of radial velocities , and to reject any targets with significant rv variation . the ongoing observations are performed with three high spectral resolution spectrographs : * sophie on the 1.93-m telescope at observatoire de haute - provence , * narval on the tlescope bernard lyot at observatoire pic - du - midi , * coralie on the euler swiss telescope at la silla . as of june 2011 we have observed 995 distinct candidates with sophie , coralie and narval . the detailed observations per instrument are : * 691 stars ( 1165 velocities ) with sophie * 669 stars ( 945 velocities ) with coralie * 93 stars ( 98 velocities ) with narval figure [ chemin : fig1 ] ( left - hand panel ) represents the spatial distribution in the equatorial frame of the sample and the number of measurements per object we have done so far with the three instruments . in addition to those new observations , we use radial velocity measurements available from the spectroscopic archives of two other high - resolution instruments : elodie , which is a former ohp spectrograph , and harps which is currently observing at the eso la silla 3.6-m telescope . the archived data allow us to recover 1057 radial velocities for 292 stars ( elodie ) and 1289 velocities for 113 stars ( harps ) . figure [ chemin : fig1 ] ( right - hand panel ) summarizes the status of the total number of measurements for the sample of 1420 candidate stars performed with all five instruments . we have derived the variation of radial velocity of each star for which we have at least two velocity measurements separated by an elapsed time of at least 100 days . these stars represent a subsample of 1044 among 1420 targets . the variation is defined as the difference between the maximum and minimum velocities , as reported in the frame of the sophie spectrograph . its distribution is displayed in figure [ chemin : fig2 ] . a candidate is considered as a reference star for the rvs calibration when its radial velocity does not vary by more than an adopted threshold of 300 m s@xmath1 . such a threshold has been defined to satisfy the condition that the variation of the rv of a candidate must be well smaller than the expected rvs accuracy ( 1 km s@xmath1 at best for the brightest stars ) . as a result , we find @xmath12 of the 1044 stars exhibiting a variation larger than 300 m s@xmath1 , as derived from available measurements performed to date . those variable stars will have to be rejected from the list of standard stars . note that about 75% of the 1044 stars have very stable rv , at a level of variation smaller than 100 m s@xmath1 . stability threshold.,width=377 ] observations of asteroids are very important for the radial velocity calibration . indeed they will be used to determine the zero - points of the rvs measured with sophie , coralie and narval ( as well as the gaia - rvs zero - point ) . those goals will be achieved by comparing the spectroscopic rvs of asteroids from ground - based measurements with theoretical kinematical rvs from celestial mechanics . the theoretical rvs are provided by imcce and are known with an accuracy better than 1 m s@xmath1 . about 280 measurements of 90 asteroids have been done so far . as an illustration , figure [ chemin : fig3 ] ( left - hand panel ) displays the residual velocity ( observed minus computed rvs ) of asteroids observed by the sophie instrument as a function of the observed rvs . the average residual of asteroids observed with sophie is 30 m s@xmath1 and the scatter is 38 m s@xmath1 . in figure [ chemin : fig3 ] ( right - hand panel ) we also show the variation of the residual rvs with time . it nicely shows how stable the rvs are as a function of time . the residual rvs are relatively constant within the quoted errors . the error - bars represent the dispersion of all measurements performed at each observational run . their amplitude is mainly related to the conditions of observations that differ from one session to another ( in particular the moonlight contamination ) . though significant ( between 10 and 50 m s@xmath1 ) those error - bars remain smaller than our target stability criterion of 300 m s@xmath1 , which will enable us to determine correctly the rv zero point of each instrument . . a horizontal dashed line represents the mean residual velocity of 30 m s@xmath1 . * right panel : * comparison of residual velocities of asteroids for sophie , coralie and narval as a function of time for the various observing runs.,title="fig:",width=302 ] . a horizontal dashed line represents the mean residual velocity of 30 m s@xmath1 . * right panel : * comparison of residual velocities of asteroids for sophie , coralie and narval as a function of time for the various observing runs.,title="fig:",width=302 ] note also we have verified that intrinsic properties of asteroids ( e.g. their size , shape , rotation velocity , albedo , etc ... ) have negligible systematic impacts on the determination of rvs zero points for the spectrographs . from now , observations of asteroids shall be performed with reduced moonlight contamination . crifo f. , jasniewicz g. , soubiran c. , et al . , 2009 , sf2a 2009 conference proceedings , 267 crifo f. , jasniewicz g. , soubiran c. , et al . , 2010 , a&a , 524 , a10 wilkinson m. , et al . , 2005 , mnras , 359 , 1306
|
the radial velocity spectrometer ( rvs ) on board of gaia will perform a large spectroscopic survey to determine the radial velocities of some @xmath0 stars .
we present the status of ground - based observations of a sample of 1420 candidate standard stars designed to calibrate the rvs .
each candidate star has to be observed several times before gaia launch ( and at least once during the mission ) to ensure that its radial velocity remains stable during the whole mission .
observations are performed with the high - resolution spectrographs sophie , narval and coralie , completed with archival data of the elodie and harps instruments .
the analysis shows that about 7% of the current catalogue exhibits variations larger than the adopted threshold of 300 m s@xmath1 .
consequently , those stars should be rejected as reference targets , due to the expected accuracy of the gaia rvs .
emphasis is also put here on our observations of bright asteroids to calibrate the ground - based velocities by a direct comparison with celestial mechanics .
it is shown that the radial velocity zero points of sophie , narval and coralie are consistent with each other , within the uncertainties . despite some scatter ,
their temporal variations remain small with respect to our adopted stability criterion .
galaxy : kinematics and dynamics galaxy : structure stars : kinematics and dynamics minor planets , asteroids : general surveys techniques : radial velocities
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
operational methods , developed within the context of the fractional derivative formalism @xcite , have opened new possibilities in the application of calculus . even classical problems , with well - known solutions , may acquire a different flavor if viewed from such a perspective . this , if properly pursued , may allow for further progress , disclosing new avenues of study and generalizations . as it is well known , the operation of integration is the inverse of that of derivation . such a statement by itself does not enable any means to establish `` practical '' rules to handle integrals and derivatives on the same footing . an almost natural environment for this specific assertion are the technicalities associated with the formalism of _ real order _ derivatives ( i.e. not necessarily positive ) , in which the distinction between integrals and derivatives becomes superfluous . the use of the real order formalism offers new computational and conceptual tools allowing e.g. the extension of the concept of integration by parts . within such an approach the integral of a function can be written in terms of the infinite sum @xmath0 where @xmath1 denotes the @xmath2th derivative of the integrand . an elementary way to verify eq . is to integrate by parts its left hand side , written down as @xmath3 , with @xmath4 . first two integrations furnish @xmath5 from which , after infinite number of steps , the general form of eq . follows . the equivalent proof is based on rewriting eq . ( [ eq : eq.1 ] ) as @xmath6 , \\ { _ { a}\hat{d}_{x}^{-1}}[h(x ) ] & = \int_a^x h(\xi ) d\xi , \quad g(x ) = 1 , \end{split}\end{aligned}\ ] ] where @xmath7 denotes the negative derivative operator @xcite . the natural extension of eq . defines the operator @xmath8 _ via _ @xmath9 = { _ { a}d_{x}^{-1}}\left[{_{a}d_{x}^{-1 } } [ h(x)]\right].$ ] the previous definition of the integral of the product of two functions suggests the use of the following , appropriately generalized , leibniz formula : @xmath10= \sum_{s=0}^{\infty } \binom{-1}{s } { _ { 0}\hat{d}_{x}^{-s-1}}[g(x ) ] f^{(s)}(x).\ ] ] the superscript denotes the order of the derivative , be it negative or positive . the identity in eq . ( [ eq : eq.1 ] ) follows from eq . ( [ eq : eq.3 ] ) for @xmath11 , for which @xmath12 = \frac{x^{s+1}}{(s+1)!},\ ] ] and @xmath13 . the interesting property of aforementioned relations is that they allow the evaluation of the primitive of a function in terms of an automatic procedure , analogous to that used in the calculation of the derivatives of a function . at the same time it marks the conceptual , even though not formal , difference between the two operations . the evaluation of the primitive of a function , using the generalized leibniz rule , gives rise to a computational procedure involving , most of the times , an infinite number of steps . ( [ eq : eq.3 ] ) becomes a truly practical tool if e.g. the function @xmath14 has special properties under the operation of derivation , like being cyclic or vanishing after a number of steps . let us combine the above formalism with the properties of the two - variable hermite - kamp de frit polynomials @xcite : @xmath15 where @xmath16 are the conventional hermite polynomials . the @xmath17 of eq . satisfy the following relations : @xmath18 using eqs . ( [ eq : eq.1 ] ) , ( [ eq : eq . 6 ] ) and , we get @xmath19 and @xmath20 observe that eqs . and imply that eqs . , , , and contain finite sums . furthermore , by taking into account the identity @xcite @xmath21 we find @xmath22 or @xmath23 eq . is a representation of known integrals ( compare eq . ( 2.33.3 ) for @xmath24 in @xcite ) in terms of series involving the hermite polynomials . these are just a few examples that underline the versatility of this method which merges new and old concepts . in the forthcoming section we will perform a step forward in this direction by combining this formalism with others approaches of the umbral and operational nature . in this way we will reveal some potentialities of this noticeable computational tool , and demonstrate that they are amenable for further implementations . in a number of previous papers @xcite it has been established that the umbral image of the bessel function is the gaussian . indeed , if we define the shift operator @xmath25 satisfing @xmath26 and @xmath27 then the relevant series expansion yields @xmath28 it is important to emphasize that the core of the previous formalism can be traced back to the work of mikusiski @xcite . the @xmath29th order bessel functions are , in the same spirit , defined as @xmath30 such a restyling allows noticeable simplifications in the theory of bessel functions themselves @xcite . other practical outcome concerns the handling of the associated integrals and of many other technicalities related to the ramanujan master theorem ( rmt ) @xcite . furthermore , the same formalism provides the possibility of recovering a large body of the properties of bessel functions and of other special functions as well , using genuine algebraic tools @xcite . without entering into details of the applications of the rmt we note that , for the purposes of this paper , it can be linked to a kind of general rule , which we call _ criterion of permanence _ , stated as it follows : _ if an umbral correspondence is established between two different functions , such a correspondence can be extended to other operations , including derivatives and integrals . _ accordingly , since the umbral correspondence between gaussian and bessel functions holds , we can use the criterion of permanence for an appropriate `` translation '' of the well - known identities of the gaussian integrals into an umbral language . in fact , using @xmath31 , @xmath32 , we get @xmath33 the extension of the criterion of permanence , under umbral correspondence , to the properties of the gamma function yields the identities @xcite @xmath34 see eq . ( 2.12.2.2 ) of @xcite , and @xmath35 other definite integrals can be obtained by a judicious application of the same principle . for example , @xmath36 furthermore , the use of eq . for @xmath37 enables us to obtain the @xmath29th derivative of the bessel function @xcite @xmath38 consequently , substituting eq . ( [ eq : eq . 20 ] ) into eq . ( [ eq : eq.1 ] ) , we get @xmath39.\ ] ] eq . is the third alternative to the formulas ( 1.10.1.1 ) and ( 1.10.1.2 ) in @xcite . likewise is the second alternative to the formula ( 1.8.1.14 ) in @xcite . the extension of such an approach to the theory of hankel transform @xcite is particularly illuminating . limiting ourselves to the @xmath40th order hankel transform ( denoted by @xmath41 ) we set @xmath42 \equiv \int_{0}^{\infty } x f(x ) j_{0}(x y ) dx = \int_{0}^{\infty } x f(x ) e^{-c_{z}(xy/2)^{2 } } dx\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0},\ ] ] and see that the hankel transform of the function @xmath43 , reduces to the integral evaluated in eq . : @xmath44 & = \int_{0}^{\infty } e^{-(1+y^{2}c_{z}/4)x^{2 } } dx\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = \frac{\sqrt{\pi}/2}{\sqrt{1 + y^{2}c_{z}/4}}\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0}.\end{aligned}\ ] ] other integral transforms can be framed within the same context . the well - known identities ( see formula ( 2.12.18.7 ) of @xcite ) @xmath45 can be easily restated as it follows : @xmath46 by noting that @xmath47 with @xmath48 being the @xmath40-th order tricomi function , satisfying the identity @xcite @xmath49 where @xmath50 we find @xmath51 compare eq . ( 2.12.18.13 ) of @xcite . by a straightforward application of this method we also obtain @xmath52 which is eq . ( 2.12.34.1 ) of @xcite . we hope to have demonstrated by now the far - reaching potentialities of the method . further examples showing its reliability will be discussed in the forthcoming sections . in the previous sections we have introduced a technique of the umbral type , which , roughly speaking , consists in the formal `` downgrading '' of bessel functions , namely higher order transcendental functions , to the elementary gaussian . a quite natural next step forward , in this process of reduction , is downgrading of the gaussian to a rational function , according to the formal prescription : @xmath53 that gives @xmath54 which represents the umbral image of the error function @xmath55 . the use of the rational image of the gaussian provided by eq . ( [ eq : eq . 30 ] ) , allows us to introduce the new functions : @xmath56 \quad \text{and } \quad sn_{\frac{1}{2}}(x ) = -im\left[\varepsilon_{\frac{1}{2}}\left(ix\right)\right],\ ] ] where @xmath57 it is evident that @xmath58 where @xmath59 is the imaginary error function . in this manner we have introduced the gaussian @xmath60 and its complementary function @xmath61 , whose behavior as functions of @xmath62 is shown in fig . [ fig1 ] . gaussian function @xmath60 ( red continuous line ) and its complement @xmath61 ( blue dashed line ) . ] @xmath60 and @xmath61 can be viewed as trigonometriclike functions and this can be strengthened by the formal identities , which follow from eqs . and : @xmath63 whose successive derivatives define new two - parameter functions : @xmath64 in general we can write @xmath65 where @xmath66 the second equality in eq . can be obtained using the properties of the gamma function and its left hand side is equal to @xmath67 , where we have used the kummer relation eq . ( 7.11.1.2 ) of @xcite for the confluent hypergeometric function @xmath68 . then , the use of ( 7.11.1.19 ) of @xcite permits one to rewrite it as @xmath69 which is precisely the right hand side of eq . . examples of the behavior of the previous functions for different values of the indices are reported in figs . 2 . ( blue dashed ) and @xmath70 ( red continuous ) for @xmath71 in fig . [ fig2]a and @xmath72 in fig . [ fig2]b.,title="fig : " ] ( blue dashed ) and @xmath70 ( red continuous ) for @xmath71 in fig . [ fig2]a and @xmath72 in fig . [ fig2]b.,title="fig : " ] the theory of integral transforms is one of the pillars of operational calculus @xcite . the heaviside operational calculus @xcite received its rigorous ( at least for mathematicians ) foundation within the context of laplace transform theory and the already quoted mikusinski operational theory relied on the formalism of convolution quotients @xcite . in addition , many integral transforms have been shown to be expressible in terms of exponential operators , as it is the case of the fractional fourier and airy transforms @xcite . in this section we make further steps in both directions . when dealing with the borel transform ( bt ) @xcite we will develop , using the formalism of exponential operators , new analytical tools to reformulate the relevant findings . we remind that the bt of a function @xmath73 is defined by the integral @xcite @xmath74 \equiv f_{b}(x ) = \int_{0}^{\infty } e^{-t}f(t x ) dt,\ ] ] already exploited tacitly in eqs . and . such a transform plays a significant role in the treating of the series resummation in the quantum field theory @xcite . in this paper we will consider its properties through the use of a different point of view , which will allow a fairly natural link to the previously discussed umbral methods . the use of the identity @xcite @xmath75 allows one to write @xmath76 , \quad \text{where}\quad \hat{b}^{(1 ) } = \int_{0}^{\infty}e^{-t}t^{x\partial_{x}}dt = \gamma(1 + x\partial_{x}),\ ] ] which is the operator form of the borel transform . the above formula also appears in connection with the inversion problem of the laplace transform , see @xcite , theorem 2.14 , eq . ( 4.74 ) . according to our procedure it follows that the borel transform of the @xmath40th order tricomi function of eq . is provided by @xmath77 \nonumber \\ & = \sum_{r=0}^{\infty } \gamma(1 + r ) \frac{(-x)^{r}}{(r!)^{2 } } \nonumber \\ & = e^{-x}.\end{aligned}\ ] ] the successive application of the borel operator to the tricomi function yields the identity @xmath78 \nonumber \\ & = \hat{b}^{(1 ) } e^{-x } \nonumber \\ & = \sum_{r=0}^{\infty } \gamma(1 + r ) \frac{(-x)^{r}}{r ! } \nonumber \\ & = \frac{1}{1+x } , \quad \text{for}\quad |x|<1,\end{aligned}\ ] ] while the further application of @xmath79 yields a divergent series , namely @xmath80 the pattern behind the successive applications of the borel operator to the tricomi function @xmath48 is the already mentioned `` downgrading '' procedure , which `` reduces '' a higher transcendental function to an elementary function . up to now we have interchanged the borel operators and series summation without taking too much caution . in the case of eq . ( [ eq : eq . 41 ] ) such a procedure is fully justified , whereas in applying to eq . ( [ eq : eq . 42 ] ) the method is limited to the convergence region . in the case of eq . ( [ eq : eq . 43 ] ) it is not justified since it gives rise to a divergent series . in the following we will adopt some flexibility in handling these problems and include in our treatment also the case of divergent series . in the previous exposition the repeated application of bt has been associated with the borel operator raised to some integer power . now , we explore the possibility of defining the _ fractional _ bt . for that purpose we introduce the operator @xmath81 which will be referred to as the borel operator of index @xmath82 . for example , @xmath83 applied to @xmath84 gives @xmath85 & = \gamma\left(1 + \frac{1}{2}x\partial_{x}\right)\sum_{r=0}^{\infty}\frac{(-1)^{r}}{(r!)^{2}}\left(\frac{x}{2}\right)^{2r } \nonumber \\ & = \sum_{r=0}^{\infty}\frac{(-1)^{r}}{r!}\left(\frac{x}{2}\right)^{2r } \nonumber \\ & = e^{-(x/2)^{2}}.\end{aligned}\ ] ] assuming that the operator @xmath86 exists and satisfies @xmath87 , and using eq . , we get @xmath88 a more rigorous definition of the inverse operator @xmath86 may be achieved through the use of the hankel contour integral @xcite , namely @xmath89 which can be extended to write @xmath90 * proposition 1 . * _ given the function @xmath14 having the integral @xmath91 , then @xmath92 dx = k \gamma(1 - \alpha ) , \quad |\alpha| < 1,\ ] ] and @xmath93 dx = \frac{k}{\gamma(1 - \alpha ) } , \quad |\alpha| < 1.\ ] ] _ the proof of eq . is fairly straightforward ; applying the previous definitions eq . we find @xmath94 dx & = \int_{-\infty}^{\infty}\left(\int_{0}^{\infty } e^{-t } f(t^{\alpha}x ) dt \right)dx \nonumber \\ & = \int_{0}^{\infty } e^{-t } \left(\int_{-\infty}^{\infty } f(t^{\alpha}x ) dx\right ) dt \nonumber \\ & = \int_{0}^{\infty } e^{-t } t^{-\alpha } dt \int_{-\infty}^{\infty } f(\sigma ) d\sigma \nonumber\\ & = k\gamma(1 - \alpha).\end{aligned}\ ] ] the same procedure can be carried out for the inverse borel transform . we shall present now how the above formalism can be employed for explicit derivation of generating functions . we use the iconic example of @xmath29th order tricomi function as a benchmark . first note that from eq . it follows for all @xmath29 ( compare eq . ( 2.12.9.3 ) of @xcite ) that @xmath95 and consequently @xmath96 if we assume that the summation and integration may be interchanged and if we set @xmath97 with the unknown function @xmath98 , then eq . can be rewritten in the form @xmath99 according to eq . ( [ eq : eq . 41 ] ) and eq . ( 5.7.6.1 ) of @xcite , eq . allows the conclusion @xmath100 a slight variation of the previous arguments applies to the standard bessel functions as well . the use of eq . ( 2.12.9.3 ) of @xcite for @xmath101 and @xmath102 , i.e. @xmath103 yields @xmath104 regarding again @xmath105 as an unknown function , we obtain from eqs . ( [ eq : eq . 60 ] ) and ( [ eq : eq . 59 ] ) @xmath106 which can easily be translated into the well - known generating function of @xmath107 , see eq . ( 5.7.6.1 ) of @xcite : @xmath108 further comments on these last points will be provided in the forthcoming sections . the application of borel transform techniques is a frequently used tool in analysis and in applied science @xcite . its use in the treatment of perturbative series in quantum field theory is tricky albeit effective and proceeds as follows : the inverse bt is used to accelerate the convergence of the perturbative expansion ; the function obtained is then bt transformed to recover the sum of the series . to better clarify the relevance of borel resummation methods to the topics of the present article we critically review what we did in the previous section . the successive application of the bt to a function with non - zero radius of convergence has led to a function whose series coefficients grow factorially with the order of expansion . its range of convergence is therefore vanishing . in quantum field theory one is faced with the opposite problem , namely that of recovering a function with non - zero radius of convergence starting from a divergent series . the problem is `` cured '' by dividing each term in the expansion by a factor @xmath109 . this is called a borel sum ; if it can be summed and analytically continued over the whole real axis , then the initial expansion is called borel summable @xcite . for @xmath110 , let us calculate @xmath111 : @xmath112 using the umbral method this can be rewritten as @xmath113 as an example we calculate the action of @xmath114 on the gaussian . that gives @xmath115 which is the inversion of eq . . the laguerre polynomials @xcite can be framed within the same context . indeed , it is easy to obtain eq . ( 7.414.6 ) of @xcite : @xmath116 where @xmath117 are two - variable laguerre polynomials , related to the standard laguerre polynomials by @xmath118 @xcite . a consequence of eqs . and is @xmath119 accordingly , the ordinary generating function of the laguerre polynomials is the inverse bt of the geometric series . namely , from eq . we get @xmath120 \nonumber \\ & = \frac{1}{(1 - y \xi ) + \xi c_{z } x}\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = \frac{1}{1 - y \xi } \exp\left(- \frac{x\xi}{1-y\xi}\right).\end{aligned}\ ] ] the last equality in eq . follows from eq . . if the borel operator acts on the @xmath121 variable we obtain @xmath122 the function @xmath123 denotes the bessel truncated polynomials @xcite . the use of eq . ( 5.11.1.5 ) of @xcite ( see also @xcite ) @xmath124 yields the generating function for the bessel truncated polynomials @xmath125 = \frac{c_{0}(x \xi)}{1 - y\xi}.\ ] ] we employ previous results to obtain the ordinary generating functions of _ lacunary _ laguerre polynomials . ( let us stress that the following example is included solely for illustrative purposes , as its result is immediately obtained from eq . . ) . according to our formalism we have @xmath126 \nonumber \\ & = \frac{1}{2 } \left[(1 - \sqrt{t } y)^{-1 } \left(1 + \frac{c_{z}\sqrt{\xi } x}{1 - \sqrt{t}y}\right)^{-1 } + ( 1 + \sqrt{t } y)^{-1 } \left(1 - \frac{c_{z}\sqrt{t } x}{1 + \sqrt{t}y}\right)^{-1}\right ] \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = \frac{1}{2}\left[\frac{\exp\left(-\frac{\sqrt{t } x}{1 - \sqrt{t}y}\right)}{1 - \sqrt{t}y } + \frac{\exp\left(\frac{\sqrt{t } x}{1 + \sqrt{t}y}\right)}{1 + \sqrt{t}y}\right].\end{aligned}\ ] ] the exponential generating function for lacunary laguerre polynomials @xmath127 through the umbral representation gives @xmath128 which reproduces eq . ( 40 ) in @xcite . we shall present now the derivation of eq . using the bt approach . the use of eq . gives @xmath129 inverting eq . and using eq . for @xmath130 we find @xmath131 which reproduces eq . . in the concluding remarks we will further comment on the lacunary laguerre generating functions and on the relevant link with previous research @xcite . to make further progress in our exposition we introduce the integral transform , known in the literature as the borel - leroy ( b - l ) transform @xcite , @xmath132 thus , the associated two - parameter differential operator can be written as @xmath133 the relevant action on the @xmath134th order tricomi function yields @xmath135 where @xmath136 which converges for @xmath137 . the inverse b - l transform of the exponential yields the bessel - wright function @xmath138 @xcite , namely @xmath139 going back to eq . ( [ eq : eq . 59 ] ) we also find that @xmath140 whose the right hand side can serve for the direct computation of the generating function in eq . ( [ eq : eq . 60 ] ) . as a further generalization we discuss the @xmath141-borel - leroy transform defined by @xmath142 which , upon the use of the euler @xmath143 function @xmath144 @xcite , can be transformed into the differential form @xmath145 it is interesting to note that the previous operator when acting on the exponential transforms it into a mittag - leffler function @xcite , namely @xmath146 ( remark that eq . is of operational nature and any modification of the argument of the exponential should be preceded by a detailed evaluation of eq . ) . generalizing * proposition 1 * with @xmath14 such that @xmath91 to the @xmath141-borel - leroy transform , we get @xmath147 where we used eq . and eq . ( 8.380.1 ) of @xcite . . for @xmath148 gives @xmath149 , \left[\frac{\alpha + \beta}{2 } , \frac{\alpha+\beta+1}{2}\right ] ; -x^2\right ) dx\\[0.3\baselineskip ] & = \sqrt{\pi } \frac{\gamma(\alpha-1)\gamma(\beta)}{\gamma(\alpha-1+\beta)},\end{aligned}\ ] ] see eq . ( 2.22.1.1 ) in @xcite , where @xmath150 is the generalized hypergeometric function . these are just few examples of the various possibilities offered by the present formalism further applications will be discussed in the forthcoming section . in the previous sections we have provided a comprehensive analysis of different formulations of the operational techniques usually adopted to treat problems associated with the handling of the properties of special functions and polynomials . we have seen that the use of the bt is a powerful unifying tool yielding the appropriate environment for a more rigorous treatment of the umbral techniques . in this paragraph we collect examples connected with orthogonal polynomials . the derivation of the ordinary generating function of @xmath151-_lacunary _ laguerre polynomials ( @xmath152 ) in the language of bt goes as follows @xmath153^{-1}\nonumber \\ & = \frac{1}{p } \sum_{k=0}^{p-1 } ( 1 - t^{1/p } e^{2\pi i k / p } y)^{-1 } \left(1 + \frac{t^{1/p } e^{2\pi i k / p } c_{z } x}{1 - t^{1/p } e^{2\pi i k / p } y}\right)^{-1 } \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = \frac{1}{p } \sum_{k=0}^{p-1 } ( 1 - t^{1/p } e^{2\pi i k / p } y)^{-1 } \sum_{n=0}^{\infty } \left(-\frac{t^{1/p } e^{2\pi i k / p } x}{1 - t^{1/p } e^{2\pi i k / p } y}\right)^{\ ! n } c_{z}^n \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = \frac{1}{p } \sum_{k=0}^{p-1 } \frac{\exp\left(-\frac{t^{1/p } e^{2\pi i k / p } x}{1 - t^{1/p } e^{2\pi i k / p } y}\right)}{1 - t^{1/p } e^{2\pi i k / p } y}.\end{aligned}\ ] ] the generalization to the case @xmath154 , @xmath155 is straightforward . for completeness it is worth to touch on the possibility of treating the hermite polynomials in a fashion borrowed from the treatment of laguerre s . for this purpose we need another choice of the function @xmath156 in eq . , namely @xmath157 . then the two - variable hermite polynomials @xmath158 can be defined as @xmath159 the right hand side of eq . gives @xmath160 \ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = \sum_{k=0}^n \binom{n}{k } x^{n - k } \frac{(2\sqrt{y})^k}{\sqrt{\pi } } \gamma\left(\frac{1+k}{2}\right ) \left|\cos\left(k\frac{\pi}{2}\right)\right| \nonumber \\ & = \sum_{r=0}^{\lfloor n/2\rfloor } \binom{n}{2r } x^{n-2r } \frac{4^r y^r}{\sqrt{\pi } } \gamma\left(\frac{1}{2 } + r\right ) \nonumber \\ & = n ! \sum_{r=0}^{\lfloor n/2\rfloor } \frac{x^{n-2r } y^r}{(n-2r ) ! } \frac{4^r \gamma(\frac{1}{2 } + r)}{\sqrt{\pi } ( 2r ) ! } \nonumber \\ & = n ! \sum_{r=0}^{\lfloor n/2\rfloor } \frac{x^{n-2r } y^r}{r ! ( n-2r)!},\end{aligned}\ ] ] compare eq . . the operator formula eq . is an effective tool to quickly derive a number of known and less known summation formulas involving the hermite polynomials . we start with @xmath161 \ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = \sum_{r=0}^{\infty } \frac{(4yx^{2})^r}{\sqrt{\pi } } \frac{\gamma(\ulamek{1}{2 } + r)}{(2r ) ! } \nonumber \\ & = \sum_{r=0}^{\infty } \frac{(y x^2)^r}{r ! } = e^{y x^2 } , \end{aligned}\ ] ] and @xmath162 \ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = \frac{1}{\sqrt{\pi } } \int_{-\infty}^{\infty } e^{-\xi^2}e^{y ( 2\xi\sqrt{x})^2 } d\xi \nonumber\\ & = \frac{1}{\sqrt{1 - 4yx } } , \quad \text{for } \quad |x| < \frac{1}{4 |y|}.\end{aligned}\ ] ] combining the previous results one can show @xmath163 \ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = e^{t(x + c_z)^2 } \ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = \frac{1}{\sqrt{1 - 4yt}}e^{\frac{x^{2}t}{1 - 4yt } } , \quad \text{for } \quad |t| < ( 4|y|)^{-1},\end{aligned}\ ] ] see eq . ( 5.12.1.4 ) of @xcite . an alternative procedure has been put forward by gessel and jayawant @xcite , who discussed a _ triple lacunary _ generating function for hermite polynomials using umbral and combinatorial techniques . moreover , it is interesting to note that from eqs . ( [ eq : eq . 83 ] ) and we can derive another operational definition of the hermite polynomials : @xmath164 which , on account of @xmath165 yields the well - known operational identity @xmath166 also used as a primary definition of @xmath17 @xcite . according to the previous identities the derivation of the classical mehler formula can be given rapidly . using eqs . and we get @xmath167^n}{n ! } h_{n}(u , v)\ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = e^{t x u + v ( t x)^2 } e^{vt^2 c_z^2 + ( 2v x t^2 + tu ) c_z}\ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = e^{-u^2/(4 v)}e^{v t^2 [ x + u/(2 v t ) + c_z]^2 } \ , \ulamek{(2\sqrt{y})^z}{\sqrt{\pi } } \gamma(\ulamek{1+z}{2 } ) |\cos(z\ulamek{\pi}{2})|\big\vert_{z=0 } \nonumber \\ & = \frac{e^{y(ut)^2}}{\sqrt{1 - 4yvt^2 } } \exp\left(\frac{u t x + v t^2 x^2}{1 - 4yvt^2}\right ) , \quad 1 - 4yvt^2>0.\end{aligned}\ ] ] we conclude this paper with a further remark , concerning the possibility of extending the method to hybrid bilateral generating functions , involving products of laguerre and hermite polynomials . this problem has already been discussed in @xcite using the procedure exploiting the gauss transform representation of hermite polynomials . the derivation we propose here is significantly simpler : @xmath168^{n}}{n ! } h_{n}(u , v)\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = e^{t(y - c_{z}x)u + [ t(y - c_{z}x)]^{2 } v}\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = e^{(ty)u + ( ty)^{2}v } e^{- t(xu + 2tyxv)c_{z } + ( tx)^{2 } v c_{z}^{2}}\ , \frac{1}{\gamma(1+z)}\big\vert_{z=0 } \nonumber \\ & = e^{(ty)u + ( ty)^{2}v}\,\sum_{r=0}^{\infty } \frac{x^r}{(r!)^2 } h_r(-tu-2yvt^2 , vt^2).\end{aligned}\ ] ] most of the topics treated in this paper may deserve a deeper and more thorough treatment . however , we believe that in spite of the heterogeneity of the argument we touched upon many implications offered by the formalism . we hope that they may provide the hints for further speculations . very recently v. strehl @xcite commented on the identities of @xcite and stressed their relevance with the remark : `` all identities are very combinatorial , and combinatorics can help to systematize and extend them considerably '' . furthermore , the following question has been addressed and answered in @xcite : `` what the lacunary laguerre series really count ? '' . evidently it is very tempting and challenging to reinterpret ensemble of the above results from the combinatorial perspective . the authors express their sincere appreciation to dr . d. babusci for interesting and enlightening discussions on the topics treated in this paper . it is also a pleasure to recognize the interest and the encouragement of prof . v. strehl . k. g. , a. h. and k. a. p. were supported by the pan - cnrs program for the french - polish collaboration . moreover , k. g. thanks for the support from mnisw , warsaw ( poland ) , under `` iuventus plus 2015 - 2016 '' , program no ip2014 013073 . g. dattoli , p. l. ottaviani , a. torre , and l. vzquez , _ evolution operator equations : integration with algebraic and finite - difference methods . applications to physical problems in classical and quantum mechanics and quantum field theory _ , riv . 20 * , 1 ( 1997 ) . j. mikusiski , _ sur les fondements du calcul operatoire _ , studia math . * 11 * , 41 ( 1949 ) ; `` operational calculus '' , ( pergamon press , london , 1959 ) ; s. okamoto , _ a simplified derivation of mikusiski s operational calculus _ , proc . japan acad . a * 55*(1 ) , 1 ( 1979 ) . t. amdeberhan and v. h. moll , _ a formula for a quartic integral : a survey of old proofs and some new ones _ , ramanujan j. * 18 * , 91 ( 2009 ) ; t. amdeberhan , o. espinosa , i. gonzalez , m. harrison , v. h. moll , and a. straub , _ ramanujan s master theorem _ , ramanujan j. * 29 * , 103 ( 2012 ) . a. a. zakharov , _ borel summation method _ , in `` encyclopedia of mathematics '' , m. hazewinkel ( ed . ) , ( springer , berlin , 2001 ) ; s. weinberg , `` the quantum theory of fields . ii '' , ( cambridge university press , new york , 2005 ) .
|
integro - differential methods , currently exploited in calculus , provide an inexhaustible source of tools to be applied to a wide class of problems , involving the theory of special functions and other subjects .
the use of integral transforms of the borel type and the associated formalism is shown to be a very effective mean , constituting a solid bridge between umbral and operational methods .
we merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
sudan and guruswami s list decoding of reed - solomon codes @xcite has developed into algebraic soft - decision decoding by koetter and vardy @xcite . as reed - solomon codes are widely used in coding applications , algebraic soft - decision decoding is regarded as one of the most important developments for reed - solomon codes . hence there have been many subsequent works to make the decoding method efficient and practical @xcite . engineers have proposed fast electronic circuits implementing the algebraic soft - decision decoder @xcite . one may say that the algebraic soft - decision decoding of reed - solomon codes is now in a mature state for deployment in applications @xcite . reed - solomon codes are the simplest algebraic geometry codes @xcite . therefore it is natural that the list decoding of reed - solomon codes was soon extended to algebraic geometry codes by shokrollahi and wasserman @xcite and guruswami and sudan @xcite . however , it seems that no algebraic geometry codes other than reed - solomon codes have been considered for algebraic soft - decision decoding . one reason for this unbalanced situation is presumably that the complexity of an algebraic soft - decision decoder for algebraic geometry codes would be prohibitively huge as the complexity for reed - solomon codes is already very large . however , algebraic geometry codes have the advantage that they are longer than reed - solomon codes over the alphabet of the same size , promising better performance . we may also expect that once we have an explicit formulation of algebraic soft - decision decoding for algebraic geometry codes , some clever ways to reduce the complexity to a practical level may be found , as has happened for reed - solomon codes @xcite . in this work , we present an algebraic soft - decision decoder for hermitian codes . hermitian codes are one of the best studied algebraic geometry codes , and they are often regarded as the first candidate among algebraic geometry codes that could compete with reed - solomon codes . to formulate an algebraic soft - decision decoder for hermitian codes , we basically follow the path set out by koetter and vardy for reed - solomon codes . thus there are three main steps of the decoding : the multiplicity assignment step , the interpolation step , and the root - finding step . for the multiplicity assignment step and the root - finding step , we may use algorithms in @xcite and @xcite , respectively . here we focus on the interpolation step , the goal of which is to construct the @xmath0-polynomial whose roots give the candidate codewords . as for mathematical contents , this work is an extension of our previous @xcite and @xcite . the core contribution of the present work is an algorithm constructing a set of generators of a certain module from which we extract the @xmath0-polynomial using the grbner conversion algorithm given in @xcite . in section 2 , we review the definitions of basic concepts and the properties of hermitian curves and codes . we refer to @xcite and @xcite for the basic theory of algebraic curves and algebraic geometry codes , and @xcite and @xcite for grbner bases and commutative algebra . in section 3 , we formulate the algebraic soft - decision decoding of hermitian codes . we present our interpolation algorithm in section 4 and a complexity analysis of the decoding algorithm in section 5 . in section 6 , we provide some simulation results of the algebraic soft - decision decoder . as this work is an extension of @xcite , we omitted some proofs that can be found in that work but allowed some similar materials included here for exposition purposes . let @xmath1 be a prime power , and let @xmath2 denote a finite field with @xmath3 elements . the hermitian curve @xmath4 is the affine plane curve defined by the absolutely irreducible polynomial @xmath5 over @xmath2 . the coordinate ring of @xmath6 is the integral domain @xmath7/{\langle{y^q+y - x^{q+1}}\rangle}={\mathbb{f}}[x , y],\ ] ] with @xmath8 and @xmath9 denoting the residue classes of @xmath10 and @xmath11 , respectively . note that every element of @xmath12 can be written uniquely as a polynomial of @xmath8 and @xmath9 with @xmath9-degree less than @xmath1 , as we have @xmath13 . so @xmath12 is also a free module of rank @xmath1 over @xmath14 $ ] . the function field @xmath15 is the quotient field of @xmath12 . for each @xmath16 , there are exactly @xmath1 elements @xmath17 such that @xmath18 . therefore there are @xmath19 rational points @xmath20 of @xmath6 with @xmath21 , which can be grouped into @xmath3 classes of @xmath1 points with the same @xmath8-coordinates . a rational point @xmath22 of @xmath6 is associated with a maximal ideal @xmath23 , and the local ring @xmath24 of @xmath6 at @xmath22 is the localization of @xmath12 at @xmath25 . for a nonzero @xmath26 , the valuation @xmath27 is the largest integer @xmath28 such that @xmath29 . the projective closure of @xmath6 is a smooth curve with a unique rational point @xmath30 at infinity . the functions @xmath8 and @xmath9 on @xmath6 have poles at @xmath30 of orders @xmath1 and @xmath31 , respectively , that is , @xmath32 and @xmath33 . the genus of @xmath6 is given by @xmath34 . it is well known that the number of rational points of the curve @xmath6 attains the maximum value possible for the genus and the size of the base field . for @xmath35 , the @xmath2-linear space @xmath36 has a basis consisting of @xmath37 for @xmath38 , @xmath39 , and @xmath40 . therefore @xmath41 recall that the hamming space @xmath42 is an @xmath2-linear space with the hamming distance function @xmath43 . for @xmath44 , let @xmath45 . the evaluation map @xmath46 defined by @xmath47 is a linear map over @xmath2 . we now fix a positive integer @xmath48 . the hermitian code @xmath49 is defined to be the image of @xmath50 by the evaluation map . if @xmath51 , then @xmath52 is injective on @xmath50 , and the dimension of @xmath49 is equal to @xmath53 , which is @xmath54 for @xmath55 by the riemann - roch theorem . note also that the minimum distance of @xmath49 is at least @xmath56 . define @xmath57 for @xmath44 . for a vector @xmath58 , define @xmath59 we can easily prove that @xmath60 if @xmath61 , and @xmath62 otherwise . therefore @xmath63 for all @xmath64 . let @xmath65 and @xmath66 . we consider the hermitian curve @xmath6 defined by @xmath67 over @xmath68 . there are @xmath69 rational points on @xmath6 , @xmath70 let @xmath71 . the linear space @xmath72 is spanned by the basis @xmath73 . hermitian code @xmath74 is a linear code over @xmath68 of length @xmath69 and dimension @xmath75 . we use the following generator matrix for encoding @xmath76 note that the positions @xmath77 form an information set of @xmath78 . our message is @xmath79 , which is encoded into the codeword @xmath80 the functions @xmath81 are as follows : @xmath82 we will continue this example throughout . the smooth surface @xmath83 has the coordinate ring @xmath84=r[z]$ ] . the function field @xmath85 is the quotient field of @xmath86 . a rational point @xmath87 of @xmath88 is a pair @xmath89 with @xmath44 and @xmath90 , and is associated with a maximal ideal @xmath91 . the local ring @xmath92 of @xmath88 at @xmath87 is the localization of @xmath86 at @xmath93 . a nonzero function @xmath94 defines a curve on the surface @xmath88 . the multiplicity of @xmath95 at a rational point @xmath87 , denoted @xmath96 , is the largest integer @xmath28 such that @xmath97 . we note the following properties of multiplicity on the surface @xmath88 . let @xmath22 be a rational point of @xmath6 . * if @xmath26 , then @xmath98 for every @xmath90 . * for @xmath26 , @xmath99 if @xmath100 , and @xmath62 otherwise . * for @xmath101 , @xmath102 for every rational point @xmath87 of @xmath88 . suppose that some codeword of @xmath49 was sent through a noisy channel . the output of the channel is some probabilistic information , for each location @xmath103 , of the plausibility of each @xmath90 . the multiplicity assignment step translates the information to a doubly indexed list @xmath104\ ] ] of nonnegative integers , where we regard @xmath105 as assigned to the point @xmath106 . the integer value @xmath105 would be chosen roughly proportional to the plausibility of the symbol @xmath107 according to the channel output . we call @xmath108 the multiplicity matrix . corresponding to @xmath108 , define @xmath109\mid\text{$\operatorname{mult}_{(p_i,\gamma)}(f)\ge m_{i\gamma}$ for $ 1\le i\le n , \gamma\in{\mathbb{f}}$}\}},\ ] ] an ideal of @xmath110 $ ] . we call @xmath111 the interpolation ideal . note that by definition @xmath112 for a vector @xmath58 , the score of @xmath113 with respect to @xmath108 is defined as @xmath114 hence @xmath115 is also the sum of the multiplicities of the points through which the curve @xmath116 passes . the task of the algebraic soft - decision decoder is to find the codeword that has the best score with respect to @xmath108 . this codeword is presumed to be the most likely to have been sent , given the channel output . [ djcqoe ] suppose that the codeword in the previous example is sent through a noisy channel and received data gives rise to the matrix of the plausibilities of symbols @xmath117 which is then translated to the multiplicity matrix @xmath118 where the rows are indexed by @xmath119 from top to bottom . note that neither of the vectors @xmath120 and @xmath121 that have the best score with respect to @xmath108 is a codeword of @xmath74 . [ lemabcd ] let @xmath122 $ ] be the multiplicity matrix . then @xmath123/i_m=\sum_{i=1}^n\sum_{\gamma\in{\mathbb{f}}}\binom{m_{i\gamma}+1}{2}.\ ] ] since @xmath111 is a zero - dimensional ideal , @xmath124/i_m&=\sum_{i=1}^n\sum_{\gamma\in{\mathbb{f}}}\dim_{{\mathbb{f}}}{\mathcal{o}}_{(p_i,\gamma)}/i_m{\mathcal{o}}_{(p_i,\gamma)}\\ & = \sum_{i=1}^n\sum_{\gamma\in{\mathbb{f}}}\dim_{{\mathbb{f}}}\hat{\mathcal{o}}_{(p_i,\gamma)}/i_m\hat{\mathcal{o}}_{(p_i,\gamma ) } , \end{split}\ ] ] where @xmath125 denotes the completion of the local ring . if @xmath126 is a uniformizing parameter of @xmath127 and @xmath128 , then @xmath129 is isomorphic to @xmath130 $ ] . so @xmath131 is isomorphic to @xmath130/(s , t)^{m_{i\gamma}}$ ] . the conclusion follows . let @xmath132 with @xmath133 . then @xmath134/(i_m+{\langle{z-\mu}\rangle})=\operatorname{score}(v).\ ] ] as in the previous proof , @xmath123/(i_m+{\langle{z-\mu}\rangle } ) = \sum_{i=1}^n\sum_{\gamma\in{\mathbb{f}}}\dim_{{\mathbb{f}}}\hat{\mathcal{o}}_{(p_i,\gamma)}/(i_m+{\langle{z-\mu}\rangle})\hat{\mathcal{o}}_{(p_i,\gamma)}.\ ] ] let @xmath126 be a uniformizing parameter of @xmath127 and @xmath128 again . we find that if @xmath135 , then @xmath136 is isomorphic to @xmath130/({\langle{s , t}\rangle}^{m_{i\gamma}}+{\langle{s - tu}\rangle})={\mathbb{f}}[[t]]/{\langle{t^{m_{i\gamma}}}\rangle}$ ] , but collapses to the zero ring otherwise . here @xmath48 is some unit in @xmath129 . the conclusion follows . for @xmath137 $ ] with @xmath138 , the @xmath48-weighted degree of @xmath95 is defined as @xmath139 for @xmath140 $ ] and @xmath141 , denote by @xmath142 the element in @xmath12 that is obtained by substituting @xmath143 with @xmath144 in @xmath95 . observe that if @xmath145 , then @xmath146 . the algebraic soft - decision decoding of hermitian codes rests upon the following [ thmakwe ] suppose @xmath147 is nonzero . if a codeword @xmath148 of @xmath49 with @xmath149 satisfies @xmath150 then @xmath151 . assume that @xmath152 is not zero in @xmath12 . then @xmath153/{\langle{f , z-\mu}\rangle})\\ & \ge\dim_{\mathbb{f}}(r[z]/(i_m+{\langle{z-\mu}\rangle})=\operatorname{score}(c ) . \end{split}\ ] ] for the first equality , see lemma 5 in @xcite . this implies that if @xmath154 , we must have @xmath151 . in the interpolation step , the decoder picks a polynomial @xmath147 . then by proposition [ thmakwe ] , all codewords whose score with respect to @xmath108 is big enough can be obtained from the roots of @xmath95 over @xmath12 . thus the decoder can find among the candidates the codeword that has the best score with respect to @xmath108 . it should be noted that for the best performance of algebraic soft - decision decoding , it is crucial for the decoder to find a polynomial in @xmath111 with the smallest @xmath48-weighted degree . having the same weighted degree , the one with smaller degree in @xmath143 is preferred because this reduces the work of the root - finding step . here the idea of grbner bases is relevant . we call the elements in the set @xmath155 monomials of @xmath110 $ ] . recall that every element of @xmath110 $ ] can be written as a unique linear combination over @xmath2 of monomials of @xmath110 $ ] . note that @xmath156 for two monomials @xmath157 , @xmath158 in @xmath159 , we declare @xmath160 if @xmath161 or @xmath162 when tied . it is easy to verify that @xmath163 is a total order on @xmath159 . notions such as the leading term and the leading coefficient of @xmath140 $ ] are defined in the usual way . for @xmath140 $ ] , the @xmath143-degree of @xmath95 , written @xmath164 , is the degree of @xmath95 as a polynomial in @xmath143 over @xmath12 . now we define the @xmath0-polynomial of @xmath111 as the unique , up to a constant multiple , element in @xmath111 with the smallest leading term with respect to @xmath163 . by the definition , the @xmath0-polynomial is an element of @xmath111 with the smallest @xmath48-weighted degree , and moreover it has the smallest @xmath143-degree among such elements . therefore we may say that the @xmath0-polynomial is an optimal choice for the interpolation step . the last step of algebraic soft - decision decoding is to compute roots of the @xmath0-polynomial over @xmath12 or the function field @xmath15 . only those roots that belong to @xmath50 yield candidate codewords . if the list of the candidate codewords is empty , the decoder may declare decoding failure or resort to hard - decision decoding directly from the channel output . if there are several codewords in the list , then the decoder chooses the codeword that has the best score , and outputs the received message by projecting the codeword on the information set . [ cjops ] the @xmath0-polynomial of @xmath111 @xmath165 is obtained by the interpolation algorithm in the next section . it turns out the @xmath0-polynomial has the factorization @xmath166 therefore a root - finding algorithm will output two roots . the first root @xmath167 gives the codeword @xmath168 whose score is @xmath169 while the second root @xmath170 gives the codeword @xmath171 whose score is @xmath172 . therefore the decoder chooses @xmath173 , and the received message is @xmath174 which is the correct sent message . we will need upper bounds on the @xmath48-weighted degree and the @xmath143-degree of the @xmath0-polynomial of @xmath111 . let @xmath0 denote the @xmath0-polynomial of @xmath111 . [ propdkwd ] if @xmath175 is a finite set of monomials of @xmath110 $ ] such that @xmath176 then there is a set of coefficients @xmath177 such that @xmath178 . lemma [ lemabcd ] implies that monomials in @xmath179 are linearly dependent over @xmath2 in @xmath110/i_m$ ] . on the other hand , they are linearly independent over @xmath2 in @xmath110 $ ] . in a table , we arrange monomials of @xmath110 $ ] such that the monomials in the same column have the same @xmath48-weighted degree and the monomials in the same row have the same @xmath143-degree . let weighted degrees increase from left to right and @xmath143-degrees from bottom to top . note that @xmath180 . so we have the following table @xmath181 the symbol @xmath182 indicates that there is no monomial for the position . the table of monomials of @xmath110 $ ] suggests the following formula . let @xmath183 if @xmath184 is a weierstrass gap at @xmath30 , and @xmath185 otherwise . note that @xmath186 for @xmath187 . the number of monomials with @xmath48-weighted degree @xmath184 is @xmath188 let @xmath189 be the smallest integer such that @xmath190 let @xmath191 . then the @xmath48-weighted degrees and the @xmath143-degrees of monomials up to the @xmath192th monomial are not greater than @xmath189 and @xmath193 , respectively . now proposition [ propdkwd ] implies @xmath194 and @xmath195 . @xmath196 , and @xmath186 for @xmath197 since @xmath198 . so we have @xmath199 for our @xmath108 , @xmath200 . therefore @xmath201 , @xmath202 . hence @xmath203 and @xmath204 . let @xmath193 be a positive integer such that @xmath195 . define @xmath205_l={\{f\in r[z]\mid { \text{$z$-$\deg$}}(f)\le l\}}.\ ] ] note that @xmath110_l$ ] is a free module over @xmath12 of rank @xmath206 with a free basis @xmath207 . define @xmath208_l.\ ] ] clearly @xmath209 is a submodule of @xmath110_l$ ] over @xmath12 . recall that the ring @xmath210 $ ] is in turn a free module over @xmath14 $ ] of rank @xmath1 , with a free basis @xmath211 . so we may view @xmath110_l$ ] as a free module of rank @xmath212 over @xmath14 $ ] with a free basis @xmath213 . the elements of @xmath214_l$ ] will be called monomials of @xmath110_l$ ] . it is clear that the total order @xmath163 is precisely a monomial order on the free module @xmath110_l$ ] over @xmath14 $ ] . we also view @xmath209 as a submodule of the free module @xmath110_l$ ] over @xmath14 $ ] . it is immediate that the @xmath0-polynomial of @xmath111 is also the element of @xmath209 with the smallest leading term with respect to @xmath163 . as a consequence of the definition of grbner bases , @xmath0 occurs as the smallest element in any grbner basis of the module @xmath209 over @xmath14 $ ] with respect to @xmath163 . we begin with let @xmath122 $ ] be a doubly indexed list of nonnegative integers . for each @xmath215 , let @xmath216 , and let @xmath217 $ ] . for each @xmath184 with @xmath218 , let @xmath219 be such that @xmath220 . let @xmath221 $ ] where @xmath222 for @xmath223 and @xmath224 . then as a module over @xmath12 , @xmath225 where @xmath226 and @xmath227 is such that @xmath228 . by the properties ( i ) , ( ii ) , ( iii ) of local multiplicity , it is clear that @xmath229 . to show the reverse inclusion , let @xmath147 . we can write @xmath230 for some @xmath231 $ ] and @xmath232 . let @xmath233 . if @xmath126 is a uniformizing parameter of @xmath127 , then @xmath126 and @xmath234 form a system of parameters of @xmath92 . recall that the completion @xmath235 is isomorphic to the power series ring @xmath236 $ ] . now if @xmath237 , then in @xmath235 , @xmath238 for some unit @xmath48 in @xmath235 . since @xmath239 and @xmath240 are algebraically independent over @xmath2 , we see that @xmath241 . then as this is true for all @xmath44 , it follows that @xmath242 . hence @xmath243 . again by the properties of local multiplicity , @xmath244 . thus we showed the reverse inclusion . recall the multiplicity matrix @xmath122 $ ] . let @xmath245 . initially let @xmath246 and @xmath247 . proceed inductively for @xmath248 . choose @xmath219 such that @xmath249 if @xmath250 . let @xmath251 such that @xmath252 . let @xmath253 now let @xmath254 $ ] and @xmath255 $ ] . observe @xmath256 for all @xmath257 , and therefore @xmath258 $ ] . by induction , we get for @xmath259 , @xmath260 as a module over @xmath12 . here @xmath261 and @xmath262 for @xmath263 . we may view the ideal @xmath226 as a module over @xmath14 $ ] . indeed @xmath264 is a free module of rank @xmath1 over @xmath14 $ ] . thus we obtain the input is an @xmath265 matrix @xmath122 $ ] of nonnegative integers . the output is the generators @xmath266 of @xmath209 as a module over @xmath14 $ ] . repeat steps b1 and b2 for @xmath267 . * let @xmath268 for @xmath44 . let @xmath269 . for each @xmath270 , let @xmath271 be such that @xmath272 . set @xmath273 for @xmath274 , where @xmath275 is a set of generators of @xmath276 as a module over @xmath14 $ ] . when @xmath277 is empty , @xmath278 so @xmath279 . * set @xmath280 and for @xmath281 , set @xmath282 notice that if we compute @xmath283 by the method in the following subsection , @xmath284 has leading term @xmath285 with respect to lex order @xmath286 . we continue from example [ djcqoe ] . we show the first few steps to compute a set of generators of @xmath209 with @xmath202 using algorithm b. initially @xmath287 . then @xmath288 as we compute in example [ jwqdd ] , @xmath289 as a module over @xmath14 $ ] . so we set @xmath290 in step b2 , we compute ( setting @xmath291 arbitrarily ) @xmath292 now the matrix of @xmath105 is @xmath293 going on to @xmath294 , we have @xmath295 then @xmath296 as a module over @xmath14 $ ] . hence @xmath297 now @xmath298 and the matrix of @xmath105 is @xmath299 proceeding this way until @xmath300 , we obtain a set of generators of the module @xmath209 . we arrange the coefficients ( polynomials in @xmath8 ) of the generators in the following matrix @xmath301 where the rows are @xmath302 , @xmath303 , @xmath304 , @xmath305 , , @xmath306 , @xmath307 in this order , and the columns are coefficients of @xmath185 , @xmath9 , @xmath143 , @xmath308 , @xmath309 , @xmath310 , , @xmath311 , @xmath312 in this order . we now tackle the task of computing a set of generators of @xmath264 as a module over @xmath14 $ ] . for this , we switch to a different indexing of the rational points of @xmath6 by grouping the @xmath19 rational points into @xmath3 classes with the same @xmath8-coordinates . thus the rational points are @xmath313 for @xmath314 and @xmath315 . let @xmath316 if @xmath317 is the point @xmath127 . also assume that for each @xmath314 , we have arranged the index @xmath318 such that @xmath319 are put in decreasing order , @xmath320 with the new notations , @xmath321 [ prop6 ] for @xmath322 , suppose that @xmath323 $ ] satisfy @xmath324 for all @xmath314 . define for @xmath325 , @xmath326 then @xmath327 as a module over @xmath14 $ ] . let @xmath328 . then for @xmath314 and @xmath315 , @xmath329 therefore @xmath330 . recall that we may view @xmath12 as a free module of rank @xmath1 over @xmath14 $ ] . let @xmath331 be the submodule of @xmath12 generated by @xmath332 over @xmath14 $ ] . then @xmath333 is isomorphic to @xmath334/{\langle{\prod_{1\le a\le q^2}(x-{\alpha}_a)^{\mu_{a , c}}}\rangle}.\ ] ] therefore @xmath335/{\langle{\prod_{1\le a\le q^2}(x-{\alpha}_a)^{\mu_{a , c}}}\rangle } = \sum_{1\le c\le q}\sum_{1\le a\le q^2}\mu_{a , c}.\ ] ] on the other hand , as @xmath336 by its definition , we have @xmath337 hence @xmath338 . together with @xmath339 , this implies that @xmath340 . [ jwqdd ] we compute generators @xmath341 , @xmath342 of @xmath343 . we arrange the points as @xmath344 so that @xmath319 are in decreasing order , @xmath345 we will see in the next subsection that @xmath346 satisfies @xmath347 therefore @xmath348 generates @xmath343 as a module over @xmath14 $ ] . as @xmath350 , we see that @xmath351 in the completion of the local ring at @xmath352 . on the other hand , if @xmath353 is a rational point of @xmath6 , then @xmath354 , @xmath355 defines an automorphism of @xmath6 taking @xmath353 to @xmath352 . hence at @xmath353 , we have @xmath356 now we consider the following problem . suppose that @xmath357 are rational points on @xmath6 with distinct @xmath358 . given some positive integers @xmath359 for @xmath360 . we want to construct @xmath361 with @xmath362 $ ] such that @xmath363 for @xmath360 . there are at least two ways to do this . [ [ first - method ] ] first method + + + + + + + + + + + + for @xmath360 , let @xmath364 be the truncation of the series expansion of @xmath9 at @xmath365 modulo @xmath366 , and let @xmath367 $ ] be defined by @xmath368 then @xmath369 satisfies the required conditions by the chinese remainder theorem . [ [ second - method ] ] second method + + + + + + + + + + + + + a somewhat more explicit way is as follows . if @xmath370 $ ] , then the condition @xmath371 is equivalent to the following linear conditions on the coefficients @xmath372 , @xmath373 for @xmath374 , where @xmath375 except @xmath376 , @xmath377 , @xmath378 for @xmath379 . now let @xmath380 . then the required @xmath95 can be determined by solving the linear system @xmath381 for the vector @xmath382 where @xmath383 is a certain vector of length @xmath192 and @xmath179 is a square matrix of size @xmath192 obtained by the horizontal join of @xmath384 matrices @xmath385_{0\le i\le{n-1},0\le j\le \mu_k-1}\ ] ] for @xmath386 . the matrix @xmath179 is called a confluent vandermonde matrix in the literature , and is known to be invertible ( actually the determinant is @xmath387 @xcite ) . therefore the linear system has a unique solution . let us compute @xmath388 in the previous example by the second method . here @xmath389 . if @xmath390 , then @xmath391 where @xmath392 and @xmath393 . the solution of this linear system was given in the previous subsection . for this task , we use the grbner conversion algorithm in @xcite that converts a set of generators of a submodule of @xmath14^n$ ] to a module grbner basis with respect to a special weighted monomial order . we review the algorithm below . let @xmath394 . tuples in @xmath395 are ordered lexicographically such that @xmath352 is the first tuple in @xmath395 and the successor of @xmath396 is @xmath397 if @xmath398 or @xmath399 if @xmath400 . thus @xmath401 is a basis for @xmath110_l$ ] as an @xmath14$]-module and the weight of the basis element @xmath402 is @xmath403 . the index of @xmath140_l$ ] is defined to be the largest tuple @xmath396 such that the coefficient of @xmath402 is nonzero . in particular , if the leading term of @xmath140_l$ ] is @xmath404 with respect to @xmath163 , then @xmath405 . note that @xmath406 for the generators @xmath407 of @xmath209 computed by algorithm b. the algorithm finds the element of @xmath209 with the smallest leading term . initially set @xmath407 to be the initial set of generators of the module @xmath209 computed by algorithm b. let @xmath408 during the execution of the algorithm . for @xmath409 and @xmath410 in @xmath395 , we abbreviate @xmath411 . * set @xmath412 . * set @xmath28 to the successor of @xmath28 . if @xmath413 , then proceed ; otherwise go to step i6 . * set @xmath414 . if @xmath415 , then go to step i2 . * set @xmath416 and @xmath417 . * if @xmath418 , then set @xmath419 if @xmath420 , then set , storing @xmath421 in a temporary variable , @xmath422 go back to step i3 . * output @xmath407 with the smallest leading term , and the algorithm terminates . algorithm i converts the initial basis given in to a grbner basis with respect to the order @xmath163 . the computed grbner basis is @xmath423 the twelve rows represent the polynomials in the grbner basis of the module @xmath209 over @xmath14 $ ] . comparing the weights of the leading coefficients of the polynomials , which lie on the diagonal , we find that the polynomial represented by the eleventh row is the required @xmath0-polynomial of the ideal @xmath111 , given explicitly in example [ cjops ] equation . elements of @xmath12 can be written uniquely as polynomials in @xmath9 of degree less than @xmath1 with coefficients in @xmath14 $ ] . we assume that for computations in @xmath12 , we use this representation of elements of @xmath12 . also we think of @xmath424 and @xmath425 for @xmath26 in this representation . note that a straightforward way of multiplying two elements @xmath426 of @xmath12 takes @xmath427 multiplications on @xmath2 and that @xmath428 if @xmath429 and @xmath430 . first we consider computing @xmath362 $ ] satisfying @xmath431 for @xmath360 as in section [ subsecc ] . this computation takes @xmath432 multiplications on @xmath2 where @xmath380 , if we use gaussian elimination to solve the linear system . note also @xmath433 . next we consider computing @xmath434 according to proposition [ prop6 ] in section [ subsecb ] . the first product @xmath435 on the right side of has at most @xmath436 linear factors . hence @xmath435 can be computed with @xmath437 multiplications on @xmath2 . note @xmath438 . on the other hand , as @xmath439 , the second product @xmath440 can be computed with @xmath441 multiplications on @xmath2 . note @xmath442 and @xmath443 . then @xmath435 and @xmath440 can be multiplied with @xmath441 multiplications on @xmath2 . hence , in total , computing @xmath434 takes @xmath441 multiplications on @xmath2 . note @xmath444 and @xmath445 . now we consider computations in steps b2 and b3 of algorithm b in section [ subseca ] . fix @xmath446 . computing @xmath283 ( @xmath447 ) , as shown above , takes @xmath448 multiplications on @xmath2 for each @xmath449 . computing @xmath450 can be done with @xmath451 multiplications on @xmath2 . note @xmath452 . let @xmath453 denote the product of the right side in . it is easy to verify @xmath454 and @xmath455 if @xmath456 . so computing @xmath457 takes @xmath458 multiplications on @xmath2 . note @xmath459 . computing @xmath460 takes @xmath461 multiplications on @xmath2 . summing up , an execution of algorithm b takes @xmath462 multiplications on @xmath2 . lastly noting @xmath463 and using a result in @xcite , we see that an execution of algorithm i takes @xmath464 multiplications on @xmath2 . therefore the algebraic soft - decision decoder of hermitian codes can be implemented in a way that takes @xmath465 multiplications on @xmath2 . we implemented the algebraic soft - decision decoder ( sdd ) for hermitian codes in software . in this section , we present some simulation results that show the performance of the algebraic soft - decision decode for half - rate hermitian codes . first we describe the general setup of our simulations . we assume the awgn channel . for qpsk and qam modulations , the signal points correspond one - to - one with the symbols in the finite field over which the code is defined , and the posterior probabilities of the symbols are simply set to those of the corresponding signal points . for bpsk , each of the symbols correspond with a bit sequence , and the posterior probabilities of the symbols are set to the products of the posterior probabilities of the bits . koetter and vardy s multiplicity assignment algorithm @xcite is used to translate the posterior probabilities to the values of the multiplicity matrix . the multiplicity assignment algorithm accepts a parameter @xmath277 that limits the @xmath143-degree of the @xmath0-polynomial , thereby the list size of the candidate codewords to at most @xmath277 . from the multiplicity matrix , our interpolation algorithm finds the @xmath0-polynomial . then wu and siegel s root - finding algorithm @xcite is used to compute the roots of the @xmath0-polynomial . the list of candidate codewords is then formed from the roots . if the list is empty , then the decoder simply output the message part of the received vector determined by hard - decision directly from the posterior probabilities of the symbols . if the list is not empty , the decoder outputs the message from the codeword that has the best score with respect to the multiplicity matrix . the smallest field over which hermitian codes are defined is @xmath68 and the length of these codes is @xmath69 . the length is extremely small , and it is perhaps unrealistic to expect the codes to be used in practice . however the codes are amenable for simulations with somewhat larger @xmath277 . figure [ fig1 ] show the performance of the half - rate @xmath466 $ ] hermitian code with qpsk . the example used in previous sections was sampled from this simulation with snr @xmath467 and @xmath468 . ( 10,0 ) ; ( 2,6 ) ( 10,6 ) ; ( 2,0 ) ( 2,6 ) ; ( 10,0 ) ( 10,6 ) ; in -1 , ... , -6 in 0.301 , 0.477 , 0.602 , 0.698 , 0.778 , 0.845 , 0.903 , 0.954,1 ( 2 , ) ( 10 , ) ; in 2 , ... , 10 ( , 0 ) ( , 6 ) ; /in 2 , ... , 10 ( 0pt,2pt ) ( 0pt,0pt ) node[below ] @xmath469 ; /in 0 , ... , -6 ( 2pt,0pt ) ( 0pt,0pt ) node[left ] @xmath470 ; plot[yshift=6 cm ] fileawgn-qpsk-hd-uncoded-ber.table ; plot[mark = oplus , yshift=6 cm ] fileawgn - qpsk - sd - hermitian[8,4]-asd - l5-fer.table ; plot[mark = o , yshift=6 cm ] fileawgn - qpsk - sd - hermitian[8,4]-asd - l5-ber.table ; ( 9.8,5.8 ) node[legend , text width=15em ] plot coordinates ( -.5,0)(0,0 ) ; ber of uncoded qpsk + plot[mark = oplus , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; fer of hermitian [ 8,4 ] algebraic sdd for @xmath468 + plot[mark = o , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; ber of hermitian [ 8,4 ] algebraic sdd for @xmath468 + ; ( 6,0 ) [ yshift=-15pt ] node[below ] @xmath471 in db ; ( 2,3 ) [ xshift=-40pt ] node[rotate=90 ] error rate ; figures [ fig3 ] and [ fig4 ] show the performance of @xmath472 $ ] hermitian code with bpsk modulation . for comparison , the figures also show the performance of the half - rate @xmath473 $ ] reed - solomon code . observe that the performance curve of hermitian code more steeply decrease than that of reed - solomon code , and from around 5 db , the hermitian code outperforms the reed - solomon code . ( 0,0 ) ( 10,0 ) ; ( 0,-7 ) ( 10,-7 ) ; ( 0,0 ) ( 0,-7 ) ; ( 10,0 ) ( 10,-7 ) ; in -1 , ... , -7 in 0.301 , 0.477 , 0.602 , 0.698 , 0.778 , 0.845 , 0.903 , 0.954,1 ( 0 , ) ( 10 , ) ; in 0 , ... , 10 ( , 0 ) ( , -7 ) ; /in 0 , 1 , ... , 10 ( 0pt,2pt ) ( 0pt,0pt ) node[below ] @xmath469 ; /in 0 , ... , -7 ( 2pt,0pt ) ( 0pt,0pt ) node[left ] @xmath470 ; plot fileawgn-bpsk-hd-uncoded.table ; plot[mark = triangle ] fileawgn - bpsk - sd - rs[16,8]-asd - l1-ber.table ; plot[mark = triangle ] fileawgn - bpsk - sd - rs[16,8]-asd - l2-ber.table ; plot[mark = triangle ] fileawgn - bpsk - sd - rs[16,8]-asd - l3-ber.table ; plot[mark = o ] fileawgn - bpsk - sd - hermitian[64,32]-asd - l1-ber.table ; plot[mark = o ] fileawgn - bpsk - sd - hermitian[64,32]-asd - l2-ber.table ; plot[mark = o ] fileawgn - bpsk - sd - hermitian[64,32]-asd - l3-ber.table ; ( .2,.2 ) node[legendsw , text width=14em ] plot coordinates ( -.5,0)(0,0 ) ; uncoded bpsk + plot[mark = triangle , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; rs [ 16,8 ] algebraic sdd for @xmath474 + plot[mark = o , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; hermitian [ 64,32 ] algebraic sdd for @xmath474 + ; ( 5,0 ) [ yshift=-15pt ] node[below ] @xmath471 in db ; ( 0,3.5 ) [ xshift=-40pt ] node[rotate=90 ] bit error rate ; ( 0,0 ) ( 10,0 ) ; ( 0,-7 ) ( 10,-7 ) ; ( 0,0 ) ( 0,-7 ) ; ( 10,0 ) ( 10,-7 ) ; in -1 , ... , -7 in 0.301 , 0.477 , 0.602 , 0.698 , 0.778 , 0.845 , 0.903 , 0.954,1 ( 0 , ) ( 10 , ) ; in 0 , ... , 10 ( , 0 ) ( , -7 ) ; /in 0 , 1 , ... , 10 ( 0pt,2pt ) ( 0pt,0pt ) node[below ] @xmath469 ; /in 0 , ... , -7 ( 2pt,0pt ) ( 0pt,0pt ) node[left ] @xmath470 ; plot[mark = triangle ] fileawgn - bpsk - sd - rs[16,8]-asd - l1-fer.table ; plot[mark = triangle ] fileawgn - bpsk - sd - rs[16,8]-asd - l2-fer.table ; plot[mark = triangle ] fileawgn - bpsk - sd - rs[16,8]-asd - l3-fer.table ; plot[mark = o ] fileawgn - bpsk - sd - hermitian[64,32]-asd - l1-fer.table ; plot[mark = o ] fileawgn - bpsk - sd - hermitian[64,32]-asd - l2-fer.table ; plot[mark = o ] fileawgn - bpsk - sd - hermitian[64,32]-asd - l3-fer.table ; ( .2,.2 ) node[legendsw , text width=14em ] plot[mark = triangle , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; rs [ 16,8 ] algebraic sdd for @xmath474 + plot[mark = o , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; hermitian [ 64,32 ] algebraic sdd for @xmath474 + ; ( 5,0 ) [ yshift=-15pt ] node[below ] @xmath471 in db ; ( 0,3.5 ) [ xshift=-40pt ] node[rotate=90 ] frame error rate ; figures [ fig5 ] and [ fig6 ] also show that the hermitian code outperforms the reed - solomon code with @xmath475-qam modulation , from around 8 db onward . ( 14,0 ) ; ( 2,-7 ) ( 14,-7 ) ; ( 2,0 ) ( 2,-7 ) ; ( 14,0 ) ( 14,-7 ) ; in -1 , ... , -7 in 0.301 , 0.477 , 0.602 , 0.698 , 0.778 , 0.845 , 0.903 , 0.954,1 ( 2 , ) ( 14 , ) ; in 2 , ... , 14 ( , 0 ) ( , -7 ) ; /in 2 , ... , 14 ( 0pt,2pt ) ( 0pt,0pt ) node[below ] @xmath469 ; /in 0 , ... , -7 ( 2pt,0pt ) ( 0pt,0pt ) node[left ] @xmath470 ; plot fileawgn-16qam-hd-uncoded-ber.table ; plot[mark = triangle ] fileawgn-16qam - sd - rs[16,8]-asd - l1-ber.table ; plot[mark = triangle ] fileawgn-16qam - sd - rs[16,8]-asd - l2-ber.table ; plot[mark = triangle ] fileawgn-16qam - sd - rs[16,8]-asd - l3-ber.table ; plot[mark = o ] fileawgn-16qam - sd - hermitian[64,32]-asd - l1-ber.table ; plot[mark = o ] fileawgn-16qam - sd - hermitian[64,32]-asd - l2-ber.table ; plot[mark = o ] fileawgn-16qam - sd - hermitian[64,32]-asd - l3-ber.table ; ( 2.2,.2 ) node[legendsw , text width=14em ] plot coordinates ( -.5,0)(0,0 ) ; uncoded @xmath475-qam + plot[mark = triangle , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; rs [ 16,8 ] algebraic sdd for @xmath474 + plot[mark = o , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; hermitian [ 64,32 ] algebraic sdd for @xmath474 + ; ( 8,0 ) [ yshift=-15pt ] node[below ] @xmath471 in db ; ( 2,3.5 ) [ xshift=-40pt ] node[rotate=90 ] bit error rate ; ( 2,0 ) ( 14,0 ) ; ( 2,-7 ) ( 14,-7 ) ; ( 2,0 ) ( 2,-7 ) ; ( 14,0 ) ( 14,-7 ) ; in -1 , ... , -7 in 0.301 , 0.477 , 0.602 , 0.698 , 0.778 , 0.845 , 0.903 , 0.954,1 ( 2 , ) ( 14 , ) ; in 2 , ... , 14 ( , 0 ) ( , -7 ) ; /in 2 , ... , 14 ( 0pt,2pt ) ( 0pt,0pt ) node[below ] @xmath469 ; /in 0 , ... , -7 ( 2pt,0pt ) ( 0pt,0pt ) node[left ] @xmath470 ; plot[mark = triangle ] fileawgn-16qam - sd - rs[16,8]-asd - l1-fer.table ; plot[mark = triangle ] fileawgn-16qam - sd - rs[16,8]-asd - l2-fer.table ; plot[mark = triangle ] fileawgn-16qam - sd - rs[16,8]-asd - l3-fer.table ; plot[mark = o ] fileawgn-16qam - sd - hermitian[64,32]-asd - l1-fer.table ; plot[mark = o ] fileawgn-16qam - sd - hermitian[64,32]-asd - l2-fer.table ; plot[mark = o ] fileawgn-16qam - sd - hermitian[64,32]-asd - l3-fer.table ; ( 2.2,.2 ) node[legendsw , text width=14em ] plot[mark = triangle , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; rs [ 16,8 ] algebraic sdd for @xmath474 + plot[mark = o , mark indices=2 ] coordinates ( -.5,0)(-.25,0)(0,0 ) ; hermitian [ 64,32 ] algebraic sdd for @xmath474 + ; ( 8,0 ) [ yshift=-15pt ] node[below ] @xmath471 in db ; ( 2,3.5 ) [ xshift=-40pt ] node[rotate=90 ] frame error rate ; we presented an algebraic soft - decision decoder for hermitian codes . software simulations show that hermitian codes perform better than reed - solomon codes for algebraic soft - decision decoding , as expected . however , for the decoder to be really practical , reduction of the computational complexity remains an important problem . one promising avenue is to generalize the idea of complexity reduction for reed - solomon codes in @xcite . designing efficient electric circuits implementing the decoder is of course an issue to explore . the extent of our computer simulations of the decoding algorithm was limited by our computing resources . it would be good to have analytic results about the performance of the decoding algorithm . there have been several analytic performance analyses for the algebraic soft - decision decoding of reed - solomon codes @xcite . similar analyses may be done for hermitian codes . our description of the decoding algorithm is interwoven with the particular structure of hermitian codes . however , the underlying principle of the decoding algorithm seems to apply to a wider class of algebraic geometry codes . in particular , plane algebraic curves with one point at infinity are immediate candidates . we leave an adequate treatment of this subject as a remaining work . v. olshevsky and m. a. shokrollahi , `` a displacement approach to decoding algebraic codes , '' in _ fast algorithms for structured matrices : theory and applications _ , math.1em plus 0.5em minus 0.4em amer . math . soc . , 2003 , vol . 265292 . j. ma , a. vardy , and z. wang , `` efficient fast interpolation architecture for soft - decision decoding of reed - solomon codes , '' in _ proc . ieee symp . circuits and systems _ , kos , greece , may 2006 , pp . 48234826 . j. ma , a. vardy , z. wang , and q. chen , `` direct root computation architecture for algebraic soft - decision decoding of reed - solomon codes , '' in _ proc . ieee symp . circuits and systems _ , new orleans , la , may 2007 , pp . 14091412 .
|
an algebraic soft - decision decoder for hermitian codes is presented .
we apply koetter and vardy s soft - decision decoding framework , now well established for reed - solomon codes , to hermitian codes .
first we provide an algebraic foundation for soft - decision decoding
. then we present an interpolation algorithm finding the @xmath0-polynomial that plays a key role in the decoding . with some simulation results ,
we compare performances of the algebraic soft - decision decoders for hermitian codes and reed - solomon codes , favorable to the former .
hermitian codes , algebraic soft - decision decoding , interpolation algorithm , grbner bases .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
gas in giant molecular clouds ( gmcs ) is distributed non - uniformly and appears to aggregate itself into isolated dense clumps or more contiguous and elongated , filament - like structures . detailed observations of potential star - forming clouds have demonstrated the ubiquitous nature of filamentary clouds . for example , giant molecular filaments on scales of a few parsecs have been reported in the inter - arm regions of the milky - way ( ragan _ et al . _ 2014 ; higuchi _ et al . _ 2014 and other references therein ) . on the other hand , relatively small ( by about an order of magnitude compared to the former ) , dense filaments have also been reported within star - forming clouds in the local neighbourhood ( e.g. schneider & elmegreen 1979 ; nutter _ et al . _ 2008 ; myers 2009 ; andr ' e _ et . _ 2010 ; jackson _ et . _ 2010 ; arzoumanian _ et . _ 2011 ; kainulainen _ et al . _ 2011 , 2013 and kirk _ et . _ 2013 are only a few authors among an exhaustive number of them ) . star - forming sites within gmcs are often found located within dense filamentary clouds or at the junctions of such clouds and further , these filamentary clouds usually show multiplicity , in other words , striations roughly orthogonal to the main filament , and form hubs ( e.g. palmeirim _ et al . _ 2013 ; hacar _ et al . _ 2013 ; schneider _ et al . _ 2012 & 2010 ; myers 2009 ) . in fact , inferences drawn from detailed observations of star - forming clouds have led some authors to suggest that turbulence - driven filaments could possibly represent the first phase in the episode of stellar - birth , followed by gravitational fragmentation of the densest filaments to form prestellar cores . filamentary clouds therefore form a crucial part of the star - formation cycle . consequently , a significant observational , theoretical and/or numerical effort has been directed towards understanding these somewhat peculiar clouds . in the last few years we have significantly improved our understanding about these clouds as a number of them have been studied in different wavebands of the infrared regime of the electromagnetic spectrum using sub - millimeter arrays on the jcmt and the herschel ( e.g. nutter & ward - thompson 2007 ; andr ' e _ et al . _ 2010 ; menschikov _ et al . _ 2010 ; also see review by andr ' e _ et al . _ 2014 ) . the stability and possible evolution of filamentary clouds has also been studied analytically in the past and in more recent times . however , these models were usually developed under simplifying assumptions . for example , ostriker ( 1964 ) , developed one of the earliest models by approximating a filamentary cloud as an infinite self - gravitating cylinder described by a polytropic equation of state and derived its density distribution . in a later contribution , bastien ( 1983 ) , bastien _ et al . _ ( 1991 ) and inutsuka & miyama ( 1992 ) studied the stability criteria of filamentary clouds under the assumption of isothermality and suggested that such clouds were more likely to form via the radial collapse of an initial cylindrical distribution of molecular gas . these models also demonstrated formation of prestellar cores along the dense axial filament via jeans - fragmentation . formation of dense filaments via interaction between turbulent fluid flows has been demonstrated numerically by a number of authors ( e.g. klessen _ et al . _ 2000 ; bate _ et al . _ 2003 ; price & bate 2008 , 2009 ; federrath _ et al . _ 2010a ; padoan & nordlund 2011 and federrath & klessen 2012 ) . similarly in other recent contributions ( e.g. heitsch _ et al . _ ( 2009 ) ; peters _ et al . _ 2012 and heitsch 2013 ) , respective authors specifically investigated the process that is likely to assemble a dense filament . the conclusion of these latter authors supports the idea of filament formation via radial collapse of gas followed by an accretional phase during which the filament acquires mass even as it continues to self - gravitate . in fact , peters _ ( 2012 ) demonstrated the formation of filamentary clouds on the cosmic scale and argued that a filament was more likely to collapse radially and form stars along its length when confined by pressure of relatively small magnitude . in another recent contribution , smith _ et al . _ ( 2014 ) , have demonstrated the formation of dense filaments in turbulent gas , however , they have not addressed the other crucial issue about the temperature profile of these filaments ; observations have revealed that gas in the interiors of dense filaments is cold at a temperature on the order of 10 k ( e.g. arzoumanian _ et al . ( 2013 ) also address the issue of tidal - forces acting on the sides of a filamentary cloud having finite length and suggest that gas accreted at the ends can possibly give rise to fan - like features often found at the ends of infrared dark clouds ( irdcs ) . in a semi - analytic calculation , jog(2013 ) demonstrated that tidal - forces also tend to raise the canonical jeans mass , though compressional force - fields are more likely to raise the local density and therefore lower the jeans mass . ( 2014 ) derived a dispersion relation for a rotating non - magnetised filamentary cloud idealised as a polytropic cylinder with localised density perturbations . under these simplifying assumptions , the authors demonstrated that the filament indeed developed jeans - type instability with propensity to fragment on the scale of the local jeans length . these conclusions are in fact , consistent with those drawn in an earlier work by pon _ ( 2011 ) , or even those by bastien _ ( 1991 ) who arrived at similar conclusions from their analytical treatment of the problem . on the other hand , inutsuka & miyama ( 1997 ) , showed that a cylindrical distribution of gas is unlikely to become self - gravitating as long as its mass per unit length was less than a certain critical value . similar conclusions were also drawn by fischera & martin ( 2012 ) when they performed a stability analysis of clouds idealised as isothermal cylinders . in the present work we study the dynamical stability of an initially non - self gravitating cylindrical distribution of molecular gas maintained initially in pressure equilibrium . in particular , we are interested to see how this distribution of gas behaves as it is allowed to cool in the presence of self - gravity . we would like to examine if a radial collapse does indeed ensue and the nature of the new state of equilibrium attained by the gas . thus , we would like to investigate the physical conditions under which radial collapse is likely to be the favoured mode of evolution . although the objective we have set ourselves is similar to that of peters _ et al . _ ( 2012 ) , our work differs on three counts : ( i ) unlike peters _ et al . _ who assumed a polytropic equation of state to calculate gas temperature , we calculate gas temperature by solving the energy equation coupled with a cooling function , ( ii ) we do not use a filament that already is dynamically evolving instead , we begin by setting up a cylindrical cloud in pressure equilibrium and then examine if it does indeed collapse to form a thin dense filamentary cloud . this relatively simple set - up represents the intermediate stage of filament - formation , in other words , the stage after supersonically moving turbulent flows in a molecular cloud ( mc ) collide to form a filamentary distribution of warm gas ( see e.g. klessen _ et al . _ 2000 ; bate _ et al . _ 2003 ; price & bate 2008,2009 ; federrath _ et al . _ 2010a ; federrath & klessen 2012 ; smith _ et al . _ 2014 ) , and more importantly , ( iii ) we investigate if gravitational supercriticality is a necessary pre - condition for filament - formation and follow the evolution of the dense filament further to study the formation of prestellar cores in it . we observe that when the magnitude of the confining pressure is relatively small such that the initial volume of gas is at least critically stable , it does indeed collapse radially to assemble a thin dense filament which then fragments via a jeans - like instability on the scale of the fastest growing unstable mode to form prestellar cores . we also investigate physical properties such as the distribution of gas density and temperature within the post - collapse filament . we demonstrate that the density profile of the post - collapse isothermal filamentary cloud is indeed similar to the classic profile suggested by ostriker ( 1964 ) ; the profile is plummer - like for a filament that is allowed to cool dynamically . the article is divided as follows : in section 2 we discuss the stability of a pressure - confined isothermal cylinder . then in section 3 we present the initial conditions used for this work and briefly describe the numerical algorithm used for the simulations . the results are presented and discussed in respectively , sections 4 and 5 before concluding in section 6 . we consider a cylindrical distribution of isothermal molecular gas composed of the usual cosmic mixture ( approximately 90% hydrogen and 10% helium ) . the cylinder has initial radius , @xmath0 , length , @xmath1 , and maintained at temperature , @xmath2 . in a purely non - magnetic case gas within this cylinder is supported by thermal pressure , @xmath3 , against self - gravity and is confined externally by finite pressure exerted by the inter - cloud medium ( icm ) , @xmath4 . we assume an initial state of equilibrium so that the external pressure , @xmath4 , balances the internal pressure , @xmath3 . the internal and external pressure are both thermal by nature since particles representing the two media are not imparted any initial momentum . the stability of a spherical gas body against self - gravity is defined in terms of the thermal jeans mass . now , for a cylindrical distribution of gas its mass line - density , @xmath5 , defined as @xmath6 this quantity is analogous to the mass of a spherical cloud and as will be seen below , is useful in describing the stability of a cylindrical cloud against self - gravity . here , @xmath7 , is the initial density of the cylindrical distribution of gas , assumed to be uniform . a gas cylinder is stable against gravitational collapse as long as its mass line - density is smaller than the maximum value @xmath8 where @xmath9 is the isothermal sound - speed for the cylinder . we define the stability factor , @xmath10 , as the ratio of the mass line - density for a cylinder against its maximum value . thus , @xmath11 when @xmath12 exceeds unity , @xmath13 , a radial collapse ensues and the cylinder may be described as _ super - critical _ ; cylinders with @xmath12 less than unity are described as _ sub - critical _ ( fischera & martin 2012 ) . we describe cylinders with @xmath14 1 as _ critically stable_. as in our earlier work investigating the stability of starless cores ( anathpindika & di francesco 2013 ) , in the following sections we will calculate the stability factor , @xmath10 , at different radii within the model cylindrical cloud over the course of its evolution to describe its stability against self - gravity . we now calculate the radius of the initial cylindrical distribution of gas . keeping in mind that the pressure exerted by the confining icm balances the thermal pressure within the cylinder , we can write @xmath15 so that its radius , @xmath16 finally , following a simple dimensional analysis we define the free - fall time , @xmath17 @xmath18 @xmath19 , which is slightly faster than the free - fall time in jeans analysis ; @xmath20 , is the density of the post - collapse filament having radius , @xmath21 , and @xmath9 , the sound - speed of gas within it . a free - fall ensues if @xmath22 , where @xmath23 is the sound - crossing time , @xmath24 . using the above inequality we obtain the condition for jeans instability as , @xmath25 the corresponding mass of the fragment is @xmath26 , which , using eqn . ( 3 ) above , can be re - cast as @xmath27 @xmath28 , being the radius of the filament over which gas density remains approximately uniform before turning over in its wings . [ cols= " > , < , < , > , < , < , < , < , < , < , < , < , > " , ] in this demonstrative work we began by placing a non - self gravitating uniform - density cylindrical distribution of molecular gas in pressure equilibrium with its surrounding medium . to this initial distribution of gas we assigned a feducial temperature ( see table 1 ) . in this work the gas pressure and pressure exerted by the external medium are purely of thermal nature . while ambient physical conditions such as the turbulent mach number in a mc could possibly influence filament properties , we leave this aspect of the problem for future investigation . other free parameters in the set - up were , the length of the cylinder , @xmath1= 5 pc , contrast , @xmath29=10 , between the initial gas density within the cylinder and its surrounding medium , the average initial gas density within the cylinder , @xmath30 = 600 @xmath31 and the initial mass line - density , @xmath32 . this choice of gas density is also consistent with densities found in a typical star - forming cloud ( e.g. andr ' e _ et al . the temperature of the confining medium is calculated by using the condition of pressure - balance at the surface of the cylinder . the initial radius of the cylinder was calculated using eqn . ( 4 ) above . though simple , this set of initial conditions is suitable for the extant purposes of studying the origin of the density profile and the temperature profile of a typical filament and the formation of prestellar cores along the length of this filament . parameters used in the realisations performed in this work have been listed in table 1 . sinusoidal density perturbations along the length of the initial distribution of gas were imposed in five simulations ; see table 1 . these perturbations were imposed by revising particle positions , @xmath33 , to @xmath34 , such that the following expression was satisfied , @xmath35 ( hubber _ et al . _ 2006 ) , where the wave - number , @xmath36 , the integer , @xmath37 = 1 , 2 for respectively , the fundamental mode and the second harmonic ; @xmath38 = 0.1 , is the amplitude of perturbations . the number of gas particles in a simulation are calculated according to - @xmath39 where , @xmath40 , is the volume of the initial distribution of gas , while the initial average smoothing length , @xmath41 , was set equal to @xmath42 ; @xmath43 being the resolution parameter listed in column 11 of table 1 and defined by eqn . ( 10 ) below ; the jeans length , @xmath44 pc , at the density threshold(@xmath45 g @xmath31 ) , adopted for representing prestellar cores in this work at a temperature of 10 k. particles representing the initial distribution of gas were assembled by extracting a cylinder of unit size and length from a pre - settled glass like distribution . this unit - cylinder was then stretched to the desired dimensions and then placed in a confining medium of external - pressure , modelled using the icm particles . the envelope of icm particles had a thickness , 30@xmath41 , along each spatial dimension of the cylinder . the icm particles are special sph particles that exert only hydrodynamic force on the normal gas particles ; gas particles on the other hand , exert both , gravitational and hydrodynamic forces . the cylinder and its confining medium were then placed in a box lined with boundary particles that prevent sph particles from escaping . unlike the gas and the icm particles , the boundary particles are dead and do not contribute to sph forces . the layer of dead boundary particles was there to simply hold the gas+icm particles in the box and prevent them from diffusing away . particles within the set - up were initially stationary as no external velocity field was introduced . simulations were performed using our well tested sph algorithm , seren , in its conservative energy mode ( hubber _ et al . the sph calculations in this work also include contributions due to the artificial conductivity prescribed by price ( 2008 ) , to avoid any spurious numerical artefacts possible due to the creation of a gap between multi - phase fluids . an sph particle , in this work , had a fixed number of neighbours , @xmath46 = 50 . gas cooling was implemented by employing the cooling function given by the equation below , @xmath47 @xmath48 ( koyama & inutsuka 2002 ) . constant background heating due to cosmic rays is provided by the heating function , @xmath49 erg s@xmath50 . the temperature of an sph particle is calculated by revising its internal energy . the equilibrium temperature , @xmath51 , attained by * a * small sph particle corresponds to equilibrium energy , @xmath52 , and the timescale , @xmath53 , over which energy is either radiated or acquired by a particle is , @xmath54 then , after a time - step @xmath55 , the energy @xmath56 of a particle is revised according to , @xmath57 ( v ' azquez - semadeni _ et al . _ 2007 ) . however , a more accurate calculation of the gas temperature demands solving the radiative transfer equation(e.g . _ 2007 ; price & bate 2009 ; bate 2012 ) , and accounting for the molecular chemistry in the cloud ( glover _ et al . we leave this for a future work . + _ the sph sink particle _ + we represent a prestellar core by an sph sink particle in this work , for it is not our intention to follow the internal dynamics of a core . an sph particle is replaced with a sink only when the particle satisfies the default criteria for sink - formation which include , negative divergence of velocity and acceleration of the seed particle ( see bate , bonnell & price 1995 ; bate & burkert 1997 and federrath _ et al . _ ( 2010b ) ) . we set the density threshold for sink - formation at @xmath58 g @xmath31 ( @xmath59 @xmath31 ) , which despite being on the lower side , is consistent with the average density of a typical prestellar core . the interested reader is referred to hubber _ ( 2011 ) for other technical details related to implementation of this prescription in our algorithm seren . the radius of a sink particle was set at , @xmath60 , where @xmath61 is the smoothing length of the sph particle that seeds a sink . sink particles in the simulation therefore have different radii . a sink interacts with normal gas particles only via gravitational interaction and accretes gas particles that come within the pre - defined sink - accretion radius , @xmath62 . + + the average initial smoothing length , @xmath41 , for the ensemble of calculations discussed in this article is calculated as - @xmath63 where the number of neighbours , @xmath64 = 50 . the diameter of an sph particle , @xmath65 , so that the resolution criterion becomes , @xmath66 @xmath67 , and @xmath68 , the typical length - scale of fragmentation has been defined by eqn . ( 5 ) above . evidently , higher the value of @xmath69 , better the resolution . value of the resolution factor , @xmath69 , for each simulation has been listed in column 10 of table 1 . the resolution criterion defined by eqn . ( 10 ) above is the sph equivalent of the truelove criterion ( truelove _ et al . _ 1997 ) , defined originally for the grid - based adaptive mesh refinement algorithm . it has been shown that the gravitational instability is reasonably well - resolved for @xmath43 = 1 , and need be @xmath182 , for a good resolution of this instability . note that some authors prefer to define the resolution factor , @xmath69 , as the inverse of the fraction on the right - hand side of eqn . ( 10 ) , in which case the criterion for good resolution becomes , @xmath70 ( e.g. hubber _ et al . all our simulations discussed in this work satisfy this resolution criterion . however , this choice of resolution is reportedly insufficient to achieve convergence in calculations of crucial physical parameters such as the energy and momentum , but sufficient to prevent artificial fragmentation of collapsing gas ( federrath _ et al . _ 2014 ) . in order to alleviate this latter problem these authors suggest a more stringent resolution criterion , @xmath71 15 . however , our extant purpose here is limited to studying the stability of a filamentary cloud and its fragmentation so that the adopted choice of numerical resolution(@xmath72 2 ) , should be sufficient . in fact , in 4.1 below we demonstrate numerical convergence over the observed fragmentation of the post - collapse filament . finally , the sink radius , @xmath62 , defined in 3.2 above is @xmath73 pc in simulations developed with the least number of particles and @xmath74 pc in cases 6 and 7 , those developed with the highest resolution in this work . although prestellar cores of smaller size are known to exist in star - forming clouds ( see for e.g. jijina _ et al . _ 1999 ) , the resolution adopted here is sufficient for a demonstrative calculation of the kind presented in this work . the typical width of the post - collapse filament , @xmath180.1 pc , is also consistently resolved by a minimum of 80 particles along its length . since this volume of gas is allowed to cool , what follows , is the fragmentation of cooling gas . the length - scale of this fragmentation is comparable to the thermal jeans length ( see for e.g. v ' azquez - semadeni _ et al . _ 2007 ) , which is much larger than the initial width of the cylinder so that there is no fragmentation in the radial direction , as will be demonstrated below . furthermore , we have set our resolution to the jeans length , @xmath68 , evaluated at 10 k , which is lower than the average gas temperature within interiors of the filament as will be evident in the following section . our simulations therefore have the desired spatial resolution to represent core - formation in the post - collapse filament . our interest lies particularly in examining the possibility of forming a dense filamentary cloud out of a cylindrical volume of gas confined by thermal pressure . this possibility is explored for different choices of initial conditions ; first , when the volume of gas is initially critically stable , then marginally super - critical and finally , sub - critical . interestingly though , the initial cylindrical distribution of gas evolves radically differently if it is at / above critical stability as compared to when it is sub - critical . we also vary the spatial resolution and compare results from simulations developed by using a cooling function with that from an isothermal calculation . the initial cylindrical distribution of gas , in all these test cases , has comparable mass and an identical magnitude of confining pressure , @xmath4 . one of these , test realisation 1 , was developed under the assumption of isothermality while the cooling function defined by eqn . ( 7 ) above , was used in the remaining test cases . the mass line - density , @xmath77 , defined by eqn . ( 2 ) is slightly less than that for the initial distribution of gas , @xmath32 , for all the test realisations , except for the first where the two are of comparable magnitudes . the initial distribution of gas is therefore marginally super - critical in all realisations ; gas in the first realisation is approximately in equilibrium in the sense , @xmath78 ( see table 1 ) , @xmath0 being the radius of the cylinder . in all these realisations the radius of the initial cylindrical distribution of gas begins to shrink due to the onset of an inwardly propagating compressional wave . this radial contraction of the initial cylindrical distribution in outside - in fashion is visible in the rendered density images of the cross - section through the mid - plane of the collapsing cylinder shown in fig . 1 . overlaid on these plots are the velocity vectors that denote the direction of local gas - motion . this contraction soon assembles a thin dense filament along the axis of the cylinder and prestellar cores , represented by sph sink particles , begin to form along the length of the filament . at this point a word of caution would be appropriate . by a compressional wave we do not imply that the external pressure compresses the initial cylindrical distribution of gas in the radial direction . that hydrostatic cylinders can not be compressed in this manner has been demonstrated by fiege & pudritz ( 2000 ) . in this work density enhancement along the cylinder - axis is the result of gas being squeezed during the radial collapse of the initial configuration . we reserve the attribute of a compressional wave to describe this process . shown on the upper - panel of fig . 2 is a rendered density plot of the post - collapse filament in realisation 1 . the black blobs on this image represent the location of prestellar cores that begin to form along the length of this filament . evidently , the cores that had formed in this filament at the time of terminating the calculation are separated by approximately a jeans length(@xmath18 0.2 pc at @xmath2 = 25 k ; avg . density @xmath45 g @xmath31 ) , defined by eqn . the image on the lower - panel of fig . 2 is a plot of the radial distribution of gas density within the collapsing cylindrical cloud in this realisation . this plot and all other similar plots in this paper were generated by taking a transverse section through the mid - plane of the filament . the inwardly propagating compressional disturbance is evident from the density plots at early epochs of the collapse . as the gas is steadily accumulated in the central plane of the cylinder , the density of gas there rises and eventually develops a profile that matches very well with the one suggested by ostriker ( 1964 ) , @xmath79^{p/2}};\ ] ] with @xmath80 = 4 . here @xmath81 ; @xmath28 being the radius at which the density profile develops a knee . the density of this filament falls - off relatively steeply in the outer regions which is consistent with that reported by malinen _ ( 2012 ) for typical filamentary clouds in the taurus mc . furthermore , this figure also shows , @xmath82 0.1 pc , which is comparable to the radius of typical filaments found in eight nearby star - forming clouds in the gould - belt ( e.g. arzoumanian _ et al_. 2011 ; andre _ et al . _ 2014 ) . another interesting conclusion that can be drawn from this realisation is that about the stability of a filamentary cloud . it can be seen from fig . 2 that the centrally assembled , post - collapse dense filament at the epoch when calculations for this realisation were terminated , has a radius on the order of 0.1 pc , the point at which the density - profile develops a knee . an average density of @xmath45 g @xmath31 , the threshold for core - formation in these calculations , would suggest a mass line - density , @xmath83 50 m@xmath84 pc@xmath50 , for this post - collapse filament which is significantly higher than , @xmath85 41 m@xmath84 pc@xmath50 . the filament would therefore be expected to be in free - fall and should collapse rapidly to form a singular line . however , such is not to be the case and fig . 3 shows that the collapsing cylindrical cloud , or indeed the post - collapse filament , in fact , remains sub - critical at all radii , @xmath86 . we will discuss later in 5.1 the implications of this finding for the classification of observed filamentary clouds in typical star - forming regions . in fig . 3 we have shown the radial variation of the stability factor , @xmath10 , defined by eqn . ( 3 ) , and integrated over the radius of the collapsing cylindrical cloud . for the purpose , using the sturges criterion for optimal bin - size ( sturges 1926 ) , we divided the cylindrical distribution of gas into a number of concentric cylinders having incremental radii in the radially outward direction . number of these concentric cylinders is given by , @xmath87 ; @xmath88 being the number of gas particles in a realisation . the innermost cylinder would therefore have the smallest radius and radii of successive cylinders in the outward direction would be incremented by a small increment , @xmath89 . thus , if @xmath90 is the radius of the @xmath91 cylinder , then the radius of the @xmath92 cylinder is simply , @xmath93 ; @xmath94 . the initial distribution of gas in this case was characterised by @xmath78 . evident from the plot shown in fig . 3 is the fact that the gravitational state of the collapsing cylinder does not vary during the process . within the cylinder gas always remains gravitationally sub - critical so that , @xmath95 , although it does rise steadily close to the centre where the density is the highest in the post - collapse filament . in other words , shells of gas within the collapsing cylinder though self - gravitating , are not essentially in free - fall . that this must be the case is also evident from the density plots shown in fig . 1 . from these plots it can be seen that gas inside the collapsing cylinder has a relatively small velocity in comparison with that in the outer regions that is pushed in by the compressional wave . also , the observed outside - in type of collapse lends greater credence to the conclusion that gas within the collapsing cylinder must be gravitationally sub - critical , for had it been super - critical , the collapse would have been of the inside - out type , i.e. , the innermost regions of the cylinder would have collapsed rapidly followed by the outer regions . we had presented a similar argument in an earlier work ( anathpindika & di francesco 2013 ) , where hydrodynamic models developed to explain why some prestellar cores failed to form stars despite having masses well in excess of their respective jeans mass , were discussed . in that work it was demonstrated that cores could indeed contract in the radial direction without becoming singular ( i.e. , a free - fall collapse was not seen in these cores ) , if the mass of infalling gas at radii within the core was smaller than the local jeans mass , or equivalently , we had suggested that the jeans condition had to be satisfied at all radii within a collapsing core for it to become singular ( protostellar ) . in the present work we have used the mass line - density instead of the thermal jeans mass to discuss the stability of a cylindrical volume of gas . we observe , the collapsing cylinder appears to evolve through quasi - equilibrium configurations where thermal pressure provides support against self - gravity and a radial free - fall does not ensue . interestingly , we observe that prestellar cores can indeed form in a filamentary cloud that is not in radial free - fall . these cores acquire mass by accreting gas and shown in fig . 4 is the accretion history of these cores . , calculated for different radii at various epochs of the collapsing filament in realisation 2.,scaledwidth=50.0% ] realisation 2 is a repetition of the previous simulation but the gas temperature was now calculated by solving the energy equation and further , the energy of a gas particle was corrected according to eqn . ( 9 ) to account for gas cooling / heating . as with the isothermal cylindrical cloud in the previous realisation , the marginally super - critical initial distribution of gas in this case also begins to collapse in the radial direction . and , as in the previous case the inwardly propagating compressional wave assembles a dense filament of gas along the axis of the initial cylindrical cloud . shown in fig . 5 is the radial distribution of gas density within the collapsing cylindrical cloud which , though remarkably similar to that shown in fig . 2 for the isothermal filament , is relatively shallow ( plummer - like ; @xmath80 = 2 in eqn . the radial variation of the stability factor , @xmath10 , within the collapsing cylinder for this realisation has been shown in fig . this plot is qualitatively similar to the one produced for simulation 1 and shown earlier in fig . 3 . as with this former plot , gas within the collapsing cylinder in this case also always remains less than unity , @xmath96 , suggesting that gas is not in free - fall . also , the filament in this case is assembled on a timescale comparable to that in the isothermal realisation . this is probably because , despite the gas - cooling , the collapsing gas is still gravitationally sub - critical . a direct comparison of the mass line - density , @xmath97 , against the threshold for stability , @xmath77 , however , suggests otherwise . this is because , implicit in a comparison of this kind is the assumption of isothermality and uniform distribution of gas density within the post - collapse filament . on the contrary , and as is evident from fig . 7 , the gas temperature within the post - collapse filament is hardly uniform . the cold central region of the filament is cocooned by the relatively warm jacket of gas . the innermost regions of this filament acquire an average temperature of 10 k which is consistent with that reported from observations of filamentary clouds in typical star - forming regions ( see for e.g. palmeirim _ et al . _ 2013 ; arzoumanian _ et al . _ 2011 ; andr ' e _ et al . _ 2010 ) . finally , shown in fig . 8 is the history of core - formation in this filament . it is qualitatively similar to that for the isothermal filament plotted in fig . to this end , following the formation of the first core , the next set of cores form relatively quickly . also , the filament in the second realisation , where gas was allowed to cool , begins to form cores earlier than the isothermal filament and in general , cores in the latter are somewhat less massive than those in the former . this is the result of a higher mass line - density of the cooling filament relative to the isothermal filament in the earlier realisation . = 1 . ) in realisation 6 ( _ only a small volume of the medium confining the filament has been shown on this plot _ ) . fine black blobs on top of the image represent the positions of cores in the post - collapse filament at the time of termination of calculations . perturbations , in this realisation , were imposed on the length - scale @xmath68 , defined by eqn . not all density perturbations have condensed at this epoch , but those that are relatively closely spaced are separated on a scale on the order of @xmath44 pc._lower - panel : _ same as the picture shown in the upper- panel , but now for the realisation 4 that was developed with a somewhat lower resolution , though sufficient to avoid artificial fragmentation along the length of the post - collapse filament.,title="fig:",scaledwidth=50.0% ] = 1 . ) in realisation 6 ( _ only a small volume of the medium confining the filament has been shown on this plot _ ) . fine black blobs on top of the image represent the positions of cores in the post - collapse filament at the time of termination of calculations . perturbations , in this realisation , were imposed on the length - scale @xmath68 , defined by eqn . not all density perturbations have condensed at this epoch , but those that are relatively closely spaced are separated on a scale on the order of @xmath44 pc._lower - panel : _ same as the picture shown in the upper- panel , but now for the realisation 4 that was developed with a somewhat lower resolution , though sufficient to avoid artificial fragmentation along the length of the post - collapse filament.,title="fig:",scaledwidth=50.0% ] ( the second harmonic ) , in realisation 7 . again , not all density perturbations have condensed at this epoch , but those that are relatively closely spaced are separated on a scale on the order of @xmath98 pc . , scaledwidth=50.0% ] in realisation 1 , for instance , where no external perturbations were imposed , we observed that the separation between cores in the post - collapse filament was on the order of the corresponding jeans length . next , we use the same set of initial conditions as those for realisation 2 in this next set of four realisations namely , 4 , 5 , 6 and 7 , but now by imposing perturbations on the initial distribution of gas as described in 3.1 . perturbations for realisation 4 were imposed on the scale of the fragmentation length defined by eqn . ( 5 ) , which at 10 k and average density , @xmath45 g @xmath31 , the threshold for core - formation , is @xmath99 0.08 pc . we define this as the fundamental mode of perturbation . in the next realisation , numbered 5 in table 1 , we imposed the second harmonic of this perturbation . we repeat this set of two calculations with a higher number of particles , labelled 6 and 7 respectively , in table 1 . as with the gas in realisation 2 that was subject to cooling , in this case also , we observe that the initial distribution of gas collapsed radially to form a thin filament aligned with the central axis of the cylinder . shown in figs . 9 and 10 are the rendered images of the post - collapse filaments that form in realisations 6 and 7 , those that have the best resolution in this set of calculations . for comparison purposes we have also shown a plot of the post - collapse filament from realisation 4 on the lower - panel of fig . the physical parameters for this realisation are the same as those for realisation 6 , but was developed with a slightly lower , albeit sufficient resolution to prevent artificial fragmentation ( truelove criterion satisfied ) . a comparison of the plots on the two panels of fig . 9 suggests little qualitative difference between the outcomes from the respective realisations . we must , however , point out that while the satisfaction of the resolution criterion defined by eqn . ( 10 ) above is sufficient to ensure there is no artificial fragmentation , as is indeed seen in this work , it is unlikely to be sufficient to obtain convergence in calculations of energy and momentum of the collapsing gas ( federrath _ et al . _ 2014 ) . in fact , below we compare the sink - formation timescales and the magnitude of gas - velocity in the radial direction within the post - collapse filament . evidently , cores in the post - collapse filament indeed , appear like beads on a string . however , we note that not all density perturbations had collapsed to form cores when calculations were terminated . a few more would be expected to form in this filament , but the extant purpose of demonstrating a jeans - type fragmentation of this post - collapse filament is served with the set of cores that can be seen on either rendered image . finally , shown in figs . 11 and 12 are the accretion histories for cores that form in realisations 6 and 7 , respectively . there is not much difference between the timescale of fragmentation , as reflected by the epoch at which the first core appears in either realisation . although , the formation of cores after the first one , in realisation 7 is somewhat delayed and form over a timescale of 10@xmath100 yrs , as against those in case 6 which form relatively quickly after the first core appears . however , it is likely that a more significant difference between timescales would be visible for a higher harmonic . also evident from the comparative plots shown in fig . 11 is the absence of convergence in the timescale of sink - formation and the mass of sink - particles in the two realisations likely due to inadequate numerical resolution as was discussed in 3.3 . crucially , the number of sink particles remains the same irrespective of the choice of resolution which , it was argued previously is sufficient to prevent artificial fragmentation even for the realisation with the lowest realisation in this work . also , the accretion histories for sink - particles in either case are qualitatively similar . in this second case we raised the gas temperature such that the initial cylindrical distribution of gas was rendered sub - critical . we performed one realisation for this choice of @xmath4 and repeated it with a higher resolution , numbered 8 in table 1 . here we discuss this latter realisation . in contrast to the simulations discussed under case 1 , the radial collapse of the cylinder could not be sustained as it became to be squashed by the confining pressure into a spheroidal globule that was in approximate equilibrium with the external medium . the other notable feature being , this spheroidal globule was assembled on a relatively longer timescale , @xmath102 2.5 myrs ; in comparison to that observed in the simulations tested under case 1 . as with the realisations discussed earlier under case 1 , shown in fig.13 is the stability factor , @xmath10 , for this realisation . this is plot is similar to the ones shown earlier in figs . 3 and 6 in the sense that gas within the cylindrical cloud always remains sub - critical as @xmath95 . however , it differs from the former plots in one respect ; the magnitude of @xmath10 , in the outer regions , away from the central axis of the globule , shows a significant fall relative to that for the initial configuration . this , as we will demonstrate later , is because gas in the outer regions is significantly warmer than that observed in realisations grouped under case 1 . , calculated at different epochs for the cylindrical distribution of gas in realisation 8 . as was seen in figs . 2 and 5 for realisations grouped under case 1 , in this realisation as well the magnitude of @xmath10 remains unchanged and gas always remains sub - critical even as the initial cylindrical distribution is squashed into a spheroidal globule.,scaledwidth=50.0% ] myrs).,scaledwidth=50.0% ] = 2.5 myrs ) . , scaledwidth=50.0% ] shown in fig . 14 is the density averaged gas temperature within this globule and was made at the time of terminating the calculations in this case ( @xmath102 2.5 myrs ) . from this plot it is clear that though the interiors of this globule are indeed cold , comparable to the central regions of the dense filament in case 1 , gas away from the central axis is significantly warmer . the relatively large gas temperature renders it gravitationally sub - critical and so , it is unable to collapse radially to assemble a thin dense filament along the axis of the initial distribution . instead , it begins to be squashed from either side and proceeds to form an elongated spheroidal gas - body as can be seen in the rendered density image shown in fig . 15 . overlaid on this image are density contours that readily reveal evidence of sub - fragmentation to form smaller cores along the cold central region of the globule ; velocity vectors on this image indicate the direction of local gas - flow within the elongated globule . the magnitude of this velocity field has been plotted in fig . 15 . note that the magnitude of this velocity field is consistent with that derived for typical cores ( e.g. motte _ et al . _ 1998 ; jijina _ et al . _ 1999 ) . however , the local direction of the velocity field is unlikely to reveal much about the boundedness of a core which reinforces our suggestion made in an earlier work related to modelling starless cores . there it was shown , a core could remain starless despite exhibiting inwardly pointed velocity field which usually , is indicative of a gravitational collapse ( anathpindika & difrancesco 2013 ) . formation of sub - structure within this globule can be identified with the aid of density contours . this indicates the onset of sub - fragmentation . the fragments , at this epoch though , have not reached the density threshold for cores adopted in this work . prestellar cores usually appear embedded within dense filamentary clouds . understanding the morphology of these clouds is therefore crucial towards unravelling the details of the star - forming process . in the present work we explore the possibility of formation of prestellar cores along the length of a filamentary cloud . to this end we developed hydrodynamic simulations starting with a cylindrical distribution of gas and separately studied its evolution when it was gravitationally sub - critical and super - critical . we identify two possible modes of evolution : one , when the initial cylindrical distribution of gas is either marginally super - critical(@xmath103 1 ) , or even critically stable(@xmath14 1 ) , a thin dense filament forms as a result of radial collapse of the initial distribution of gas . the post - collapse filament continues to accrete gas during in - fall and prestellar cores form along the length of this filament via a jeans - like instability . in literature this mode has been identified as a tendency to form a spindle ( e.g. inutsuka & miyama 1998 ) ; and two , when the initial distribution of gas is gravitationally sub - critical(@xmath1041 ) , it tends to get squashed and forms a spheroidal globule that is in approximate pressure equilibrium with its confining medium . the relatively cooler regions of this globule show evidence for further fragmentation into smaller cores . similar examples of fragmentation within larger cores to form smaller ones has been identified in for instance , the serpens north ( duarte - cabral et al . 2010 ) , or within a number of infrared dark clouds ( e.g. wilcock et al . 2011 , 2012 ) . previous numerical work by a number of authors on the formation of filamentary clouds , and their stability against self - gravity , has culminated in at least three possible scenarios : ( i ) filamentary clouds are often seen in magneto-/hydrodynamic simulations of turbulent gas . these clouds are believed to be generated due to interaction between turbulent gas , followed by enhancement of self - gravity once the gas becomes sufficiently dense ( see for e.g. klessen & burkert 2000 ; klessen et . al . 2000 ; federrath & klessen 2013 ) . this model favours filament formation via interaction between turbulent flows so that there is no global collapse ; dense filamentary clouds instead , appear locally ( e.g. federrath et al . 2010 ; peters et al . 2012 ) , ( ii ) heitsch et al . ( 2010 , 2013 ) , have demonstrated that filamentary clouds could also form within a self - gravitating mc , and ( iii ) fragmentation of gas - sheets is also a possible mode of formation for elongated clouds . this has been demonstrated in both , the simple case of an isothermal sheet ( e.g. schmid - burgk 1967 ; myers 2009 ) , as well as in the one confined by shocks ( e.g. v ' azquez - semadeni et al . 2007 ; anathpindika 2009 ; heitsch 2010 ) . although there is little doubt about the fact that dense filaments are a result of fragmentation of larger clouds and that prestellar cores usually form within such filaments like beads strung on a wire , the process(es ) leading to the formation of a thin filament are somewhat unclear and is still a matter of debate . heitsch ( 2013 ) , for instance has suggested that filaments , during their formation , do exhibit density enhancement in their central regions , but this density enhancement is not associated with a corresponding reduction in the filament - width . smith et al . ( 2014 ) , partially agree with this conclusion but have suggested that dense filaments are likely to have a static density - profile , one that does not vary with time . in other words filament - formation is probably a one step process and once created , dense filaments remain as they are . in simulations grouped under case 1 of this work we have demonstrated the formation of a thin dense filamentary cloud via radial collapse of the initial cylindrical distribution of molecular gas . et al._(2012 ) argue that the magnitude of the externally confining pressure probably determines if whether a dense filament could possibly form . these authors have suggested that a relatively small magnitude of the external pressure is likely to support the formation of a dense filament . on the contrary , a filamentary distribution of gas is more likely to end up as a spheroidal globule when the magnitude of confining pressure is relatively large . however , the possible cause(s ) for such occurrence is(are ) not clear from their work . conclusions drawn from the simulations presented in this work are broadly consistent with those of peters _ et al._(2012 ) , but with an added qualification about the gravitational state of the distribution of gas that precedes a dense filament . this leads us to the next question , that about the propensity of a cylindrical cloud to collapse radially and assemble a dense filament . we observe that when the initial distribution of gas is gravitationally sub - critical , apart from a higher magnitude of confining pressure , the gas does not collapse in the radial direction , instead , it forms a spheroidal globule . we therefore believe , the gravitational state of the initial distribution of gas is likely to hold the key to its future evolution . we argue , a radial collapse leading to the formation of a dense filament is possible when the gas is at least critically stable(@xmath14 1 ) , initially . interestingly , the mass line - density for the post - collapse filament in this case exceeds its maximum value , @xmath77 , for stability under the assumption of a uniform temperature of 10 k , as is indeed seen in the interior of the filament ; though gas in the wings of the filament is warmer . under this assumption of isothermality and uniform density the filament could be described as gravitationally _ super - critical _ ( fischera & martin 2012 ) . although the two sets of simulations discussed in this work demonstrate that an initially sub - critical cylindrical volume of gas can not be induced to collapse in the radial direction , plots shown in figs . 2 and 5 suggest that gas in a radially collapsing cylindrical cloud does not become gravitationally super - critical ( @xmath105 1 ) . in other words , the mass line - density , @xmath106 , within the cylinder never exceeds its maximum value , @xmath77 , at any radius within the cylinder . simulations grouped under this case(case 1 ) also demonstrate that a filament need not be in radial free - fall in order to form prestellar cores along its length . interestingly , evidence for global in - fall has recently been reported in the filamentary cloud dr21 ( schneider et al . 2010 ) , while palmeirim et al . ( 2013 ) and andre et al . ( 2014 ) also report observations of some filamentary clouds that exhibit radial collapse . inutsuka & miyama ( 1997 ) , by assuming super - critical initial conditions and an isothermal gas had demonstrated such a collapse . however , this latter set of initial conditions would always be predisposed to radial collapse leading to the formation of a thin dense filament . despite their propensity to collapse radially , globally super - critical filaments are unlikely to be in free - fall . the difficulty with the idea of a radially free - falling filament is that that filament would collapse rapidly to form a thin line and it would be a challenge to explain the formation of cores in it . it is therefore difficult to reconcile the suggestion of free - falling filaments . we will revisit this point in the context of observations of filaments in the following subsection . that filamentary clouds are unlikely to experience a free - fall collapse can simply be shown by deriving an expression for the radial component of gravitational acceleration , @xmath107 , within a typical filamentary cloud . for the sake of argument , it would be safe to adopt a plummer - density distribution for gas within a filamentary cloud so that the radial distribution of thermal pressure within this cloud may be written as , @xmath108 from the plummer - density profile we have , @xmath109 combining eqns . ( 12 ) and ( 13 ) , we wind up with an expression for the gravitational acceleration as , @xmath110 then in the limit of @xmath111 , eqn . ( 14 ) reduces to @xmath112 which suggests that filamentary clouds are unlikely to have a true super - critical state , i.e. , filaments are unlikely to be in free - fall even if they do collapse radially . we have shown that prestellar cores are likely to form via a jeans - type fragmentation of the filamentary cloud and have demonstrated this by developing simulations with and without initial perturbations to the density field . this scenario is consistent with the argument presented by freundlich & jog ( 2014 ) . we also note that a plummer - like profile ( @xmath80 = 2 ) , fits the radial density distribution of the post - collapse filament very well when gas temperature is calculated by accounting for gas - cooling . on the other hand , the steeper ostriker - profile ( @xmath80 = 4 ) , fits the radial density profile for the isothermal filament . furthermore , the typical radius of the post - collapse filament in our simulations at the epoch when calculations were terminated , is on the order of @xmath180.1 pc . both these findings are consistent with corresponding values derived for filamentary clouds in the gould - belt ( e.g. arzoumanian _ _ 2011 ; malinen _ et al . _ 2012 ; palmeirim _ et al . _ 2013 ; andr ' e _ et al . although a few handful number of exceptions have been reported with relatively steep density profiles in outer regions and have slopes in excess of 2 , e.g. the b211/3(@xmath80 = 2.27 ) filament in the taurus mc that also has a relatively small radius of @xmath180.04 pc ( malinen _ et al . _ 2012 ; palmeirim _ et al . however , it remains to be seen if these post - collapse dense filaments would continue to self - gravitate and acquire even smaller radii ; investigation into this question is best deferred for a future work . we also note that purely hydrodynamic simulations such as the ones discussed in this work succeed in reproducing the typical plummer - like radial density profile for filamentary clouds ( also see smith _ et al . _ 2014 ) . this is contrary to some of the earlier suggestions about the propensity of formation of such filaments in magneto - hydrodynamic simulations ( e.g. fiege & pudritz 2000 ; tilley & pudritz 2003 ; hennebelle 2003 ) . it appears that the slope of the radial profiles of filamentary clouds is unlikely to be influenced by the presence / absence of the magnetic field . also , unlike much of the earlier work we have also derived the radial distribution of gas density and temperature for the post - collapse filament and showed that is indeed consistent with that derived observationally for typical filamentary clouds observed in nearby star - forming regions . an interesting point to which attention must be drawn is about the temporal evolution of the density profile of filamentary clouds . radial density profiles shown in figs . 2 and 5 demonstrate that the collapse of the initial cylindrical distribution of gas is accompanied with a rising central density . this is consistent with the finding of heitsch ( 2013 ) . however , we also observe that the width of the filament shrinks during the process which is inconsistent with the other finding by the same author and smith _ et al . _ as can be seen from these plots , the problem could be circumvented by adopting a value of the @xmath28 , the point where the density - profile turns - over , smaller than where it is actually seen to lie . ( 2014 ) have argued that rotation tends to stabilise a filamentary cloud against self - gravity and leads to a density - profile shallower than the ostriker - profile . on the contrary , we have shown that even a simple prescription of gas - cooling can indeed reproduce a plummer - like density distribution for the post - collapse filamentary cloud . on the other hand , upon raising the magnitude of external pressure , @xmath113 , such that the initial distribution of gas was rendered sub - critical ( because the condition of pressure balance at the gas - icm interface demands that gas temperature be raised ) , we observed , such a volume of gas was unable to sustain a collapse in the radial direction . instead , after initially shrinking along the radial direction , gas was squashed laterally and therefore tended to rebound . confined by external pressure , this gas then assembled a pressure - supported spheroidal globule on a time - scale larger than that for simulations discussed above . interestingly , arzoumanian et al . ( 2013 ) have recently identified filamentary clouds with a greater internal pressure in the ic5146 , aquila and the polaris molecular clouds . these filaments have a relatively large velocity dispersion and therefore , are gravitationally unbound . some of these in fact , exhibit signs of lateral expansion with little evidence for core - formation along the filament - axis as is usually envisaged in the _ beads - on - string _ analogy . on the basis of results obtained from our simulations we suggest , factors controlling the efficiency with which molecular gas cools would determine the propensity to form dense filamentary clouds which is simply a restatement of the stability argument presented in 2 . a larger thermal pressure could have an impact on the molecular chemistry of a cloud and therefore , on the efficiency with which it could possibly cool . however , in the absence of a complete treatment of the molecular chemistry we can not address this issue in the present work . irrespective of whether the initial cylindrical distribution of gas ends up as a dense filament , we observe that gas in the collapsing cylinder remains gravitationally sub - critical , characterised by the stability factor , @xmath114 , at all epochs ( see figs . 3 and 6 ) . this suggests , the gravitational state of the gas remains unaltered over the course of its evolution . of particular interest is the implication of this finding for dense filaments . we have seen that the formation of a dense filament is the result of a radial contraction of the initial cylindrical distribution of gas . this contraction is associated with a rise in the central density along the axis of the cylinder whence a filamentary cloud is assembled . although this is our observation in the simulations discussed here , we note that the radially collapsing gas is not in a free - fall and is therefore characterised by @xmath96 , at all radii within the volume of gas . this sequence supports the hypothesis that filamentary clouds are probably assembled via gravitational contraction during which mass is steadily accreted by the centrally located filamentary cloud . this scenario is further reinforced by observational findings of accreting filaments found in taurus and cygnus x ( goldsmith _ et al . _ 2008 ; nakamura & li 2008 ; schneider _ et al . _ 2010 ) . a comparison of the radial velocity field in the two cases discussed in this work would be instructive . for case 1 we take the sixth realisation , one of the two in this set with the highest resolution , as a representative calculation . the radial velocity field in the post - collapse filamentary cloud has been shown in fig . we note that the magnitude of velocity is relatively large in the central region and peters - off in the wings of the filament . in fact , the radial distribution of gas velocity can be approximated by a power - law of the type , @xmath115 , which is very close to that for a bound object ( @xmath116 ) . a similar plot for simulation 8 under case 2 was shown previously in fig . the difference between these plots is evident ; in this latter case , the gas close to the centre is moving radially outward where as that in the outer regions is transonic and moving inwards with approximately constant velocity ( sound speed @xmath180.55 km / s in this region ) , as a consequence of being squashed . the findings reported here from simulations in case 1 are consistent with those for typical filamentary clouds ( see for e.g. arzoumanian _ _ 2013 ) . filaments of the type seen in case 1 are conventionally described as gravitationally bound . this brings us to the next question - as to whether gravitationally bound filaments are also likely to experience a free - fall collapse . for the purpose of illustration we consider the examples of four filamentary clouds . ( i ) ic5146 in the cygnus region . this filament with a mass line - density , @xmath83 152 m@xmath84 pc@xmath50 , which is an order of magnitude in excess of the maximum mass line - density , @xmath77 , evaluated at 10 k , and therefore should be expected to be in a free - fall collapse in the radial direction . however , it seems , this is not really the case as the filament does not appear to be in radial free - fall ( arzoumanian _ et al . _ 2011 ; fischera & martin 2012 ) . ( ii ) the filamentary cloud dr21 is the next example . this is a massive filament that has two sub - filaments that are gravitationally super - critical and show signs of in - fall , but no apparent signs of radial free - fall ( schneider _ et al . this filament appears more akin with the outcome of simulations in case 2 of this work . ( iii ) next , we consider the example of the perseus mc . it has a mass line - density , @xmath83 50 m@xmath84 pc@xmath50 to 100 m@xmath84 pc@xmath50 , a good factor of 3 - 5 larger than the maximum mass line - density , @xmath77 , for the deduced isothermal temperature of @xmath1812 k ( hatchell _ et al . _ 2005 ) . again , a direct comparison of @xmath106 against @xmath77 for the perseus mc would suggest that the filament is gravitationally bound and must collapse radially . however , there is no corroborative evidence to suggest that the filament is in free - fall though several cores have been detected along its length . in fact , estimates of line - width of the optically thick c@xmath117o emission suggests that that filament is probably thermally supported . ( iv ) finally , we consider the example of the serpens mc , recent observations of which , have revealed evidence of radially infalling gas onto the dense filaments in this cloud . yet , there is no evidence to suggest that these filaments are in free - fall , however , prestellar cores have been detected along their length and appear to have formed on the scale of the local jeans length ( friesen _ et al . results from our simulations grouped under case 1 corroborate these observations . it therefore appears to us that the condition , @xmath118 , i.e. , @xmath119 , is probably necessary to induce radial collapse in a cylindrical volume of gas . such a volume of gas could become super - critical by accreting mass from its surroundings ( heitsch & hartmann 2014 ) . however , this aspect of the problem is beyond the scope of present investigation and therefore best left for a future work . significantly though , this collapsing gas is unlikely to be in free - fall and @xmath120 which is consistent with the suggestion of mckrea(1957 ) ; see also toci & galli ( 2014 ) . in fact , if indeed this gas is in radial free - fall , filamentary clouds would rapidly end up in a thin line and formation of cores in such a collapsing cloud would be difficult to reconcile . on the contrary , a steady in - fall , as gas within the collapsing filament is allowed to cool appears to be a more promising mode of evolution of filamentary clouds where gas is initially at least critically stable . as we have seen in the simulations grouped under case 1 , such filaments can indeed form cores along their length via a jeans - like fragmentation . this evolutionary scenario also circumvents the uneasy question as to why filaments with extremely high column densities , on the order of a few times 10@xmath121 @xmath122 , are not found . the likely solution perhaps is that filaments are not in radial free - fall . those that are gravitationally bound , shrink in the radial direction and eventually acquire a thermally supported configuration . if indeed filaments were to become pressure - supported during their evolutionary sequence , it would also explain why filaments across star - forming regions tend to have comparable widths , on the order of @xmath180.1 pc ( e.g. arzoumanian _ et al . _ 2011 ) . at 10 k , the temperature typically found in filamentary clouds and an average density @xmath1810@xmath123 g @xmath31 , a representative density for prestellar cores , the thermal jeans length , @xmath99 0.075 pc , which is comparable to the typical radius of filamentary clouds . we make a similar observation in simulations grouped under case 1 . for instance , figs . 2 and 5 , where the radial density profile for the post - collapse filament in simulations 1 and 2 have been plotted , demonstrate that the filament radius , @xmath124 , at the epoch when calculations were terminated in respective simulations . although , we note that magnetic field could also have a key role to play in the evolution of filamentary clouds . the point though , is beyond the scope of this article . in this work we have demonstrated that an initial cylindrical distribution of molecular gas can indeed collapse radially to assemble a thin dense filament along its axis when the magnitude of confining pressure is relatively small such that gas is initially at least critically stable . in this case a radial collapse ensues and the gas is able to cool on a relatively short timescale . equivalently , it may also be argued that a collapse of this kind is likely when the line mass exceeds its critical value required to maintain stability . however , there is a caveat to this argument . this argument is valid only if gas is assumed to be isothermal and has uniform density as was the case with our initial cylindrical distribution of gas . for a typical dense filament this argument can not be applied since its density and temperature distribution is far from uniform . we argue , a filamentary cloud even if gravitationally bound , is unlikely to be in radial free - fall , instead it is likely to contract in the radial direction and consequently , the density along its central axis steadily rises as gas is accreted on to it . we argue that this evolutionary sequence of a typical filamentary cloud could possibly explain - ( i ) the dearth of filamentary clouds that have extremely high extinction , and ( ii ) why the width of filamentary clouds shows little variation across star - forming regions . we suggest that filamentary clouds are likely to be thermally - supported and their radii(width ) are likely to be on the order of the local thermal jeans length ( twice the local thermal jeans length ) . a plummer - like density profile appears to fit very well the radial distribution of gas density in the post - collapse filament so that magnetic field may not be necessary to generate filaments with a relatively shallow slope , as has been suggested in the past ( e.g. fiege & pudritz 2000 ) . we also demonstrate that prestellar cores are likely to form via a jeans - like instability while it is accreting gas . it also appears , cores are likely to form along the length of a dense filament if it is at least ( gravitationally)critically stable . processes via which filaments acquire mass from their parent cloud are therefore likely to hold the key to determining the efficiency of star - formation ( see also tafalla & hacar 2014 ) on the other hand , when the magnitude of the confining pressure was increased such that the initial distribution of gas was rendered gravitationally sub - critical , we observed that a radial collapse was unsustainable . instead , the gas distribution was squashed from either side and demonstrated a propensity to expand laterally . the result was an elongated spheroidal globule which then showed signs of sub - fragmentation . such objects are more likely to form out of dynamic interactions between gas flows within molecular clouds . the upshot therefore is that potential star - forming dense filaments are likely to be found in molecular clouds where ambient conditions contrive to render the volume of gas immediately preceding the dense filament critically stable , at least ( i.e. mass line - density approximately equal to its maximum stable value ) . the possible effect of the ambient conditions on the chemistry within a molecular cloud must directly impact the gas thermodynamics and therefore , the dynamical stability of that cloud . we will examine this issue in a future contribution . the authors thank an anonymous referee for a helpful and prompt report . j.f . acknowledges support from the indo - french centre for the promotion of advanced research ( ifcpar / cefipra ) through a raman - charpak fellowship over the course of which this project was conceived . anathpindika , s , 2009 , a&a , 504 , 437 anathpindika , s & di francesco , j. , 2013 , mnras , 430 , 1854 andr ' e , ph . , 2010 , a&a , 518 , l102 andr ' e , ph . , di francesco , j. , ward - thompson , d. , inutsuka , s. , pudritz , r & pineda , j. , astroph 1312.6232 , to appear in _ protostars and planets vi , university of arizona press ( 2014 ) _ , eds . h. beuther , r. klessen , c. dullemond , th . henning arzoumanian , d. , andr ' e , ph . , peretto , n & k@xmath125nyves , v. , 2013 , a&a , 553 , 119a arzoumanian , d. , andr ' e , ph . , didelon , p. , k@xmath125nyves , v. , schneider , n. , menshchikov , a _ et al . _ , 2011 , a&a , 529 , l6 bastien , p. , 1983 a&a , 119 , 109 bastien , p. , arcoragi , j. , benz , w. , bonnell , i & martel , h. , 1991 , apj , 378 , 255 bate , m. r. , bonnell , i. a & price , n. m. , 1995 , mnras , 277 , 362 bate , m. r. & burkert , a. , 1997 , mnras , 288 , 1060 bate , m. r. , bonnell , i. a. , bromm , v. , 2003 , mnras , 339 , 577 bate , m. r. , 2012 , mnras , 419 , 3115 duarte - cabral , a. , fuller , g. a. , peretto , n. , hatchell , j. , ladd , e. f. , buckle , j. , richer , j & graves , s. , f. , 2010 , a&a , 519 , a27 federrath , c. , roman - duval , j. , klessen , r. , s. , schmidt , w & mac low , -m . , m. , 2010a , a&a , 512 , a81 federrath , c. , banerjee , r. , clark , p. c. & klessen , r. s. , 2010b , apj , 713 , 269 federrath , c & klessen , r. , 2012 , apj , 761 , 156 fischera , j & martin , p. g. , 2012 , a&a , 542 , a77 federrath , c & klessen , r. , 2013 , apj , 763 , 51 federrath , c. , schr@xmath126n , m. , banerjee , r & klessen , r. , 2014 , apj , 790 , 128 freundlich , j. , jog , c. j. & combes , f. , 2014 , _ to appear in a&a _ , astroph 1402.0977 friesen , r. k. , medeiros , l. , schnee , s. , bourke , t. l. , di francesco , j. , gutermuth , r. & myers , p. , 2013 , mnras , 436 , 1513 glover , s. c. o. , federrath , c. , mac low , m. m. & klessen , r. s. , 2010 , mnras , 404 , 2 goldsmith , p. f. , heyer , m & narayanan , g. , 2008 , apj , apj , 680 , 428 hacar , a. , tafalla , m. , kauffmann , j. , & kova@xmath127s , a. 2013 , a&a , 554 , a55 hatchell , j. , richer , j. s. , fuller , g. , qualtrough , c. j. , ladd , e. f & chandler , c. j. , 2005 , a&a , a&a , 440 , 151 heitsch , f. , ballesteros - paredes , j & hartmann , l. , 2010 , apj , 704 , 1735 heitsch , f. , 2013 , apj , 769 , 115 heitsch , f & hartmann , l. , 2014 , mnras , 443 , 230 hennebelle , p. , 2003 , a&a , 397 , 381 higuchi , a. , chibueze , j.o . , habe , a. , tasker , e. , takahira , k & takaino , s. , 2014,_to appear in apj _ , astroph 1403.4734 hubber , d. , goodwin , s & whitworth , a. p. , 2006 , a&a , astroph0512247 hubber , d. , batty , c. , mcleod , a & whitworth , a.p . , 2011 , a&a , 529 , 28 inutsuka , s & miyama , s. , 1992 , apj , 388 , 392 inutsuka , s & miyama , s. , 1997 , apj , 480 , 681 jackson , j. m. , finn , s. c. , chambers , e. t. , et al . 2010 , apj , 719 , l185 jijina , j. , myers , p. c & adams , f. c. , 1999 , apjss , 125 , 161 jog , c. j. , 2013 , mnras , 434 , l56 kainulainen , j. , alves , j. , beuther , h. , henning , t & schuller , f. , 2011 , a&a , 536 , a48 kainulainen , j. , ragan , s. e. , henning , t & stutz , a. , 2013 , a&a , 557 , a120 kirk , h. , myers , p. c. , bourke , t. l _ et al . _ , 2013 , apj , 766 , 115 kirk , j. m. , ward - thompson , d. , palmeirim , p. , et al . 2013 , mnras , 432 , 1424 klessen , r & burkert , a. , 2000 , apjss , 128 , 287 klessen , r. , heitsch , f & mac low , m. , m. , 2000 , apj , 535 , 887 koyama , h. & inutsuka , s. , i. , 2002 , apj , 564 , l97 malinen , j. , juvela , m. , rawlings , m. g. , ward - thompson , d. , palmeirim , p & andr ' e , ph . , 2012 , a&a , 544 , a50 mckrea , w. h. , 1957 , mnras , 117 , 562 menshikov , a. , andr ' e , p. , didelon , d _ et al . _ , 2010 , a&a , 518 , l103 motte , f. , andr ' e , ph & neri , r. , 1998 , a&a , 336 , 150 myers , p. c. , 2009 , apj , 700 , 1609 myers , p. c. , 2013 , apj , 764 , 140 nakamura , f & li , z. , 2008 , apj , 687 , 854 nutter , d & ward - thompson , d. , 2007 , mnras , 374 , 1413 nutter , d. , kirk , j. m. , stamatellos , d & ward - thompson , d. , 2008 , mnras , 384 , 755 nguyn - lng , q. , motte , f. , hennemann , m. , hill , t. , rygl , k. l. j. , schneider , n _ et al . _ , 2011 , a&a . , astroph 1109.3584 ostriker , j , 1964 , apj , 140 , padoan , p & nordlund , 2011 , , apj , 730 , 40 palmeirim , p. , andr ' e , ph . , hirk , j. , ward - thompson , d. , arzoumanian , d__et al . _ _ , 2013 , a&a , 550 , a38 peters , t. , schleicher , d. , klessen , r. s. , banerjee , r. , federrath , c. , smith , r & sur , s. , 2012 , apjl , 760 , l28 pon , a. , johnstone , d & heitsch , f. , 2011 , apj , 740 , 88 price , d. , 2008 , jcoph , 227 , 10040 price , d. & bate , m.r . , 2008 , mnras , 385 , 1820 price , d. & bate , m.r . , 2009 , mnras , 398 , 33 ragan , s. e. , henning , th . , tackenberg , j. , beuther , h. , johnston , k. , kainulainen , j & linz , h. , 2014 , _ to appear in a&a _ , astroph 1403.1450 recchi , s. , hacar , a & palestini , a. , _ to appear in a&a _ , 1408.0007 schmid - burgk , j. , 1967 , apj , 149 , 727 schneider , n. , csengeri , t. , bomtemps , s. , motte , f. , simon , r. , hennebelle , p. , federrath , c & klessen , r. , 2010 , a&a , 520 , a49 schneider , n. , csengeri , t. , hennemann , m. _ et al . _ , 2012 , a&a , 540 , l11 smith , r. j. , glover , s. c. o. & klessen , r. s , 2014 , _ astroph . sturges , h. a. , 1926 , _ jour . of american stat . _ , 65 - 66 tafalla , m. & hacar , a. , 2014 , astroph 1412.1083 tilley , d. a. & pudritz , r. e. , 2003 , apj , 593 , 426 toci , c & galli , d. , _ to appear in mnras _ , astroph 1410.6091 truelove , j. k. , klein , r. i. , mckee , c. f. , holliman , j. h.ii , howell , l. k & greenhough , j. a. , 1997 , apj , 489 , l179 v ' azquez - semadeni , e. , g ' omez , g. , katharina jappsen , a. , ballesteros - paredes , j. , gonz ' alez , r. f & klessen , r. , 2007 , apj , 657 , 870 whitworth , a. p. , bhattal , a. s. , chapmas , s. , disney , m & turner , j. a. , 1994 , a&a , 290 , 421 wilcock , l. , kirk , j. m. , stamatellos , d. , ward - thompson , d. , whitworth , a. , battersby , c _ et al . _ , 2011 , a&a , 526 , a159 wilcock , l. , ward - thompson , d. , kirk , j. m. , stamatellos , d. , whitworth , a. , elia , d. , fuller , g _ et al . _ , 2012 , mnras , 424 , 716
|
it is now widely accepted that dense filaments of molecular gas are integral to the process of stellar birth and potential star - forming cores often appear embedded within these filaments . although numerical simulations have largely succeeded in reproducing filamentary structure in dynamic environments such as in turbulent gas and while analytic calculations predict the formation of dense gas filaments via radial collapse , the exact process(es ) that generate / s such filaments which then form prestellar cores within them , is unclear . in this work
we therefore study numerically the formation of a dense filament using a relatively simple set - up of a uniform - density cylinder in pressure equilibrium with its confining medium . in particular , we examine if its propensity to form a dense filament and further , to the formation of prestellar cores within this filament bears on the gravitational state of the initial volume of gas .
we report a radial collapse leading to the formation of a dense filamentary cloud is likely when the initial volume of gas is at least critically stable ( characterised by the approximate equality between the mass line - density for this volume and its maximum value ) .
though self - gravitating , this volume of gas , however , is not seen to be in free - fall .
this post - collapse filament then fragments along its length due to the growth of a jeans - like instability to form prestellar cores like _ beads on a string_. we suggest , dense filaments in typical star - forming clouds classified as gravitationally super - critical under the assumption of : ( i ) isothermality when in fact , they are not , and ( ii ) extended radial profiles as against one that is pressure - truncated , thereby causing significant over - estimation of their mass line - density , are unlikely to experience gravitational free - fall .
the radial density and temperature profile derived for this post - collapse filament is consistent with that deduced for typical filamentary clouds mapped in recent surveys of nearby star - forming regions .
this profile is also in agreement with a plummer - like density profile . for an isothermal filament
though , the density profile is much steeper , consistent with the classic density profile suggested by ostriker ( 1964 ) . on the other hand , increasing the magnitude of the confining pressure such that the initial volume of gas is rendered gravitationally sub - critical is unable to collapse radially and tends to expand laterally which could possibly explain similar gas filaments found in recent surveys of some molecular clouds .
simulations were performed using smoothed particle hydrodynamics ( sph ) and convergence of results is demonstrated by repeating them at higher resolution .
unlike some of the earlier work reported in literature , here we calculate gas temperature by solving the sph energy equation and allow it to cool according to a cooling function .
gravitation hydrodynamics ism : structure ism : clouds prestellar cores
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of wave scattering in disordered media is an active field of research , stimulated both by innovative applications in imaging and sensing @xcite and by fundamental questions in mesoscopic physics @xcite . in the last few years , the possibility to control the propagation of optical waves in complex media in the multiple scattering regime has been demonstrated using wavefront shaping techniques @xcite . this breakthrough offers new perspectives for imaging and communication through complex media @xcite . the initial schemes make use of optimization techniques requiring intensity measurements in the transmitted speckle , which in terms of practical applications is a serious drawback . finding a way to control the transmission and focusing of light through a strongly scattering medium from measurements of the reflected speckle only is an issue of tremendous importance . progresses have been made recently by taking advantage of the memory effect @xcite , with imaging capabilities limited to relatively small optical thicknesses . nevertheless , the connection between the reflected and the transmitted speckle patterns generated by a disordered medium in the multiple scattering regime has not been addressed theoretically so far . in this paper , we make a step in this direction by studying theoretically and numerically the statistical correlation between the intensities measured in the transmitted and the reflected speckle patterns . the spatial intensity correlation function @xmath0 is defined as @xmath1 where the notation @xmath2 denotes a statistical average over disorder and @xmath3 is the intensity fluctuation . this correlation function has been extensively studied in the context of wave scattering and mesoscopic physics @xcite . theoretical approaches often make use of the canonical slab or waveguide geometries ( for a review see @xcite and references therein ) , where either transmitted or reflected intensity is considered , or consider point sources in an infinite or open medium and compute intensity correlations for two points inside or outside the medium @xcite . it seems that the intensity correlation function for two points lying on different sides of a slab medium has not been studied , and that the existence of a correlation has only been mentionned through passing @xcite . in this work , we study the correlation between the intensities in the input and output planes of a strongly scattering slab , as sketched in fig . [ schema ] . using numerical simulations and analytical calculations , we show that for optically thick slabs a correlation persists and takes negative values . moreover , at smaller optical thicknesses , short and long - range correlations coexist , with relative weights that depend on the optical thickness . we believe these results to be a step forward for the control of transmission through strongly scattering media , thus finding applications in sensing , imaging and information transfer . the spatial intensity correlation in a speckle pattern can be split into three contributions , historically denoted by @xmath4 , @xmath5 and @xmath6 @xcite , as follows : @xmath7 the first term @xmath4 corresponds to the gaussian statistics approximation for the field amplitude , and is a short - range contribution , whose width determines the average size of a speckle spot @xcite . @xmath5 and @xmath6 are non - gaussian long - range correlations that decay on much larger scales @xcite . in the diffusive regime , and for two observation points lying on the same side of the scattering medium , these three contributions have different weights such that @xmath8 . in the reflection / transmission configuration considered here , and at large optical thickness , we show that an intensity correlation persists and is dominated by @xmath5 because of the short - range behavior of @xmath4 . in this case , the long - range character of @xmath5 is confered by its algebraic decay with respect to the distance between the two observation points . moreover , this correlation is negative , a result that may have implications in the context of wave control by wavefront shaping . for smaller optical thicknesses , we also show that a crossover can be found between regimes dominated by @xmath4 and @xmath5 , respectively . the paper is organized as follows . in section [ correlation_large_b ] , we study the reflection / transmission correlation function at large optical thicknesses . first , we use numerical simulations to compute the correlation function without approximation and to describe its main features . second , we present the analytical calculation of the @xmath5 contribution to the reflection / transmission correlation function in the multiple scattering regime , for both two - dimensional and three - dimensional geometries , and show that @xmath5 is the leading contribution at large optical thicknesses . in section [ correlation_small_b ] , we study the correlation function at small optical thicknesses , a regime in which the @xmath4 contribution dominates . in section [ stats ] , as a consequence of the negative value of the correlation that is found in the multiple scattering regime , we describe some peculiarities of the statistical distribution of reflected / transmitted intensities . finally , in section [ conclusion ] we summarize the main results and discuss some implications for the control of wave transmission through disordered media . in this section we present exact numerical simulations of wave scattering in the multiple scattering regime . we restrict ourselves to a 2d geometry for the sake of computer memory and time . we consider a slab of scattering material , characterized by its thickness @xmath9 and its transverse size @xmath10 ( we keep @xmath11 in order to avoid finite size effects ) , as depicted in fig . [ schema ] . our purpose is the calculation of the correlation function @xmath12 , where @xmath13 is a point located on the input surface ( reflection ) and @xmath14 is a point located on the output surface ( transmission ) . [ c]@xmath10 [ c]@xmath9 [ c]@xmath15 [ c]@xmath13 [ c]@xmath14 [ c]@xmath16 and transverse size @xmath10 is illuminated from the left by a monochromatic plane - wave at normal incidence . the correlation between the reflected and transmitted speckle patterns is characterized by the correlation function between the intensities at points @xmath13 ( reflection ) and @xmath14 ( transmission).,title="fig : " ] to proceed , we use the coupled dipoles method @xcite to calculate numerically the intensity in the transmitted and reflected speckle patterns . repeating the calculations for a large number of configurations of disorder ( positions of scatterers ) allows us to compute statistics . the system contains @xmath17 randomly distributed non - overlapping point scatterers , and is illuminated by a plane wave from the left at normal incidence . we deal with te - polarized waves with an electric field oriented along the invariance axis of the system ( scalar waves ) . the resonant point scatterers are described by their electric polarizability @xmath18 where @xmath19 is the resonance frequency and @xmath20 the linewidth . this specific form of the polarizability fulfils the optical theorem ( i.e.energy conservation ) . from the polarizability the scattering cross section @xmath21 and the scattering mean - free path @xmath22^{-1}$ ] can be deduced , where @xmath23 is the number density of scatterers . in the following , we consider scatterers at resonance ( @xmath24 ) in order to reach large optical thicknesses with a reasonable number of scatterers ( typically a few hundreds ) . in the coupled dipoles formalism , the exciting field @xmath25 on scatterer number @xmath26 is written as @xcite @xmath27 where @xmath28 is the 2d free - space green function given by @xmath29 , @xmath30 being the hankel function of first kind and order zero . equation ( [ linear_system ] ) defines a set of @xmath17 linear equations that are solved by a standard matrix inversion procedure . once the exciting field is known on each scatterer , the field @xmath31 and the intensity @xmath32 at any position @xmath33 inside or outside the scattering medium can be calculated by a direct summation , using @xmath34 we have carried out numerical simulations in the multiple scattering regime with an optical thickness @xmath35 . this choice of optical thickness is limited by the number of configurations that can be calculated in a reasonable computer time in order to get a sufficiently accurate statistics to compute averaged values ( requiring typically @xmath36 configurations ) . the correlation function @xmath37 obtained from numerical calculation is plotted in fig . [ c_2_large_b ] ( red solid line ) versus the lateral shift @xmath15 between the observation points in the reflected and transmitted speckles ( see the geometry in fig . [ schema ] ) . [ c]@xmath38 [ c]@xmath39 [ l]@xmath40 [ l]@xmath41 given by the numerical simulations ( red solid line ) and analytical correlation @xmath5 given by eq . ( blue dashed line ) . multiple scattering regime with @xmath42 and @xmath43.,title="fig : " ] surprisingly an intensity correlation subsists even for large optical thicknesses ( multiple scattering regime ) . moreover the reflection / transmission correlation function at large optical thickness takes a negative value around @xmath44 . this means that @xmath45 , showing that the probability to have a dark spot in the transmitted speckle in lateral coincidence with a bright spot in the reflection speckle ( and _ vice - versa _ ) should be substantial . this property , that might have implications for the control of wave transmission by wavefront shaping , is investigated more precisely in section [ stats ] . to get more insight on the numerical result presented above , we present the calculation of the @xmath5 contribution to the intensity correlation function for scalar waves in both two - dimensional ( 2d ) and three - dimensional ( 3d ) geometries . intuitively , we expected @xmath5 to be the leading contribution at large optical thicknesses , since @xmath6 is always negligible compared to @xmath5 and @xmath4 vanishes exponentially with the optical thickness in this particular reflection / transmission configuration . for a dilute system such that @xmath46 , where @xmath47 with @xmath48 the speed of light and @xmath49 the wavelength in vacuum , the intensity correlation functions can be calculated analytically using a perturbative approach . since the calculation for reflection - reflection or transmission - transmission correlations is detailed in textbooks or review articles @xcite , we do not give all details here but rather focus on the specificity of the reflection - transmission geometry . for the analytical derivation , we consider that the transverse size @xmath10 of the slab is infinite . to compute the correlation function @xmath5 , we first need to compute the average intensity . the starting point is the bethe - salpeter equation that reads @xcite @xmath50 in this equation , @xmath51 is the average green function that links the average field @xmath52 to a source dipole @xmath53 located at position @xmath54 via @xmath55 the operator @xmath56 is the irreducible vertex that contains all multiple scattering sequences connecting ( @xmath57 , @xmath58 ) to ( @xmath59 , @xmath60 ) . the exact calculation of this complex object is out of reach , but an approximate expression to first - order in the small parameter @xmath61 and for independent scattering , known as the ladder approximation , can be derived @xcite . omitting the frequency dependence for simplicity in the following , it reads @xmath62 where @xmath63 plugging eq . ( [ vertex ] ) into eq . ( [ bethe - salpeter ] ) yields @xmath64 physically , this closed equation means that only contributions for which @xmath65 and @xmath66 follow the same scattering path have a significant weight in the average intensity ( all cross terms vanish ) . the first term @xmath67 in eq . is the ballistic ( or coherent ) component of the average intensity and the second term @xmath68 is the diffuse part . in order to get explicit expressions for these two quantities , we need to compute the average green function , or equivalently the average field . as a consequence of the dyson equation @xcite , in the independent scattering limit , the average green function obeys a propagation equation in an effective homogeneous medium , defined by an effective wavevector @xmath69 . we have @xmath70 where @xmath71 to first order in @xmath61 and @xmath72 is the coordinate along the direction normal to the slab . the ballistic intensity is readily deduced : @xmath73.\ ] ] regarding the diffuse intensity @xmath74 , we can rewrite eq . in the following way @xmath75{\mathrm{d}}{\boldsymbol{\mathbf{r}}}'.\ ] ] an analytical expression of the diffuse intensity can be obtained in the diffusive limit where @xmath76 . making use of the translational invariance of the medium , the fourier transform of eq . reads @xmath77\ ] ] where @xmath78 as we consider large distances , we can perform a second order taylor expansion of @xmath79 for @xmath80 which leads in the real space to a diffusion - type equation @xmath81 where @xmath82 is the space dimension . solving eq . in a slab geometry , we obtain @xmath83 -i_0 d\exp\left(-\frac{z}{\ell}\right)\end{gathered}\ ] ] where @xmath84 is the extrapolation length needed to account for the boundary conditions at the input and exit surfaces of the slab @xcite . for a dilute index - matched slab , its expression is given by @xcite @xmath85 this expression of the diffuse intensity @xmath68 is _ a priori _ valid for distances @xmath72 such that @xmath86 , for which the diffusion approximation holds , but it surprisingly gives reasonably reliable results even for @xmath87 . adding the ballistic term , it also gives reliable results for the full average intensity @xmath88 even for relatively small optical thicknesses . the intensity correlation function is a fourth order correlation in terms of field amplitude . physically , a correlation is created when the two pairs of fields that constitute the intensities in the correlation function share a common history in the scattering process . regarding @xmath5 , the crossing occurs during the propagation of the intensities inside the system and is described by a complex object known as a hikami vertex @xcite , and denoted by @xmath89 in the following . the propagation of the intensity between the slab surfaces and the crossing is described by the ladder operator , denoted by @xmath9 in the following . the expression of @xmath5 is given by @xcite @xmath90}\int { \mathrm{d}}{\boldsymbol{\mathbf{r}}}_{1}{\mathrm{d}}{\boldsymbol{\mathbf{r}}}_{2}{\mathrm{d}}{\boldsymbol{\mathbf{r}}}_{3}{\mathrm{d}}{\boldsymbol{\mathbf{r}}}_{4 } \\ \times{\mathrm{d}}{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{1 } { \mathrm{d}}{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{2 } { \mathrm{d}}{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{3 } { \mathrm{d}}{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{4 } \lvert{\left\langle}g({\boldsymbol{\mathbf{r}}}_r,{\boldsymbol{\mathbf{r}}}_{2}){\right\rangle}\rvert^2\lvert { \left\langle}g({\boldsymbol{\mathbf{r}}}_t,{\boldsymbol{\mathbf{r}}}_{4}){\right\rangle}\rvert^2 \\ \times l({\boldsymbol{\mathbf{r}}}_{2},{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{2 } ) l({\boldsymbol{\mathbf{r}}}_{4},{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{4})h({\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{1},{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{2},{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{3},{\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{4 } ) \\ \times l({\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{1},{\boldsymbol{\mathbf{r}}}_{1 } ) l({\boldsymbol{\mathbf{{\boldsymbol{\mathbf{\rho}}}}}}_{3},{\boldsymbol{\mathbf{r}}}_{3 } ) \lvert { \left\langle}e({\boldsymbol{\mathbf{r}}}_{1}){\right\rangle}\rvert^2 \lvert{\left\langle}e({\boldsymbol{\mathbf{r}}}_{3}){\right\rangle}\rvert^2\end{gathered}\ ] ] which can be rewritten diagrammatically in the following form : @xmath91(0,-9)(6,-9 ) } { \psline[linewidth=0.5,linestyle = dashed](0,-3)(6,-3 ) } { \psline[linewidth=0.5,linestyle = dashed](0,9)(6,9 ) } { \psline[linewidth=0.5,linestyle = dashed](0,3)(6,3 ) } { \psframe[fillstyle = none , dimen = middle](6,-9)(18,-3 ) \setcounter{tempa}{18 } \addtocounter{tempa}{6 } \divide \value{tempa } by 2 \setcounter{tempb}{-3 } \addtocounter{tempb}{-9 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psframe[fillstyle = none , dimen = middle](6,3)(18,9 ) \setcounter{tempa}{18 } \addtocounter{tempa}{6 } \divide \value{tempa } by 2 \setcounter{tempb}{9 } \addtocounter{tempb}{3 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psframe[fillstyle = none , dimen = middle](30,-9)(42,-3 ) \setcounter{tempa}{42 } \addtocounter{tempa}{30 } \divide \value{tempa } by 2 \setcounter{tempb}{-3 } \addtocounter{tempb}{-9 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psframe[fillstyle = none , dimen = middle](30,3)(42,9 ) \setcounter{tempa}{42 } \addtocounter{tempa}{30 } \divide \value{tempa } by 2 \setcounter{tempb}{9 } \addtocounter{tempb}{3 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psframe[fillstyle = vlines , dimen = middle](18,-9)(30,9 ) \setcounter{tempa}{30 } \addtocounter{tempa}{18 } \divide \value{tempa } by 2 \setcounter{tempb}{9 } \addtocounter{tempb}{-9 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){h } } \rput[bl](49,5){${\boldsymbol{\mathbf{r}}}_r$ } \rput[bl](49,-7){${\boldsymbol{\mathbf{r}}}_t$ } { \psline[linewidth=0.5](42,-9)(48,-6 ) } { \psline[linewidth=0.5](42,9)(48,6 ) } { \psline[linewidth=0.5](42,-3)(48,-6 ) } { \psline[linewidth=0.5](42,3)(48,6 ) } \end{dddiag}\end{gathered}\ ] ] where the hatched box is the hikami vertex and the other boxes are the ladder operators . thick lines represent the average green function and thick dashed lines stand for the average electric field . the ladder operator is defined as @xmath92 and is represented diagrammatically as @xmath93(1,-3)(7,-3 ) } { \psline[linewidth=0.5](1,3)(7,3 ) } { \psline(1,3)(1,-3 ) } { \pscircle(1,-3){1 } } { \pscircle(1,3){1 } } { \psline(7,3)(7,-3 ) } { \pscircle(7,-3){1 } } { \pscircle(7,3){1 } } \end{ddiag}+ \begin{ddiag}{14 } { \psline[linewidth=0.5](1,-3)(7,-3 ) } { \psline[linewidth=0.5](1,3)(7,3 ) } { \psline[linewidth=0.5](7,-3)(13,-3 ) } { \psline[linewidth=0.5](7,3)(13,3 ) } { \psline(1,3)(1,-3 ) } { \psline(7,3)(7,-3 ) } { \psline(13,3)(13,-3 ) } { \pscircle(1,-3){1 } } { \pscircle(1,3){1 } } { \pscircle(7,-3){1 } } { \pscircle(7,3){1 } } { \pscircle(13,-3){1 } } { \pscircle(13,3){1 } } \end{ddiag}+\ldots\ ] ] where circles and thick horizontal solid lines represent scattering events and average green functions , respectively , the top line standing for the field amplitude @xmath65 and the bottom line for its complex conjugate @xmath66 . thin vertical solid lines link scattering events involving identical scatterers . an analytical expression of the ladder operator can be obtained in the diffusive limit previously used for the computation of the average intensity . making use of the translational invariance along the direction of the slab interface , the fourier transform of the ladder operator with respect to transverse variables can be obtained from eq . , and reads @xmath94\sinh[k(l+z_0-z_>)]}{\sinh[k(l+2z_0)]}\end{gathered}\ ] ] where @xmath95 , @xmath96 . the hikami box can also be calculated in the limit @xmath97 , and its expression reduces to @xcite @xmath98 where @xmath99 to have an explicit expression of @xmath5 , we first compute the integrals involving @xmath59 and @xmath57 using @xmath100 as we deal with large optical thicknesses , we can replace the average intensity by its diffuse component . regarding the integrals involving @xmath60 and @xmath58 , we assume that the ladder operators vary slowly at the scale of the scattering mean - free path @xmath101 . this amounts to replacing @xmath60 by @xmath13 and @xmath58 by @xmath14 in the ladder positions . we end up with an explicit expression of the @xmath5 contribution to the correlation function , given by @xmath102 \\ \times\frac{l^2}{(l+2z_0)^2}\frac{\sinh(qz_0/l)^2}{\sinh[q(1 + 2z_0/l)]^2}\end{gathered}\ ] ] where @xmath103 ^ 2 \times\begin{dcases } \frac{16}{\pi k_0\ell}\frac{\cos(q\delta r / l)}{q } & \text{(2d ) , } \\ \frac{27}{k_0 ^ 2\ell l}{\operatorname{j}_0}(q\delta r / l ) & \text{(3d ) . } \end{dcases}\ ] ] the expression of the @xmath5 contribution to the reflection / transmission correlations function , together with its comparison to full numerical simulations , is the main result of this paper . the dependence of the amplitude on the system size @xmath9 is approximatively @xmath104 , with @xmath105 the dimension of space . this shows that even at large optical thicknesses , a correlation subsists between the intensities measured in the transmitted and the reflected speckles , and that this correlation is dominated by a contribution of the @xmath5 type . it is important to keep in mind that this expression has been derived in the framework of the diffusion approximation ( in particular bulk average green functions have been used ) . its validity in the geometry considered here where both reflected and transmitted intensities contribute is checked by comparison to full numerical simulations in fig . [ c_2_large_b ] . it is interesting to note that the analytical and numerical calculations are in very good quantitative agreement , showing that the diffusion approximation , used to derive the analytical results , is very accurate even for an optical thickness @xmath42 ( that is not very large ) and a geometry involving a reflected intensity ( that always involves short scattering paths ) . we have seen that @xmath5 decays algebraically with the system size @xmath9 . thus it is interesting to analyse the behavior of the full correlation function at smaller optical thicknesses , and in the crossover between the multiple and single scattering regimes . this problem can be addressed numerically , by using the numerical method described previously in various scattering regimes , and is the subject of the following section . we consider a small optical thickness @xmath106 , corresponding to the single - scattering regime . the reflection / transmission intensity correlation function calculated numerically in this regime is shown in fig . [ c_2_small_b ] ( red solid line ) . the large oscillations observed on a scale on the order of the wavelength are not described by the @xmath5 contribution , and are expected to be a signature of the short - range @xmath4 contribution . to check the validity of this assumption , we have to compute the @xmath4 contribution to the correlation function . at small optical thickness , a quantitative calculation would require to go beyond the diffusion approximation and to account properly for the boundary conditions @xcite . since our purpose in this section is only to support qualitatively the analysis of the general trends observed in the numerical simulations using a simple model , we keep using the diffusion approximation to estimate the @xmath4 contribution . [ c]@xmath38 [ c]@xmath39 [ l]@xmath107 [ l]@xmath41 given by the numerical simulations ( red solid line ) and analytical correlation including all terms @xmath107 ( blue dashed line ) . single scattering regime with @xmath106 and @xmath43.,title="fig : " ] in the scattering sequences picture , the @xmath4 correlation is created by interchanging the amplitudes between two independent ladders at the last scattering event . the @xmath4 contribution reads @xmath108 in terms of diagrams , eq . can be rewritten as @xmath109(31,-1){${\boldsymbol{\mathbf{r}}}_r$ } \rput[bl](25,-1){${\boldsymbol{\mathbf{r}}}_t$ } { \psframe[fillstyle = none , dimen = middle](6,-9)(18,-3 ) \setcounter{tempa}{18 } \addtocounter{tempa}{6 } \divide \value{tempa } by 2 \setcounter{tempb}{-3 } \addtocounter{tempb}{-9 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psframe[fillstyle = none , dimen = middle](6,3)(18,9 ) \setcounter{tempa}{18 } \addtocounter{tempa}{6 } \divide \value{tempa } by 2 \setcounter{tempb}{9 } \addtocounter{tempb}{3 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psline[linewidth=0.5](18,-9)(30,0 ) } { \psline[linewidth=0.5](18,9)(30,0 ) } { \psline[linewidth=0.5](18,-3)(24,0 ) } { \psline[linewidth=0.5](18,3)(24,0 ) } { \psline[linewidth=0.5,linestyle = dashed](0,-9)(6,-9 ) } { \psline[linewidth=0.5,linestyle = dashed](0,-3)(6,-3 ) } { \psline[linewidth=0.5,linestyle = dashed](0,9)(6,9 ) } { \psline[linewidth=0.5,linestyle = dashed](0,3)(6,3 ) } \end{dddiag}.\ ] ] in eq . , the scattering of both pairs of fields from points @xmath59 to @xmath60 and from @xmath57 to @xmath58 , respectively , is described by a ladder propagator . the mixing of amplitudes at the last scattering event is represented by four different average green functions . the integrals in eq . can be factorized , leading to @xmath110 where @xmath111 equation shows that @xmath4 is the square of the scattered field correlation function . indeed , @xmath4 can also be seen as the correlation that would be observed for a field with gaussian statistics , for which this factorization holds @xcite . starting from eqs . and , the usual way to derive the analytical expression of @xmath4 consists in replacing @xmath60 by @xmath14 ( for a speckle computed in transmission ) or @xmath60 by @xmath13 ( for a speckle computed in reflection ) in the ladder positions . in the reflection / transmission geometry , this simplification can not be performed because the relative distance between the points @xmath14 and @xmath13 can be very large compared to the scattering mean - free path @xmath101 . however , in order to get an explicit expression , we can make use of eq . which leads to @xmath112 the integration over @xmath60 is then performed using the residue theorem ( the details are given in appendix [ c_1_calculation ] ) . this leads to @xmath113}{4(k'^2+k''^2 ) } \\ \times\left[\frac{d\left\{(1+z_0/\ell)m_1+(1-z_0/\ell)\exp(-l/\ell)m_2\right\}}{l+2z_0}\right . \\ \left.\vphantom{\frac{1}{2}}-(d-1)m_3\right]\frac{{\mathrm{d}}{\boldsymbol{\mathbf{k}}}}{(2\pi)^{d-1}}\end{gathered}\ ] ] where @xmath114 , @xmath115 , @xmath116 , @xmath117 and @xmath118 are given by eqs . , and and eq . , respectively . the last integral over @xmath119 is performed numerically . in the specific geometry considered here , another contribution has to be added , in which the ballistic intensity contributes as one of the intensities involved in the correlation function @xcite . indeed , the ladder operator involves at least one scattering event , and does not account for situations in which there is no scattering event before the field interchange . such contributions to the correlation function can be important for small optical thicknesses , where the ballistic contribution is not negligible . this leads to a correction to the @xmath4 correlation function , that we denote by @xmath120 , and whose expression is @xmath121 in terms of diagrams , the above expression can be rewritten as @xmath122(31,-1){${\boldsymbol{\mathbf{r}}}_r$ } \rput[bl](25,-1){${\boldsymbol{\mathbf{r}}}_t$ } { \psframe[fillstyle = none , dimen = middle](6,-9)(18,-3 ) \setcounter{tempa}{18 } \addtocounter{tempa}{6 } \divide \value{tempa } by 2 \setcounter{tempb}{-3 } \addtocounter{tempb}{-9 } \divide \value{tempb } by 2 \pscircle(\value{tempa},\value{tempb}){2 } \rput[c](\value{tempa},\value{tempb}){l } } { \psline[linewidth=0.5](18,-9)(30,0 ) } { \psline[linewidth=0.5,linestyle = dashed](18,9)(30,0 ) } { \psline[linewidth=0.5](18,-3)(24,0 ) } { \psline[linewidth=0.5,linestyle = dashed](18,3)(24,0 ) } { \psline[linewidth=0.5,linestyle = dashed](0,-9)(6,-9 ) } { \psline[linewidth=0.5,linestyle = dashed](0,-3)(6,-3 ) } { \psline[linewidth=0.5,linestyle = dashed](0,9)(18,9 ) } { \psline[linewidth=0.5,linestyle = dashed](0,3)(18,3 ) } \end{dddiag } + \text{c.c.}\ ] ] making use of the quantity @xmath123 defined in eq . , we obtain @xmath124 } { { \left\langle}i({\boldsymbol{\mathbf{r}}}_r){\right\rangle}{\left\langle}i({\boldsymbol{\mathbf{r}}}_t){\right\rangle}}.\end{gathered}\ ] ] it is important to note that the @xmath120 contribution introduced here , and that involves the average field , is not negligible compared to the usual @xmath4 contribution . this correction should be added to the @xmath4 contribution when the product of the ballistic fields @xmath125 can not be neglected , as in the reflection / transmission geometry at low optical thickness , or in the reflection / reflection geometry at any optical thickness . we can show from eq . that the @xmath4 and @xmath120 contributions to the correlation function decrease exponentially with the optical thickness @xmath126 . this behavior explains why @xmath5 dominates at large optical thickness . but in the single scattering regime , an important contribution of the @xmath127 term is observed . this is clearly seen in fig . [ c_2_small_b ] ( blue dashed line ) , in which the sum of the three contributions @xmath107 calculated analytically in the same two - dimensional geometry used for the numerical simulation is plotted . qualitatively , the behavior observed in the numerical simulation is fairly reproduced by the analytical approach . moreover , since the @xmath127 contribution decays exponentially with the medium thickness @xmath9 , a crossover is expected towards a regime dominated by @xmath5 when the optical thickness increases . this also shows that the reflection / transmission geometry studied here may be relevant to put forward experimentally the influence of the @xmath5 contribution ( in the pure reflection or transmission geometries , the @xmath4 contribution is always the leading contribution ) . analytical and numerical results at large optical thickness have shown that a correlation between reflected and transmitted intensities exists . surprisingly , this correlation function takes negative values around @xmath44 . having @xmath45 for @xmath44 qualitatively suggests a high probability to have a bright ( dark ) spot in the transmitted speckle in coincidence with a dark ( bright ) spot in the reflected speckle . in order to address this question in more quantitative terms , we have studied the full statistical distribution of the intensities . more precisely , from the numerical simulations , we have extracted the statistical distributions of the product of the fluctuating part of the intensities at @xmath44 , defined as @xmath128\ ] ] whose average value is the intensity correlation @xmath12 at @xmath44 . [ c]@xmath129 [ c]@xmath130 at @xmath44 . multiple scattering regime with @xmath42 and @xmath131.,title="fig : " ] the statistical distribution @xmath130 obtained for @xmath42 and @xmath131 is shown in fig . [ histo ] . the distribution exhibits an asymmetric shape , with a most likely value at @xmath132 . due to this asymmetric shape , the distribution can not be simply characterized by its first moments . in particular , for @xmath42 and @xmath131 , we find that the probability to have transmitted and reflected intensity fluctuations with opposite signs @xmath133 , while we could have expected a much larger value ( above @xmath134 ) from a naive argument based on the negative sign of the correlation @xmath12 . nevertheless , it is interesting to study statistics under some constraints . in particular , we have studied the probability of having @xmath135 ( a spot in the transmitted speckle darker than the average intensity ) under the assumption that @xmath136 , with @xmath137 ( i.e.for a coinciding spot in the reflected speckle with increasing brightness ) . the results are summarized in table [ proba ] . interestingly , we find that the probability @xmath138 increases substantially with the brightness of the reflected speckle spot , an information that is not contained in the intensity correlation function . in consequence , if the reflected intensity in a speckle spot is large compared to the average reflected intensity , the transmitted intensity in the coinciding spot in the transmitted speckle is smaller than the average transmitted intensity with a large probability . this result may have implications in the context of light focusing through opaque scattering media by wavefront shaping . indeed , maximizing the intensity in a reflected speckle spot might , with a high probability , leads to a minimization of the intensity in the corresponding transmitted spot . a precise study with optimized wavefronts ( beyond plane - wave illumination ) is left for future work . .probability of having @xmath139 ( @xmath53 ) , of having @xmath135 ( @xmath140 ) at different optical thicknesses ( @xmath141 ) and under some constraints ( @xmath142 with @xmath143 ) for @xmath44 and @xmath131 . [ cols="^,^,^,^,^,^ " , ] in summary , we have studied analytically and numerically the spatial correlation between intensities measured in the reflected and transmitted speckles generated by a slab of disordered scattering medium . we have demonstrated the existence of a reflection / transmission correlation . at large optical thicknesses , the spatial correlation persists and is dominated by the @xmath5 contribution , thus exibiting a long - range behavior . interestingly , this correlation takes negative values . at small optical thicknesses , the correlation is dominated by a @xmath4-type contribution , which contains the usual @xmath4 term and an additionnal term @xmath120 involving the ballistic intensity . the statistical connection between transmitted and reflected speckles might be of interest for wavefront shaping methods used to focus and image through scattering media . since for practical implementations only the reflected speckle can be measured and controlled , a knowledge of the probability to get a bright ( dark ) spot in the transmitted speckle in coincidence with a dark ( bright ) spot in the optimized reflected speckle could be a great advantage . as a first step towards this goal , we have studied the statistical distributions of reflected and transmitted intensities , and have identified situations in which the probability of coincidence of bright and dark spots on opposite sides of the medium is high . finally , a refinement of the analytical model would be beneficial to deal with optical thicknesses for which the diffusion approximation fails to give quantitative results . one possibility could be to developp a semi - analytical approach ( coupling analytical expressions and numerical calculations ) based on the radiative transfer equation @xcite that described accuratly short and long scattering paths @xcite . this work has been initiated by stimulating discussions with demetri psaltis and ye pu . we acknowledge arthur goetschy for illuminating inputs . the research was supported by labex wifi ( laboratory of excellence within the french program `` investments for the future '' ) under references anr-10-labx-24 and anr-10-idex-0001 - 02 psl*. n.f . acknowledges financial support from the french `` direction gnrale de larmement '' ( dga ) . in the reflexion / transmission configuration the usual approximation used to calculate @xmath4 breaks . indeed in transmission / transmission or in reflexion / reflexion one usually manages to separate in eq . integrals over @xmath60 and @xmath59 because of the small distance between the two points where we compute the correlation . in our configuration , because of the large distance between these two points , we have to calculate explicitly @xmath123 . using the symmetry of the problem , taking the fourier transform of the two green functions over their transverse coordinates , we simplify eq . the following way : @xmath145 { \mathrm{d}}z_2 \frac{{\mathrm{d}}{\boldsymbol{\mathbf{k}}}}{(2\pi)^{d-1}}\end{gathered}\ ] ] where @xmath146 is the depth at which the two fields separate . to calculate the expression of the green function in this mixed domain , one can see that : @xmath147\frac{{\mathrm{d}}k}{2\pi } \\\label{greentf } & = \int\frac{\exp[ikz_2]}{k^2+k^2-k_{\text{eff}}^2}\frac{{\mathrm{d}}k}{2\pi}.\end{aligned}\ ] ] equation can be calculated using the residue theorem using the fact that @xmath148 and noting that : @xmath149 where @xmath150 is the pole with a positive imaginary part ( i.e.@xmath151 ) . . can be calculated with the residue theorem by integrating over a contour composed of the real axis and a upper - half - circle the radius of which tends to infinity . we obtain that @xmath152 following the same steps we can show that @xmath153\ ] ] leading to @xmath154}{4(k'^2+k''^2)}\exp[-ik^*_+l ] \\ \times\exp[-i{\boldsymbol{\mathbf{k}}}\cdot{\boldsymbol{\mathbf{\delta r}}}]{\mathrm{d}}z_2 \frac{{\mathrm{d}}{\boldsymbol{\mathbf{k}}}}{(2\pi)^{d-1}}.\end{gathered}\ ] ] using the expression of the average intensity we have to perform three integrations denoted respectively by @xmath114 , @xmath115 and @xmath116 : @xmath155{\mathrm{d}}z_2 \\\label{m_1 } & = \frac{1 + 2ik'(l+z_0)-\exp[2ik'l](1 + 2ik'z_0)}{4k'^2 } , \\\nonumber m_2 & = \int_0^l(z_0+z_2)\exp[2ik'z_2]{\mathrm{d}}z_2 \\\label{m_2 } & = \frac{2ik'z_0 - 1+\exp[2ik'l][1 - 2ik'(l+z_0)]}{4k'^2 } , \\\nonumber m_3 & = \int_0^l \exp(-\frac{z_2}{\ell})\exp[2ik'z_2]{\mathrm{d}}z_2 \\\label{m_3 } & = \ell \frac{1-\exp[2ik'l]\exp[-l/\ell]}{1 - 2ik'\ell}.\end{aligned}\ ] ] with these expressions , we can rewrite the final expression of @xmath123 as a fourier transform given by eq . .
|
we study theoretically the spatial correlations between the intensities measured at the input and output planes of a disordered scattering medium .
we show that at large optical thicknesses , a long - range spatial correlation persists and takes negative values . for small optical thicknesses ,
short - range and long - range correlations coexist , with relative weights that depend on the optical thickness .
these results may have direct implications for the control of wave transmission through complex media by wavefront shaping , thus finding applications in sensing , imaging and information transfer .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
leptogenesis @xcite is one of the most attractive scenarios to explain the origin of the observed matter - antimatter asymmetry of the universe . it follows from the seesaw mechanism @xcite , which gives a natural and simple explanation to the small neutrino masses observed in neutrino experiments , and relies on the conversion of a lepton asymmetry to a baryon asymmetry thanks to the non - perturbative sphaleron processes @xcite . for a recent review on leptogenesis , see @xcite . if neutrinos are massive majorana particles , then lepton number must be violated . since neutrino masses are observed experimentally to be tiny , a slightly broken lepton number symmetry , e.g. a global @xmath4 , could provide the explanation . in this case , small neutrino masses are not explained by a ` seesaw ' mechanism , but rather by a cancellation mechanism @xcite . such a symmetry was introduced in the context of the @xmath5msm @xcite in @xcite to explain at the same time the kev scale of the dark matter sterile neutrino @xcite and the quasi - degeneracy of the heavier two rh neutrinos , supposed to explain the baryon asymmetry of the universe by means of leptogenesis via neutrino oscillations @xcite . the work @xcite also made use of a slightly broken lepton symmetry to motivate large yukawa couplings with tev masses for the rh neutrinos , making them in principle accessible at the lhc . one consequence of the slightly broken @xmath4 symmetry is the existence of two quasi - degenerate rh neutrinos . this is interesting in the context of leptogenesis because it allows for an enhancement of the @xmath6 asymmetry parameter @xcite , and hence successful leptogenesis is possible much below the usually quoted bounds on the mass scale and on the reheat temperature of the universe after inflation assuming hierarchical rh neutrinos , @xmath7 @xcite . the tension with the gravitino overproduction in msugra scenarios @xcite is thus relaxed . the second consequence of the broken symmetry is the presence of large washout parameters , implying that the asymmetry will be completely independent of the initial conditions , i.e. both the initial number of rh neutrinos and any previously generated asymmetry , even taking into account flavor effects @xcite . in this paper , we study in detail the mechanism of leptogenesis in the presence of a slightly broken @xmath4 symmetry within the supersymmetric ( susy ) and non - supersymmetric two rh neutrino ( 2rhn ) model @xcite . note that the 2rhn model is physically equivalent to the @xmath0 decoupling limit ( @xmath8 ) @xcite . a related study was performed in @xcite with three quasi - degenerate rh neutrinos , which have to be motivated by a larger symmetry group , e.g. @xmath9 . there , the focus was to find numerical examples where resonant leptogenesis was possible with at the same time phenomenological consequences like observable lepton - flavor - violating signals in the non - susy setup . here , we introduce only a @xmath4 symmetry , and tackle the problem with only two rh neutrinos in both susy and non - susy cases . proceeding analytically , we keep under control the full parameter space of the problem , and we do not focus on resonant leptogenesis , which corresponds to the maximal possible enhancement of the @xmath6 asymmetry for quasi - degenerate heavy neutrinos . the 2rhn model implies one massless light neutrino and a reduction of the numbers of parameters compared to the model with three rh neutrinos from 18 to 11 , among which 7 ( 2 neutrino masses , 3 mixing angles and 2 @xmath6-violating phases ) are accessible in experiments . the lower number of parameters will allow us to have a perfect handle on the problem , and thus we will be able to derive an expression for the baryon asymmetry predicted by leptogenesis where the dependence on each parameter is simple . in particular , the high - energy parameters will factorize from the low - energy ones and from flavor effects altogether . this will make possible to study in detail the dependence of the predicted baryon asymmetry on the dirac and majorana @xmath6-violating phases as well as the unknown angle @xmath10 . note that , since the high - energy phase will be required to be non - zero for successful leptogenesis , leptogenesis from exclusively low - energy @xmath6 violation @xcite is not viable here . finally , we will use the maximal enhancement of the @xmath6 asymmetry in the resonant limit @xcite to obtain a constraint on one of the breaking parameters . interestingly , this constraint implies that successful leptogenesis is incompatible with the possible observation of rh neutrinos at future colliders @xcite as well as sizable lepton - flavor - violating signals . this conclusion holds also in the susy case , if we assume the scale of leptogenesis to be below @xmath11 gev to avoid the gravitino problem . in section 2 , we introduce the parametrization of the 2rhn model which we will use throughout the paper . in section 3 , we describe how the lepton number symmetry affects the structure of the neutrino yukawa matrix and the majorana mass matrix . then , we discuss how to parametrize the small breaking of this symmetry . in section 4 , we turn to leptogenesis , and estimate the baryon asymmetry predicted in this model with a special emphasis on the role of the pmns phases . we also constrain the size of the breaking parameters using the maximal enhancement of the @xmath6 asymmetry . in section 5 , we extend the results to the supersymmetric version of the model . finally , we summarize our main results and conclude in section 6 . consider the seesaw model with two rh neutrinos . in the basis where the charged leptons and rh majorana neutrino mass matrices are both diagonal , the `` mass '' basis , the seesaw mass matrix is given by m_=-m_d m_m^-1m_d^t , where m_md_n = diag(m_1,m_2 ) . this matrix can be diagonalized as u^m_u^d_=diag(m_1,m_2,m_3 ) , where @xmath12 is the pmns matrix . we will adopt the parametrization @xcite @xmath13 where @xmath14 , @xmath15 and , neglecting statistical errors , we will use @xmath16 and @xmath17 , compatible with the results from neutrino oscillation experiments . moreover , we will adopt the @xmath18 range @xmath19 @xcite . neutrino oscillation experiments measure two neutrino mass - squared differences , @xmath20 ev@xmath21 and @xmath22 ev@xmath21 @xcite . recall that with only two rh neutrinos the lightest active neutrino is massless . therefore , if the neutrino mass hierarchy is normal , one has @xmath23 , @xmath24 and @xmath25 , whereas if it is inverted , @xmath26 , @xmath27 and @xmath28 . let us now introduce the so - called casas - ibarra parametrization @xcite @xmath29 the @xmath30 matrix is a @xmath31 matrix which can be parametrized as @xcite @xmath32 in the normal and inverted hierarchy , respectively , and where @xmath33 is a complex parameter . @xmath34 is a discrete parameter that accounts for a discrete indeterminacy in @xmath35 . following @xcite ( see also @xcite ) in the limit of only two rh neutrinos , we introduce a global @xmath4 lepton number symmetry for the leptonic fields , lepton doublets @xmath36 and right - handed neutrinos @xmath37 . the @xmath4 charge assignments are as follows : @xmath38 the charges of all the other standard model fields are zero . note that we work in the basis where the mass matrix of the charged leptons is real and diagonal . from now on , we shall call the basis with @xmath39 the `` flavor '' basis . in this basis , the relevant terms in the lagrangian are @xmath40_{ij } \overline{\tilde{n}_i^c}\tilde{n}_j + h.c.\quad ( i=1,2,\quad \a = e,\m,\t),\ ] ] where @xmath41 and @xmath42 are the yukawa coupling matrix and the majorana mass matrix in the flavor basis , respectively . first of all , let us consider the @xmath4 symmetry to be exact . in this case , the allowed structure of @xmath41 is given by @xmath43 without loss of generality , by using the field redefinition of @xmath44 , the non - zero components @xmath45 can be taken real and positive . on the other hand , the allowed form of @xmath46 is @xmath47 where the mass parameter @xmath48 can be taken real and positive by field redefinitions of @xmath49 and @xmath50 . it should be noted that , when @xmath4 is exact , the masses of the active neutrinos are exactly zero . this is because lepton doublets carry non - zero charges , and hence a majorana mass term for them is forbidden . let us now consider the effects of @xmath4 symmetry breaking . such effects induce additional contributions to @xmath41 and @xmath46 : @xmath51 where @xmath52 are complex parameters with modulus of order one , @xmath53 is real and positive , @xmath54 is an arbitrary phase , and @xmath3 and @xmath55 are breaking parameters , which are supposed to be much smaller than one . note that we will not deal here with the issue of the @xmath4 breaking mechanism , and hence just parametrize the breaking parameters as given . we go now from the flavor basis to the mass basis for the rh neutrinos . the majorana mass matrix @xmath42 is symmetric , and therefore can be diagonalized using a unitary matrix @xmath56 : @xmath57 thus , the flavor eigenstates @xmath39 are related to the mass eigenstates @xmath58 by @xmath59_{ij } n_j$ ] . it is easy to show that , to first order in @xmath3 , the eigenvalues of @xmath42 are given by @xmath60 where @xmath61 . so the degeneracy parameter @xmath62 is equal to the breaking parameter @xmath3 up to a factor @xmath63 supposed to be of order one . for simplicity we will assume in the remainder of the paper that @xmath64 . next , we turn to the neutrino yukawa matrix @xmath41 , which we want to express in the mass basis , i.e. @xmath65 . in a first approximation , we take @xmath66 , so that the mixing matrix of @xmath67 and @xmath68 is simply given by @xmath69 where the second matrix ensures that the two eigenvalues of the majorana mass matrix @xmath70 are positive . using the casas - ibarra parametrization eq . ( [ h ] ) , one can translate eq . ( [ tildef ] ) into [ condom ] |_2|~_h|_1| . where @xmath71 . from eq . ( [ omega ] ) for normal hierarchy , and choosing the ` ' sign and @xmath72 , one obtains in the limit @xmath73 @xmath74 note that the case ` + ' and @xmath75 would have given the same structure , though with different signs . on the other hand , for the two other choices , ` + ' and @xmath72 , and ` ' and @xmath75 , the columns 1 and 2 would have been exchanged . the case of inverted hierarchy is obtained by exchanging the rows @xmath76 and @xmath77 . it is easy to see now that the condition ( [ condom ] ) is indeed satisfied when @xmath73 , for ` ' and @xmath72 as well as ` + ' and @xmath75 , and the breaking parameter @xmath55 corresponds to _ h~14||^2 . it is widely known that , in order to generate a baryon asymmetry in the early universe , one needs to satisfy the three sakharov s conditions @xcite . baryon number violation is provided by the non - perturbative sphaleron processes @xcite . the @xmath6 symmetry is violated in the decay of the heavy rh neutrinos @xcite @xmath78}\nonumber\\ & = & \frac{3}{16 \p ( h^{\dag}h)_{ii } } \sum_{j\neq i } \left\ { { \rm im}\left[h_{\a i}^{\star } h_{\a j}(h^{\dag}h)_{i j}\right ] \frac{\x(x_j / x_i)}{\sqrt{x_j / x_i}}\right.\nonumber \\ & & \hspace{3 cm } \left.+ \frac{2}{3(x_j / x_i-1)}{\rm i m } \left[h_{\a i}^{\star}h_{\a j}(h^{\dag}h)_{j i}\right]\right\ } , \end{aligned}\ ] ] where @xmath79 and [ xi ] ( x)= 23x . as for the third condition , namely departure from thermal equilibrium , it can be conveniently described by the so - called decay parameter @xmath80 , given by the ratio of the decay widths to the expansion rate when the rh neutrinos start to become non - relativistic at @xmath81 . in terms of yukawa couplings , the decay parameters can be written as k_i = v^2 m _ m_i(h^h)_ii , where @xmath82 is the equilibrium neutrino mass , given by @xcite @xmath83 with @xmath84 gev and @xmath85 . notice that , using the casas - ibarra parametrization , eq . ( [ h ] ) , the decay parameters @xmath86 can be expressed as linear combinations of the neutrino masses @xcite [ k ] k_i = _ jm_jm_|_ji^2| . we will assume in the following that the generation of asymmetry takes place at temperatures @xmath87 , in which case flavor matters in the leptogenesis process , and three flavors are distinguished , denoted @xmath88 @xcite . important parameters in flavored leptogenesis are the flavored decay parameters , given by [ kflav ] k_i = v^2 m _ m_i|h_i|^2 . for quasi - degenerate rh neutrinos it will prove useful to calculate the sum of the decay parameters . using eqs . ( [ k ] ) and ( [ kflav ] ) with the matrix @xmath30 given in eq . ( [ omega ] ) fixing the sign to @xmath89 and @xmath72 compatible with the lepton symmetry , one obtains @xmath90 and @xmath91 the case of inverted hierarchy is obtained by changing the labels @xmath77 and @xmath76 . in the limit of large @xmath92 required by the symmetry , one obtains @xmath93 and @xmath94 which does not depend on the phase of @xmath95 , and where we defined the dimensionless quantity @xmath96/m_{\star}.\ ] ] it is interesting to notice that for a normal hierarchy the small entry @xmath97 makes possible a cancellation in @xmath98 such that it can be much smaller than @xmath99 ; the other two flavors can not be smaller than roughly @xmath100 . on the other hand , when the hierarchy is inverted , a cancellation can occur in all flavors . for quasi - degenerate rh neutrinos and with a strong washout in each flavor , the solution to the flavored boltzmann equations can be written in the form @xcite @xmath101 where @xmath102 is the so - called efficiency factor , given by @xcite @xmath103 where @xmath104 note that in the strong washout regime , for @xmath105 , the following approximation holds @xcite : @xmath106 as for the @xmath6 asymmetries , in the limit @xmath107 and for the particular @xmath30 matrix in eq . ( [ omega ] ) ] , they are given by @xmath108+{\rm im}\left[h_{\a 1}^{\star } h_{\a 2 } ( h^{\dag}h)_{2 1}\right]\right),\\ \ve_{2\alpha}&\simeq & \frac{3}{16 \p ( h^{\dag}h)_{22 } } \frac{1}{3\d_2 } \left({\rm i m } \left[h_{\a 1}^{\star } h_{\a 2}(h^{\dag}h)_{1 2}\right]+{\rm im}\left[h_{\a 1}^{\star } h_{\a 2 } ( h^{\dag}h)_{2 1}\right]\right),\end{aligned}\ ] ] where @xmath109 . using the casas - ibarra parametrization eq . ( [ h ] ) the flavored @xmath6 asymmetries are given by @xmath110 in the limit of large @xmath92 and defining @xmath111 , one obtains @xmath112 where we defined another dimensionless quantity , [ f ] f_(m_2,m_3 , u_2,u_3 ) & & g_2m_2-m_3m_2+m_3 . it is interesting to notice first that the @xmath6 asymmetry is independent of @xmath92 , and second that the high - energy contribution is a necessary ingredient to have a non - zero flavored @xmath6 asymmetry ; in other words , @xmath95 has to be complex . hence , leptogenesis from exclusively low - energy phases @xcite is not possible in our model . we have pointed out earlier that @xmath113 can be much smaller than @xmath114 in certain situations , which could lead to a big enhancement of the predicted baryon asymmetry compared to an unflavored calculation . however , it turns out that in all such cases the flavored @xmath6 asymmetry is suppressed as well , so that the final effects are never much larger than the typical enhancement of a factor three in the three - flavor regime , as we will show more precisely below . since the lepton number symmetry implies @xmath115 , the asymmetry will be typically produced in the strong washout regime , with no dependence on the initial number of rh neutrinos and on any previously generated asymmetry . with an account of flavor effects , the second feature is only rigourously possible in the three - flavor regime , where the washout occurs in all directions in flavor space @xcite . the only exception will be in some very marginal regions of the parameter space where huge flavor effects imply @xmath116 , even though @xmath117 , so that the strong washout and independence of the initial conditions are no longer guaranteed . we will come back to this point at the end of the section . since the final asymmetry will be almost exclusively produced in the strong washout , using eqs . ( [ nf ] ) , ( [ kappaapprox ] ) and ( [ epslargeo ] ) , one obtains the simple expression @xmath118 which depends on the fundamental quantities @xmath48 , @xmath119 , @xmath92 and @xmath120 in a very simple way . in terms of the breaking parameters @xmath3 and @xmath55 introduced in the last section , one has @xmath121 interestingly , all low - energy parameters ( neutrino masses , pmns matrix elements ) appear exclusively in the factor @xmath122 , and are thus decoupled from the high - energy parameters ( @xmath48 , @xmath119 , @xmath92 and @xmath120 ) . actually , flavor effects altogether only appear in this factor , and we can thus easily estimate the maximal difference with an unflavored calculation , where we would have instead a factor @xmath123 on the other hand , one can find numerically the maximal factor in the flavored calculation : @xmath124 i.e. maximal flavor effects lead to a factor 56 enhancement of the final asymmetry both for normal and for inverted hierarchy . as we said earlier , larger effects are not possible here . finally , assuming a standard thermal history and accounting for the sphaleron conversion coefficient @xmath125 , the baryon - to - photon ratio can be calculated as [ etab ] _ b = a_sphn_b - l^fn_^rec 0.9610 ^ -2n_b - l^f , to be compared with the measured value @xcite [ etabobs ] _ b^cmb = ( 6.2 0.15)10 ^ -10 . it can be useful to make a numerical estimation of @xmath126 using eq . ( [ resultanal ] ) , with @xmath127 , and @xmath128 : @xmath129 this result tells us that a hierarchy in the breaking parameters @xmath130 is needed if one wants to relax the scale of leptogenesis . for instance , with the breaking parameters @xmath131 and @xmath132 , one obtains @xmath133 it can be easily seen from eq . ( [ resultanal ] ) that the scale of the rh neutrinos @xmath48 can be lowered if we decrease by the same factor @xmath3 . hence , with @xmath134 , we can reach the tev scale for the heavy neutrinos , making them at least in principle accessible at the lhc . the @xmath6 asymmetry is enhanced in the degenerate limit inversely proportionally to @xmath119 , as can be seen from eq . ( [ epslargeo ] ) . however , this effect is not unlimited . there is a maximal enhancement which leads to resonant leptogenesis @xcite . the condition to be on the resonance is given by @xmath135 @xcite , where @xmath136 is the decay width of @xmath137 . in our model and for the case of normal hierarchy , this condition can be conveniently translated into @xmath138 where we introduced the parameter @xmath139 as in @xcite to account for a controversy in the literature about whether it is allowed or not to reach the resonance @xcite . in the following , we will use conservatively @xmath140 , so that the validity of eq . ( [ veia ] ) is ensured . note that in the case of inverted hierarchy @xmath141 occurs at a value twice as large . in the following , we will assume that @xmath142 . plugging the condition ( [ condres ] ) in eq . ( [ resultanal2 ] ) , we find for normal hierarchy @xmath143 where we used eq . ( [ maxfoverg ] ) in the second step . since we want to be consistent with the @xmath18 range of eq . ( [ etabobs ] ) , we find that @xmath144 in the case of inverted hierarchy , we find @xmath145 this implies @xmath146 remember that the lepton number symmetry implied @xmath73 , so that the allowed range in the case of inverted hierarchy is quite constrained ! note also that the case of inverted hierarchy is more constrained than normal hierarchy due to the smallness of the factor @xmath147 , never larger than @xmath2 . this behavior is different from the 2rhn model in the hierarchical limit @xmath148 , where huge flavor effects make possible that inverted hierarchy yields almost the same bounds as normal hierarchy @xcite , contrary to the unflavored result @xcite . the results of eqs . ( [ resnormal ] ) and ( [ resinverted ] ) show explicitly that , in our simple model with two rh neutrinos or when @xmath0 is decoupled , successful leptogenesis is only possible for relatively large values of the breaking parameter @xmath55 , especially in the case of inverted hierarchy . this is particularly interesting in view of the possible observation of rh neutrinos at future colliders , as recently investigated in a number of papers @xcite . to have even a small chance of observing rh neutrinos , some of the active - sterile mixing angles @xmath149 should not be much smaller than 0.01 . in our case , with the constraint from successful leptogenesis , we have at most @xmath150 for @xmath151 gev , which is much too small . lepton - flavor - violating signals in the non - supersymmetric case under discussion are expected to be very suppressed for the same reason . finally , we would like to present examples compatible with the constraints ( [ resnormal ] ) and ( [ resinverted ] ) which show the explicit dependence of @xmath126 on the angle @xmath10 and on the pmns phases @xmath152 and @xmath153 , which only appear in the factor @xmath154 , as already mentioned . we present in fig . [ fig : correlnormal ] two such examples , for a normal hierarchy of light neutrinos and two choices of @xmath155 , 0.2 and 0.02 . note that the results are compatible with our rough estimation in eq . ( [ estimation ] ) . in fig . [ fig : correlinverted ] we display the case of inverted hierarchy . one notices from the figures that the two @xmath6-violating phases in the pmns matrix only yield small corrections to the predicted baryon asymmetry . when the hierarchy of light neutrinos is normal and for the maximal allowed value @xmath156 , these phases can change the final asymmetry by 40% . when the hierarchy is inverted , the effect can be more than a factor three , but only in a very restricted region of the parameter space . before concluding the section , one comment concerning figs . [ fig : correlnormal ] and [ fig : correlinverted ] is in order . we found marginal regions where @xmath116 for some flavor @xmath157 , implying that the strong washout regime and thus eq . ( [ kappaapprox ] ) no longer hold . in these regions of dependence on the initial conditions , we used the more general eq . ( [ kappa ] ) , which is also valid for @xmath158 in the case of thermal initial @xmath67 and @xmath68 abundances . in practice , these regions of huge flavor effects on the washout are found very close to the ones where the asymmetry is maximal , and they can be seen in the left panels of figs . [ fig : correlnormal ] and [ fig : correlinverted ] . note that a very small @xmath159 as in the right panels forbids such huge suppressions of the washout . more specifically , in the left panel of fig . [ fig : correlnormal ] , @xmath160 can be as low as @xmath161 in the middle of the diagonal strips , implying some dependence on the initial conditions . in the left panel of fig . [ fig : correlinverted ] , @xmath160 can be as low as @xmath162 in the darker regions ( red to black ) , so the dependence on the initial conditions is even larger there . but apart from these marginal regions , the strong washout holds , and one can safely use eq . ( [ resultanal ] ) . we turn now to the supersymmetric version of the model . one can introduce four different @xmath6 asymmetries , which by supersymmetry are all equal : @xmath163}=-{\g(n_i\to \tilde{\ell}_{\alpha}\tilde{h}_u-\g(n_i\to \tilde{\ell}^{\dagger}_{\alpha } \bar{\tilde{h}}_u ) ) \over \sum_{\a}\left[\g(n_i\to \tilde{\ell}_{\alpha } \tilde{h}_u)+\g ( n_i\to \tilde{\ell}^{\dagger}_{\alpha } \bar{\tilde{h}}_u)\right ] } \nonumber\\ & = & -{\g(\tilde{n}_i\to \ell_{\alpha}\tilde{h}_u)-\g(\tilde{n}^{\star}_i\to \bar{\ell}_{\alpha } \bar{\tilde{h}}_u ) \over \sum_{\a}\left[\g(\tilde{n}_i\to \ell_{\alpha}\tilde{h}_u)+\g(\tilde{n}^{\star}_i\to \bar{\ell}_{\alpha } \bar{\tilde{h}}_u)\right]}=-{\g(\tilde{n}_i\to \tilde{\ell}_{\alpha}h_u)-\g(\tilde{n}^{\star}_i\to \tilde{\ell}^{\dagger}_{\alpha } h_u^{\dagger } ) \over \sum_{\a}\left[\g(n_i\to \tilde{\ell}_{\alpha } h_u)+\g ( \tilde{n}^{\star}_i\to \tilde{\ell}^{\dagger}_{\alpha } h_u^{\dagger})\right]}\nonumber \ , , \end{aligned}\ ] ] where @xmath164 , @xmath165 and @xmath166 denote sleptons , higgsinos and rh sneutrinos , respectively . the @xmath6 asymmetries @xmath167 were calculated in @xcite to be @xmath168 g(x_j / x_i)+ \frac{2}{(x_j / x_i-1)}{\rm i m } \left[h_{\a i}^{\star}h_{\a j}(h^{\dag}h)_{j i}\right]\right\ } , \ ] ] where @xmath79 and [ xi ] g(x)= . in the limit @xmath107 we are interested in and for the particular @xmath30 matrix in eq . ( [ omega ] ) , it can be easily obtained that @xmath169+{\rm im}\left[h_{\a 1}^{\star } h_{\a 2 } ( h^{\dag}h)_{2 1}\right]\right),\\ \tilde{\ve}_{2\alpha}&\simeq & \frac{1}{8 \p ( h^{\dag}h)_{22 } } \frac{1}{\d_2 } \left({\rm i m } \left[h_{\a 1}^{\star } h_{\a 2}(h^{\dag}h)_{1 2}\right]+{\rm im}\left[h_{\a 1}^{\star } h_{\a 2 } ( h^{\dag}h)_{2 1}\right]\right),\end{aligned}\ ] ] where we recall that @xmath109 . so the @xmath6 asymmetries are a factor of two larger than in the non - supersymmetric case , just like in the hierarchical limit @xmath170 . let us see how the efficiency factor is affected by supersymmetry . assuming maxwell - boltzmann distributions , rh neutrinos and rh sneutrinos follow exactly the same evolution . the end result is that the source term in the boltzmann equation for the @xmath171 asymmetry ( see e.g. @xcite ) is a factor 4 larger than in the non - susy case , whereas the washout term is a factor 2 larger . the reason is that , first , the @xmath6 asymmetry is a factor 2 larger , and second there are twice as many decay modes as in the non - susy case . the latter of course affects both source and washout terms . finally , the equilibrium neutrino mass in eq . ( [ d ] ) is numerically different since @xmath172 , and @xmath173 , implying @xmath174 altogether , multiplying the @xmath6 asymmetry by a factor 2 , and replacing @xmath175 , the semi - analytical expression for the final asymmetry becomes [ cf . ( [ resultanal ] ) ] @xmath176 where @xmath177 and @xmath178 are the functions defined in eqs . ( [ f ] ) and ( [ g ] ) , but with the replacement @xmath179 . note that the dependence on @xmath180 in this expression is extremely mild , @xmath181 . the extension of the results presented in the last section is then straightforward . from the maximal enhancement of the @xmath6 asymmetry in the degenerate limit [ cf . ( [ condres ] ) ] , we obtain slightly modified bounds on the breaking parameters , @xmath182 in the case of normal hierarchy , and @xmath183 in the case of inverted hierarchy . since the limits on the breaking parameters are very similar to the non - susy case , the active - sterile mixing angles are necessarily small here as well . hence , the conclusion about the incompatibility of leptogenesis with the observation of rh neutrinos at colliders is still valid . in the case of lepton - flavor violation , the discussion is somewhat different since new diagrams contribute that are not suppressed by the small mixing between active and sterile neutrinos @xcite . for example , the rate of @xmath184 will actually depend on the combination @xmath185 . however , for the range of rh neutrino masses we are interested in to avoid the gravitino problem , i.e. @xmath186 gev , the yukawa couplings are too small ( @xmath187 ) to give an observable signal . we have studied the mechanism of leptogenesis in the presence of a slightly broken lepton number symmetry . two almost degenerate right - handed neutrinos result from the symmetry , with the small breaking parameter @xmath3 essentially describing the mass splitting , and @xmath55 fixing the size of the ( inverse of the ) washout . two well - known consequences follow : first , the scale of leptogenesis can be as low as the electroweak scale . second , the baryon asymmetry predicted through leptogenesis is independent of the initial conditions , i.e. both the initial number of rh neutrinos and any previously generated asymmetry . with only two heavy neutrinos , which is equivalent to the @xmath0 decoupling limit , the model contains few unknown parameters in addition to the two breaking parameters : one rh neutrino mass , one ` high - energy ' phase , the angle @xmath10 and two @xmath6-violating phases in the pmns matrix . the relatively low number of parameters has allowed us to have a perfect handle on the problem , and we have been able to obtain semi - analytical formulae for the final asymmetry , eqs . ( [ resultanal ] ) and ( [ resultanalsusy ] ) in the non - susy and susy case , respectively , which disclose a simple dependence on each one of these parameters . in particular , we have studied in detail the role of the pmns phases and @xmath10 ( see figs . [ fig : correlnormal ] and [ fig : correlinverted ] ) . interestingly , the high - energy phase is required to be non - zero for successful leptogenesis , implying that leptogenesis from exclusively low - energy @xmath6-violation is not viable in this context . finally , we derived from successful leptogenesis and the maximal enhancement of the @xmath6 asymmetry in the resonant limit that the breaking parameter @xmath55 must be relatively large , @xmath188 for normal and @xmath2 for inverted hierarchy ( both in susy and non - susy cases ) . as a consequence , leptogenesis is not compatible with the observation of rh neutrinos at future colliders and with a sizable lepton - flavor violation signal . the other breaking parameter , @xmath3 , can be much smaller , and actually needs to be so in order to have low - scale leptogenesis . it is a pleasure to thank pasquale di bari , jrn kersten , and georg raffelt for useful discussions . s. b. is grateful to the max - planck - institute for physics in munich where most of this work was done . m. fukugita and t. yanagida , _ baryogenesis without grand unification _ , _ phys . lett . _ * b174 * ( 1986 ) 45 . p. minkowski , _ mu @xmath189 e gamma at a rate of one out of 1-billion muon decays ? _ , _ phys . lett . _ * b67 * ( 1977 ) 421 . t. yanagida in _ workshop on unified theories , kek report 79 - 18 _ , p. 95 , 1979 . r. barbieri , d. v. nanopoulos , g. morchio and f. strocchi , _ neutrino masses in grand unified theories _ * b90 * ( 1980 ) 91 . r. n. mohapatra and g. senjanovic , _ neutrino masses and mixings in gauge models with spontaneous parity violation _ _ * d23 * ( 1981 ) 165 . v. a. kuzmin , v. a. rubakov and m. e. shaposhnikov , _ on the anomalous electroweak baryon number nonconservation in the early universe _ , _ phys . * b155 * ( 1985 ) 36 . s. davidson , e. nardi and y. nir , _ leptogenesis _ , _ phys . rept . _ * 466 * ( 2008 ) 105177 [ http://arxiv.org/abs/0802.2962 [ 0802.2962 ] ] . d. wyler and l. wolfenstein , _ massless neutrinos in left - right symmetric models _ , _ nucl . phys . _ * b218 * ( 1983 ) 205 . j. bernabeu , a. santamaria , j. vidal , a. mendez and j. w. f. valle , _ lepton flavor nonconservation at high - energies in a superstring inspired standard model _ , _ phys * b187 * ( 1987 ) 303 . g. c. branco , w. grimus and l. lavoura , _ the seesaw mechanism in the presence of a conserved lepton number _ , _ nucl . _ * b312 * ( 1989 ) 492 . t. asaka , s. blanchet and m. shaposhnikov , _ the numsm , dark matter and neutrino masses _ , _ phys . lett . _ * b631 * ( 2005 ) 151156 [ http://arxiv.org/abs/hep-ph/0503065[hep-ph/0503065 ] ] . m. shaposhnikov , _ a possible symmetry of the numsm _ , _ nucl . phys . _ * b763 * ( 2007 ) 4959 [ http://arxiv.org/abs/hep-ph/0605047 [ hep - ph/0605047 ] ] . t. asaka and m. shaposhnikov , _ the numsm , dark matter and baryon asymmetry of the universe _ , _ phys . lett . _ * b620 * ( 2005 ) 1726 [ http://arxiv.org/abs/hep-ph/0505013[hep-ph/0505013 ] ] . e. k. akhmedov , v. a. rubakov and a. y. smirnov , _ baryogenesis via neutrino oscillations _ , _ phys . * 81 * ( 1998 ) 13591362 [ http://arxiv.org/abs/hep-ph/9803255[hep-ph/9803255 ] ] . j. kersten and a. y. smirnov , _ right - handed neutrinos at lhc and the mechanism of neutrino mass generation _ , _ * d76 * ( 2007 ) 073005 [ http://arxiv.org/abs/arxiv:0705.3221 [ hep - ph ] [ arxiv:0705.3221 [ hep - ph ] ] ] . l. covi , e. roulet and f. vissani , _ cp violating decays in leptogenesis scenarios _ , _ phys . _ * b384 * ( 1996 ) 169174 [ http://arxiv.org/abs/hep-ph/9605319[hep-ph/9605319 ] ] . a. pilaftsis , _ cp violation and baryogenesis due to heavy majorana neutrinos _ , _ phys . * d56 * ( 1997 ) 54315451 [ http://arxiv.org/abs/hep-ph/9707235[hep-ph/9707235 ] ] . s. davidson and a. ibarra , _ a lower bound on the right - handed neutrino mass from leptogenesis _ _ * b535 * ( 2002 ) 2532 [ http://arxiv.org/abs/hep-ph/0202239[hep-ph/0202239 ] ] . w. buchmuller , p. di bari and m. plumacher , _ cosmic microwave background , matter - antimatter asymmetry and neutrino masses _ , _ nucl . _ * b643 * ( 2002 ) 367390 [ http://arxiv.org/abs/hep-ph/0205349 [ hep - ph/0205349 ] ] . s. blanchet and p. di bari , _ flavor effects on leptogenesis predictions _ , _ jcap _ * 0703 * ( 2007 ) 018 [ http://arxiv.org/abs/hep-ph/0607330[hep-ph/0607330 ] ] . s. antusch and a. m. teixeira , _ towards constraints on the susy seesaw from flavour- dependent leptogenesis _ , _ jcap _ * 0702 * ( 2007 ) 024 [ http://arxiv.org/abs/hep-ph/0611232[hep-ph/0611232 ] ] . m. y. khlopov and a. d. linde , _ is it easy to save the gravitino ? _ , _ phys . lett . _ * b138 * ( 1984 ) 265268 . j. r. ellis , d. v. nanopoulos , k. a. olive and s .- j . rey , _ on the thermal regeneration rate for light gravitinos in the early universe _ , _ astropart . phys . _ * 4 * ( 1996 ) 371386 [ http://arxiv.org/abs/hep-ph/9505438[hep-ph/9505438 ] ] . t. moroi , h. murayama and m. yamaguchi , _ cosmological constraints on the light stable gravitino _ * b303 * ( 1993 ) 289294 . j. pradler and f. d. steffen , _ constraints on the reheating temperature in gravitino dark matter scenarios _ , _ phys . _ * b648 * ( 2007 ) 224235 [ http://arxiv.org/abs/hep-ph/0612291[hep-ph/0612291 ] ] . e. nardi , y. nir , e. roulet and j. racker , _ the importance of flavor in leptogenesis _ , _ jhep _ * 01 * ( 2006 ) 164 [ http://arxiv.org/abs/hep-ph/0601084[hep-ph/0601084 ] ] . a. abada , s. davidson , f .- x . josse - michaux , m. losada and a. riotto , _ flavour issues in leptogenesis _ , _ jcap _ * 0604 * ( 2006 ) 004 [ http://arxiv.org/abs/hep-ph/0601083[hep-ph/0601083 ] ] . p. h. frampton , s. l. glashow and t. yanagida , _ cosmological sign of neutrino cp violation _ , _ phys . _ * b548 * ( 2002 ) 119121 [ http://arxiv.org/abs/hep-ph/0208157[hep-ph/0208157 ] ] . a. ibarra and g. g. ross , _ neutrino properties from yukawa structure _ , _ phys . * b575 * ( 2003 ) 279289 [ http://arxiv.org/abs/hep-ph/0307051[hep-ph/0307051 ] ] . a. ibarra and g. g. ross , _ neutrino phenomenology : the case of two right handed neutrinos _ * b591 * ( 2004 ) 285296 [ http://arxiv.org/abs/hep-ph/0312138[hep-ph/0312138 ] ] . p. h. chankowski and k. turzynski , _ limits on t(reh ) for thermal leptogenesis with hierarchical neutrino masses _ , _ phys . _ * b570 * ( 2003 ) 198204 [ http://arxiv.org/abs/hep-ph/0306059 [ hep - ph/0306059 ] ] . a. pilaftsis and t. e. j. underwood , _ electroweak - scale resonant leptogenesis _ , _ phys . * d72 * ( 2005 ) 113001 [ http://arxiv.org/abs/hep-ph/0506107[hep-ph/0506107 ] ] . s. pascoli , s. t. petcov and a. riotto , _ connecting low energy leptonic cp - violation to leptogenesis _ , _ phys . * d75 * ( 2007 ) 083511 [ http://arxiv.org/abs/hep-ph/0609125[hep-ph/0609125 ] ] . s. pascoli , s. t. petcov and a. riotto , _ leptogenesis and low energy cp violation in neutrino physics _ , _ nucl . phys . _ * b774 * ( 2007 ) 152 [ http://arxiv.org/abs/hep-ph/0611338[hep-ph/0611338 ] ] . g. c. branco , r. gonzalez felipe and f. r. joaquim , _ a new bridge between leptonic cp violation and leptogenesis _ , _ phys . _ * b645 * ( 2007 ) 432436 [ http://arxiv.org/abs/hep-ph/0609297 [ hep - ph/0609297 ] ] . e. molinaro , s. t. petcov , t. shindou and y. takanishi , _ effects of lightest neutrino mass in leptogenesis _ , _ nucl . * b797 * ( 2008 ) 93116 [ http://arxiv.org/abs/0709.0413[0709.0413 ] ] . a. anisimov , s. blanchet and p. di bari , _ viability of dirac phase leptogenesis _ , _ jcap _ * 0804 * ( 2008 ) 033 [ http://arxiv.org/abs/0707.3024[0707.3024 ] ] . a. pilaftsis and t. e. j. underwood , _ resonant leptogenesis _ , _ nucl . phys . _ * b692 * ( 2004 ) 303345 [ http://arxiv.org/abs/hep-ph/0309342[hep-ph/0309342 ] ] . a. anisimov , a. broncano and m. plumacher , _ the cp - asymmetry in resonant leptogenesis _ , _ nucl . * b737 * ( 2006 ) 176189 [ http://arxiv.org/abs/hep-ph/0511248[hep-ph/0511248 ] ] . f. del aguila , j. a. aguilar - saavedra , a. martinez de la ossa and d. meloni , _ flavour and polarisation in heavy neutrino production at e+ e- colliders _ , _ phys . lett . _ * b613 * ( 2005 ) 170180 [ http://arxiv.org/abs/hep-ph/0502189[hep-ph/0502189 ] ] . t. han and b. zhang , _ signatures for majorana neutrinos at hadron colliders _ , _ phys . * 97 * ( 2006 ) 171804 [ http://arxiv.org/abs/hep-ph/0604064[hep-ph/0604064 ] ] . f. del aguila , j. a. aguilar - saavedra and r. pittau , _ heavy neutrino signals at large hadron colliders _ , _ jhep _ * 10 * ( 2007 ) 047 [ http://arxiv.org/abs/hep-ph/0703261[hep-ph/0703261 ] ] . w _ , _ review of particle physics _ , _ journal of physics g _ * 33 * ( 2006 ) 1 + . m. c. gonzalez - garcia and m. maltoni , _ phenomenology with massive neutrinos _ , _ phys . _ * 460 * ( 2008 ) 1129 [ http://arxiv.org/abs/0704.1800[0704.1800 ] ] . j. a. casas and a. ibarra , _ oscillating neutrinos and mu @xmath189 e , gamma _ , _ nucl . _ * b618 * ( 2001 ) 171204 [ http://arxiv.org/abs/hep-ph/0103065[hep-ph/0103065 ] ] . s. t. petcov , t. shindou and y. takanishi , _ majorana cp - violating phases , rg running of neutrino mixing parameters and charged lepton flavour violating decays _ , _ nucl . phys . _ * b738 * ( 2006 ) 219242 [ http://arxiv.org/abs/hep-ph/0508243[hep-ph/0508243 ] ] . a. d. sakharov , _ violation of cp invariance , c asymmetry , and baryon asymmetry of the universe _ , _ pisma zh . eksp . fiz . _ * 5 * ( 1967 ) 3235 . w. buchmuller , p. di bari and m. plumacher , _ leptogenesis for pedestrians _ , _ ann . * 315 * ( 2005 ) 305351 [ http://arxiv.org/abs/hep-ph/0401240[hep-ph/0401240 ] ] . m. fujii , k. hamaguchi and t. yanagida , _ leptogenesis with almost degenerate majorana neutrinos _ rev . _ * d65 * ( 2002 ) 115012 [ http://arxiv.org/abs/hep-ph/0202210[hep-ph/0202210 ] ] . s. blanchet and p. di bari , _ leptogenesis beyond the limit of hierarchical heavy neutrino masses _ , _ jcap _ * 0606 * ( 2006 ) 023 [ http://arxiv.org/abs/hep-ph/0603107[hep-ph/0603107 ] ] . di bari , _ leptogenesis , neutrino mixing data and the absolute neutrino mass scale _ , http://arxiv.org/abs/hep-ph/0406115 [ hep - ph/0406115 ] . g. f. giudice , a. notari , m. raidal , a. riotto and a. strumia , _ towards a complete theory of thermal leptogenesis in the sm and mssm _ , _ nucl . * b685 * ( 2004 ) 89149 [ http://arxiv.org/abs/hep-ph/0310123[hep-ph/0310123 ] ] . g. engelhard , y. grossman , e. nardi and y. nir , _ the importance of n2 leptogenesis _ , _ phys . * 99 * ( 2007 ) 081802 [ http://arxiv.org/abs/hep-ph/0612187[hep-ph/0612187 ] ] . collaboration , e. komatsu _ et . _ , _ five - year wilkinson microwave anisotropy probe ( wmap ) observations : cosmological interpretation _ , http://arxiv.org/abs/0803.0547[0803.0547 ] . s. blanchet and p. di bari , _ new aspects of leptogenesis bounds _ , http://arxiv.org/abs/0807.0743[0807.0743 ] . s. t. petcov , w. rodejohann , t. shindou and y. takanishi , _ the see - saw mechanism , neutrino yukawa couplings , lfv decays l(i ) @xmath189 l(j ) + gamma and leptogenesis _ , _ nucl . * b739 * ( 2006 ) 208233 [ http://arxiv.org/abs/hep-ph/0510404[hep-ph/0510404 ] ] . s. antusch , s. f. king and a. riotto , _ flavour - dependent leptogenesis with sequential dominance _ , _ jcap _ * 0611 * ( 2006 ) 011 [ http://arxiv.org/abs/hep-ph/0609038[hep-ph/0609038 ] ] . a. abada , p. hosteins , f .- x . josse - michaux and s. lavignac , _ successful leptogenesis in so(10 ) unification with a left- right symmetric seesaw mechanism _ , http://arxiv.org/abs/0808.2058[0808.2058 ] . f. borzumati and a. masiero , _ large muon and electron number violations in supergravity theories _ , _ phys . * 57 * ( 1986 ) 961 .
|
seesaw models with a slightly broken lepton number symmetry can explain small neutrino masses , and allow for low - scale leptogenesis .
we make a thorough analysis of leptogenesis within the simplest model with two right - handed ( rh ) neutrinos ( or with @xmath0 decoupled ) .
we obtain a semi - analytical formula for the final asymmetry in both supersymmetric and non - supersymmetric cases with a simple dependence on each parameter .
the low - energy parameters factorize from the high - energy ones , and the high - energy phase must be non - zero . the role of the pmns phases is carefully studied . moreover , we find that the breaking parameter in the yukawa coupling matrix must be relatively large , @xmath1 for normal and @xmath2 for inverted hierarchy
. therefore , leptogenesis in our simple model is incompatible with rh neutrino signals at future colliders or sizable lepton - flavor violation .
the other breaking parameter , @xmath3 , which appears in the rh neutrino mass matrix , can be much smaller , and actually needs to be so in order to have low - scale leptogenesis .
c i u
# 1 # 1#1 # 1 # 1#2#3phys .
lett .
* b # 1 * ( # 2 ) # 3 # 1#2#3nucl .
phys .
* b # 1 * ( # 2 ) # 3 # 1#2#3phys .
rev .
lett . *
# 1 * ( # 2 ) # 3 # 1#2#3phys . rev . * d # 1 * ( # 2 ) # 3 # 1#2#3z .
phys . * c # 1 * ( # 2 ) # 3 # 1#2#3class . and quantum grav . *
# 1 * ( # 2 ) # 3 # 1#2#3commun .
math .
phys . *
# 1 * ( # 2 ) # 3 # 1#2#3j . math .
phys . *
# 1 * ( # 2 ) # 3 # 1#2#3ann . of phys .
* # 1 * ( # 2 ) # 3 # 1#2#3phys .
rep . * # 1c * ( # 2 ) # 3 # 1#2#3progr .
theor .
phys . *
# 1 * ( # 2 ) # 3 # 1#2#3int .
j. mod .
phys . * a # 1 * ( # 2 ) # 3 # 1#2#3mod .
phys .
lett . * a # 1 * ( # 2 ) # 3 # 1#2#3nuovo cim .
* # 1 * ( # 2 ) # 3 # 1#2#3_ibid .
_ * # 1 * ( # 2 ) # 3
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
nearly all gaseous objects that shine are also observed to generate gaseous flows . this includes the jets of protostars , the stellar winds of massive o and b stars , the gentle solar wind of our sun , the associated absorption of bright quasars , and the spectacular jets of radio - loud agn . these gaseous outflows regulate the metal and dust content and distribution within the objects and their surroundings , moderate the accretion of new material , and inject energy and momentum into gas on large scales . developing a comprehensive model for these flows is critical to understanding the evolution of the source and its impact on the surrounding environment . starburst galaxies , whose luminosity is dominated by regions and massive stars , are also observed to drive gaseous outflows . these flows are generally expected ( and sometimes observed ) to have multiple phases , for example a hot and diffuse phase traced by x - ray emission together with a cool , denser phase traced by h@xmath6 emission ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? several spectacular examples in the local universe demonstrate that flows can extend to up to @xmath7 kpc from the galaxy @xcite carrying significant speed to escape from the gravitational potential well of the galaxy s dark matter halo ( e.g. * ? ? ? galactic outflows are also revealed by uv and optical absorption lines , e.g. , , and transitions . with the galaxy as a backlight , one observes gas that is predominantly blue - shifted which indicates a flow toward earth and away from the galaxy . these transitions are sensitive to the cool (; @xmath8k ) and warm (; @xmath9k ) phases of the flow . the incidence of cool gas outflows is nearly universal in vigorously star - forming galaxies ; this includes systems at low @xmath10 which exhibit and absorption @xcite , @xmath11 star - forming galaxies with winds traced by and transitions @xcite , and @xmath12 lyman break galaxies ( lbgs ) that show blue - shifted , , and transitions @xcite . the observation of metal - line absorption is now a well - established means of identifying outflows . furthermore , because the x - ray and h@xmath6 emission generated by winds is faint , absorption - line analyses have traditionally been the only way to probe outflows in distant galaxies . however , very little research has been directed toward comparing the observations against ( even idealized ) wind models ( e.g. * ? ? ? instead , researchers have gleaned what limited information is afforded by direct analysis of the absorption lines . the data diagnose the speed of the gas relative to the galaxy , yet they poorly constrain the optical depth , covering fraction , density , temperature , and distance of the flow from the galaxy . in turn , constraints related to the mass , energetics , and momentum of the flow suffer from orders of magnitude uncertainty . both the origin and impact of galactic - scale winds , therefore , remain open matters of debate @xcite . recent studies of @xmath13 star - forming galaxies have revealed that the cool outflowing gas often exhibits significant resonant - line emission ( e.g. , ) in tandem with the nearly ubiquitous blue - shifted absorption @xcite . the resultant spectra resemble the p - cygni profile characteristic of stellar winds . this phenomenon was first reported by @xcite , who observed blue - shifted absorption and red - shifted emission for the transition in the spectrum of the local starburst galaxy ngc 1808 . more recently , @xcite , who studied absorption in @xmath14 galaxies , reported emission in a small subset of the individual galaxy spectra of their large sample . these were excluded from the full analysis on concerns that the emission was related to agn activity . the stacked spectra of the remaining galaxies , however , also indicated emission , both directly and when the authors modeled and ` removed ' the @xmath15 absorption component . the authors suggested the emission could be related to back - scattered light in the wind , but presumed that it was related to weak agn activity . very similar emission was observed by @xcite who repeated the analysis of @xcite on a set of lower redshift galaxies . bright line emission has also been reported for individual galaxies at @xmath16 by ( * ? ? ? * see also rubin et al 2011 , in prep . ) . in their analysis of a single galaxy spectrum , @xcite further demonstrated that the line emission is spatially extended , and used the size of the emission to infer that the wind extends to at least 7kpc from the galaxy . these authors additionally detected line emission from non - resonant @xmath17 transitions , and attributed the emission to fluorescence powered by resonant absorption . in other words , these photons are re - emitted by the wind into our sightline , and are analogous to the emitted photons in a p - cygni profile . line - emission that may be related to outflows is also observed for @xmath4 lbgs in the resonant transition and non - resonant @xmath17 transitions . ] this emission likely arises from a completely different physical process than those generating x - ray and h@xmath6 emission ( e.g. , shocks ) , and presumably probes both the cool gas at the base of the wind and the outskirts of the flow ( i.e. , wherever a given transition is optically thick ) . a comprehensive analysis of the scattered and fluorescent emission related to galactic - scale outflows ( e.g. via deep integral - field - unit [ ifu ] observations , @xcite ) may offer unique diagnostics on the spatial extent , morphology , and density of the outflow from distant galaxies , eventually setting tighter constraints on the energetics of the flow . although astronomers are rapidly producing a wealth of observational datasets on galactic - scale winds , a key ingredient to a proper analysis is absent . just as comparisons between observed supernova lightcurves and spectra and radiative transfer calculations of of sne models have provided crucial insight into the physics driving , e.g. , standard candle relationships for both type ia and iip sne @xcite , modeling of the observable signatures of galactic outflows is necessary for understanding the physical properties of the gas and ultimately the physics driving the flows . in this paper , we take the first steps toward modeling the absorption and emission properties of cool gas outflows as observed in one dimensional spectra and by ifus . using monte carlo radiative transfer techniques , we study the nature of and absorption and emission for winds with a range of properties , accounting for the effects of resonant scattering and fluorescence . although the winds are idealized , the results frequently contradict our intuition and challenge the straightforward conversion of observables to ( even crude ) physical constraints . these findings have a direct bearing on recent and upcoming surveys of galactic outflows , particularly those which make use of and transitions to probe outflow properties . the paper is organized as follows . in @xmath18 [ sec : method ] , we describe the methodology of our radiative transfer algorithms . these are applied to a fiducial wind model in @xmath18 [ sec : fiducial ] and variations of this model in @xmath18 [ sec : variants ] . in @xmath18 [ sec : alternate ] , we explore wind models with a broader range of density and velocity laws . we discuss the principal results and connect to observations in @xmath18 [ sec : discuss ] . a brief summary is given in @xmath18 [ sec : summary ] . this section describes our methodology for generating emission / absorption profiles from simple wind models . lcccccc uv1 & 38458.98 & 0.00 & 9/2 & 9/2 & 2600.173 & 2.36e+08 + & 38458.98 & 384.79 & 9/2 & 7/2 & 2626.451 & 3.41e+07 + & 38660.04 & 0.00 & 7/2 & 9/2 & 2586.650 & 8.61e+07 + & 38660.04 & 384.79 & 7/2 & 7/2 & 2612.654 & 1.23e+08 + & 38660.04 & 667.68 & 7/2 & 5/2 & 2632.108 & 6.21e+07 + & 38858.96 & 667.68 & 5/2 & 5/2 & 2618.399 & 4.91e+07 + & 38858.96 & 862.62 & 5/2 & 3/2 & 2631.832 & 8.39e+07 + + [ -1.5ex ] & 35760.89 & 0.00 & 3/2 & 0 & 2796.351 & 2.63e+08 + & 35669.34 & 0.00 & 1/2 & 0 & 2803.528 & 2.60e+08 + doublet and the uv1 multiplet of transitions ( based on figure 7 from @xcite ) . each transition shown is labeled by its rest wavelength ( ) and einstein a - coefficient ( s@xmath19 ) . black upward arrows indicate the resonance - line transitions , i.e. those connected to the ground state . the 2p@xmath203p configuration of mg@xmath21 is split into two energy levels that give rise to the @xmath0 doublet . both the 3d@xmath204s ground state and 3d@xmath204p upper level of fe@xmath21 exhibit fine - structure splitting that gives rise to a series of electric - dipole transitions . the downward ( green ) arrows show the * transitions that are connected to the resonance - line transitions ( i.e. they share the same upper energy levels ) . we also show a pair of transitions ( @xmath22 ; red and dashed lines ) that arise from higher levels in the z @xmath20d@xmath23configuration . these transitions have not yet been observed in galactic - scale outflows and are not considered in our analysis . , width=336 ] in this paper , we focus on two sets of radiative transitions arising from fe@xmath21 and mg@xmath21 ions ( table [ tab : atomic ] , figure [ fig : energy ] ) . this is a necessarily limited set , but the two ions and their transitions do have characteristics shared by the majority of low - ion transitions observed in cool - gas outflows . therefore , many of the results that follow may be generalized to observational studies that consider other atoms and ions tracing cool gas . the mg@xmath21 ion , with a single 3s electron in the ground - state , exhibits an alkali doublet of transitions at @xmath24 analogous to the doublet of neutral hydrogen . figure [ fig : energy ] presents the energy level diagram for this @xmath0 doublet . in non - relativistic quantum mechanics , the 2p@xmath203p energy level is said to be split by spin - orbit coupling giving the observed line doublet . these are the only electric - dipole transitions with wavelengths near 2800 and the transition connecting the @xmath25 and @xmath26 states is forbidden by several selection rules . therefore , an absorption from 2p@xmath203s @xmath27 2p@xmath203pis followed @xmath28 of the time by a spontaneous decay ( @xmath29s ) to the ground state . our treatment will ignore any other possibilities ( e.g. absorption by a second photon when the electron is at the 2p@xmath203p level ) . in terms of radiative transfer , the @xmath0 doublet is very similar to that for , the @xmath30 doublet , and many other doublets commonly studied in the interstellar medium ( ism ) of distant galaxies . each of these has the ground - state connected to a pair of electric dipole transitions with nearly identical energy . the doublets differ only in their rest wavelengths and the energy of the doublet separation . for , the separation is sufficiently small ( @xmath31 ) that most radiative transfer treatments actually ignore it is a doublet . this is generally justifiable for because most astrophysical processes have turbulent motions that significantly exceed the doublet s velocity separation and effectively mix the two transitions . for ( @xmath32 ) , ( @xmath33 ) , and most of the other doublets commonly observed , the separation is large and the transitions must be treated separately . iron exhibits the most complex set of energy levels for elements frequently studied in astrophysics . the fe@xmath21 ion alone has millions of energy levels recorded @xcite , and even this is an incomplete list . one reason for iron s complexity is that the majority of its configurations exhibit fine - structure splitting . this includes the ground - state configuration ( a @xmath20d@xmath23 ) which is split into 5 levels , labeled by the total angular momentum @xmath34 , with excitation energies @xmath35 ranging from @xmath36k ( figure [ fig : energy ] ) . transitions between these fine - structure levels are forbidden ( magnetic - dipole ) and have spontaneous decay times of several hours . in this paper , we examine transitions between the ground - state configuration and the energy levels of the z @xmath20d@xmath23configuration . this set of transitions ( named the uv1 multiplet ) have wavelengths near 2600 . there are two resonance - line transitions . ] associated with this multiplet ( @xmath37 ) corresponding to @xmath38 ; these are indicated by upward ( black ) arrows in figure [ fig : energy ] . the solid ( green ) downward arrows in figure [ fig : energy ] mark the non - resonant * transitions that are connected to the upper energy levels of the resonance lines . these transitions may occur following the absorption of a single photon by fe@xmath21 in its ground - state . this process may also be referred to as fluorescence . note that two of these transitions ( * @xmath39 ) are close enough in energy that their line profiles can overlap . the figure also shows ( as dashed , downward arrows ) two of the * transitions that connect to higher energy levels of the z @xmath20d@xmath23configuration . ignoring collisions and recombinations , these transitions may only occur after the absorption of two photons : one to raise the electron from the ground - state to an excited state and another to raise the electron from the excited state to one of the z @xmath20d@xmath23 levels with @xmath40 . the excitation of fine - structure levels by the absorption of uv photons is termed indirect uv pumping ( e.g * ? ? ? * ; * ? ? ? * ) and requires the ion to lie near an intense source of uv photons . even a bright , star - forming galaxy emits too few photons at @xmath41 to uv - pump fe@xmath21 ions that are farther than @xmath42pc from the stars . in the following , we will assume that emission from this process is negligible . our calculations also ignore collisional processes , i.e. collisional excitation and de - excitation of the various levels . for the fine - structure levels of the a @xmath20d@xmath23 configuration , the excitation energies are modest ( @xmath43k ) but the critical density @xmath44 is large . for the a @xmath20d@xmath23@xmath45a @xmath20d@xmath23@xmath46 transition , the critical density @xmath47 . at these densities , one would predict detectable quantities of which has not yet been observed in galactic - scale outflows . if collisional excitation is insignificant then one may also neglect collisional de - excitation . furthermore , observations rarely show _ absorption _ from the fine - structure levels of the a @xmath20d@xmath23 configuration and that material is not significantly blue - shifted ( rubin et al . , in prep ) . in the following , we assume that electrons only occupy the ground - state , i.e. the gas has zero opacity to the non - resonant lines . regarding the doublet , its excitation energy is significantly higher implying negligible collisional processes at essentially any density . nearly all of the absorption studies of galactic - scale outflows have focused on intensely , star - forming galaxies . the intrinsic emission of these galaxies is a complex combination of light from stars and regions that is then modulated by dust and gas within the ism . for the spectral regions studied here , the hottest stars show a featureless continuum , but later spectral types do show significant and absorption . in addition , asymmetric and/or blue - shifted absorption is exhibited in these transitions in a and f stars driving stellar winds @xcite . p cygni profiles are observed in a handful of f stars by @xcite , who attribute the emission to chromospheric activity rather than mass - loss effects . regions , meanwhile , are observed to emit weakly at the @xmath0 doublet , primarily due to recombinations in the outer layers @xcite . it is beyond the scope of this paper to properly model the stellar absorption and region emission , but the reader should be aware that they can complicate the observed spectrum , independently of any outflow , especially at velocities @xmath48 . in the following , we assume a simple flat continuum normalized to unit value . the size of the emitting region @xmath49 is a free parameter , but we restrict its value to be smaller than the minimum radial extent of any gaseous component . lastly , the source does not absorb any scattered or emitted photons . we calculated spectra using a 3d monte carlo radiation transport code originally designed for supernova outflows @xcite but modified to treat resonant line transport on a galactic scale @xcite . our methods are similar to those used in several other codes which focus on ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , except that here we include the effects of multiple line scattering and fluorescence . for our 3-dimensional calculations , the wind properties ( density , temperature and velocity ) were discretized on a @xmath50 cartesian grid ( for certain 1-d calculations , the wind properties were simply computed on the fly from analytic formulae ) . the radiation field was represented by @xmath51 photon packets ( typically @xmath52 ) which were initially emitted isotropically and uniformly throughout the source region ( @xmath53 ) . the wavelengths of the packets were sampled from a flat spectral energy distribution . photon packets were tracked through randomized scattering and absorption events until they escaped the computational domain . the distance to the next packet interaction event was determined by doppler shifting packets to the comoving frame and sampling the mean free path to each resonance line ( and dust , if present ) . the resonant line opacity followed a voigt profile which was determined using the analytic fits of @xcite . non - resonant lines were assumed to be completely optically thin , and the dust opacity was assumed to be wavelength independent and completely absorbing ( @xmath18 [ sec : dust_method ] ) . because of the relatively low optical depths encountered in these models , no `` core - skipping '' scheme was applied to accelerate the resonant line transport , and recoil effects were ignored . in a resonant line interaction , a photon excites an atom from the ground state to an excited level . the end result is either scattering ( i.e. , de - excitation back to the ground state ) or fluorescence ( de - excitation to another excited level ) . the probability that the atom de - excites to lower level @xmath54 is @xmath55 , where the @xmath56 are the einstein spontaneous emission coefficients and the sum runs over all levels accessible from the upper state @xmath57 . for each interaction event , a random number was drawn to sample the final state from this probability distribution . if the result was a scattering , the process was assumed to be coherent in the comoving frame . if the result was fluorescence , the packet was reemitted at the line center wavelength of the new transition . natural line broadening in fluorescence was ignored given the high velocity gradients in our models . in all cases , the angular redistribution was assumed to be isotropic . to generate multi - dimensional images and spectra of the system , an escape probability method was used . at the initiation of a packet , and at every subsequent interaction event , we calculated the probability @xmath58 , that the packet de - excited into state @xmath54 and escaped the domain in some pre - specified direction . here @xmath59 is the optical depth to infinity , which was constructed by integration along the path to escape , taking into account the relevant doppler shifts and possible absorption by lines or dust . the contribution of the packet to the final spectrum was then added in for every possible final state , each shifted to the proper observer - frame wavelength and weighted by the probability @xmath60 . lying above an opaque emitting surface at velocity coordinate 100 km s@xmath19 . the wind extended to 2,000 km s@xmath19 and the sobolev optical depth had values of @xmath61 ( black lines ) , @xmath62 ( red lines ) , and @xmath63 ( blue lines ) . discrepancies in the @xmath63 case are likely due to the breakdown of the sobolev approximation ( @xmath18 [ sec : sobolev ] ) at high optical depth . , width=336 ] to validate the monte carlo code , we calculated a series of test problems consisting of a point source embedded in a spherical , homogenous medium of varying resonant line optical depth . the resulting resonant line profiles ( see * ? ? ? * ) were found to be in good agreement with the standard analytic solutions @xcite . as a further test , we calculated line profiles for the case of an extended spherical source embedded in a homologously expanding wind with large velocity gradient . these model spectra ( figure [ fig : oneline_test ] ) are in good agreement with ones determined by direct numerical integration of the radiation transport equation under the sobolev approximation ( @xmath18 [ sec : sobolev ] ; e.g. , * ? ? ? * ) . for the majority of models studied in this paper , we assume the gas contains no dust . this is an invalid assumption , especially for material associated with the ism of a galaxy . essentially all astrophysical environments that contain both cool gas and metals also show signatures of dust depletion and extinction . this includes the ism of star - forming and -selected galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , strong metal - line absorption systems @xcite , and the galactic winds traced by low - ion transitions @xcite . extraplanar material likely associated with a galactic - scale outflow has been observed to emit ir radiation characteristic of dust ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition , @xcite have argued from a statistical analysis that dust is distributed to many tens of kpc from @xmath64 galaxies and have suggested it was transported from the galaxies by galactic - scale winds . although the galactic winds traced specifically by and transitions have not ( yet ) been demonstrated to contain dust , it is reasonable to consider its effects . for analysis on normalized spectra ( i.e. absorption lines ) , the effects of dust are largely minimized ; dust has a nearly constant opacity over small spectral regions and all features are simply scaled together . for scattered and resonantly trapped photons , however , the relative effect of dust extinction can be much greater . these photons travel a much longer distance to escape the medium and may experience a much higher integrated opacity from dust . indeed , dust is frequently invoked to explain the weak ( or absent ) emission from star - forming galaxies ( e.g. * ? ? ? although the transitions studied here have much lower opacity than , dust could still play an important role in the predicted profiles . in a few models , we include absorption by dust under the following assumptions : ( i ) the dust opacity scales with the density of the gas ( i.e. we adopt a fixed dust - to - gas ratio ) ; ( ii ) the opacity is independent of wavelength , a reasonable approximation given the small spectral range analyzed ; ( iii ) dust absorbs but does not scatter photons ; ( iv ) the photons absorbed by dust are re - emitted at ir wavelengths and are ` lost ' from the system . the dust absorption is normalized by @xmath65 , the integrated opacity of dust from the center of the system to infinity . the ambient ism of a star - forming galaxy may be expected to exhibit @xmath65 values of one to a few at @xmath66 ( e.g. * ? ? ? a photon propagating through a differentially expanding medium interacts with a line only when its comoving frame wavelength is doppler - shifted into near resonance with the line center rest wavelength . for a wind with a steep velocity law ( i.e. a large gradient @xmath67 ) and/or a narrow intrinsic profile ( i.e. a small doppler parameter ) , the spatial extent of the region of resonance may be much smaller than the length scale of the wind itself . the interaction can then be considered to occur at a point . in this case , @xcite introduced a formalism that gives the line optical depth @xmath68 at a given point in terms of the density and velocity gradient of the flow at that radius . for a wind in homologous expansion , this optical depth is independent of the direction of propagation and is given by @xmath69 where @xmath70 is the integrated line opacity , @xmath71 the line - center rest wavelength , @xmath72 the oscillator strength and @xmath73 the density in the lower level of the transition . we have neglected corrections for stimulated emission . this optical depth applies to a photon with wavelength @xmath74 where @xmath75 is the direction of propagation . the probability that such a photon is scattered / absorbed at the point of resonance is simply @xmath76 $ ] . we find that the sobolev approximation applies for nearly all of the models presented in this paper , and therefore provides a convenient approach to estimating the optical depth . in this section , we study a simple yet illustrative wind model for a galactic - scale outflow . the properties of this wind were tuned , in part , to yield a absorption profile similar to those observed for @xmath11 , star - forming galaxies @xcite . we emphasize , however , that we do not favor this fiducial model over any other wind scenario nor do its properties have special physical motivation . its role is to establish a baseline for discussion . the fiducial wind is isotropic , dust - free , and extends from an inner wind radius @xmath77 to an outer wind radius @xmath78 . it follows a density law , @xmath79 and a velocity law with a purely radial flow @xmath80 for the fiducial case , the hydrogen density at the inner radius is @xmath81 and the velocity at the outer radius is @xmath82 . turbulent motions are characterized by a doppler parameter only tends to modify the widths and modestly shift the centroids of emission lines . ] we convert the hydrogen density @xmath84 to the number densities of mg@xmath21 and fe@xmath21 ions by assuming solar relative abundances with an absolute metallicity of 1/2 solar and depletion factors of 1/10 and 1/20 for mg and fe respectively , i.e. @xmath85 and @xmath86=@xmath87/2 . at the center of the wind is a homogeneous source of continuum photons with size @xmath49 . the parameters for the fiducial wind model are summarized in table [ tab : fiducial ] . ccl density law & @xmath88 & @xmath89 + velocity law & @xmath90 & @xmath91 + inner radius & @xmath77 & 1kpc + outer radius & @xmath78 & 20kpc + source size & @xmath49 & 0.5kpc + density normalization & @xmath92 & @xmath93 at @xmath77 + velocity normalization & @xmath94 & 1000at @xmath78 + turbulence & @xmath95 & 15 + mg@xmath21 normalization & @xmath87 & @xmath96 + fe@xmath21 normalization & @xmath86 & @xmath87/2 + in figure [ fig : fiducial_nvt ] we plot the density and velocity laws against radius ; their simple power - law expressions are evident . the figure also shows the optical depth profile for the @xmath97 transition ( @xmath98 ) , estimated from the sobolev approximation first as a function of velocity by summing the opacity for a series of discrete and small radial intervals between @xmath77 and @xmath78 . we then mapped @xmath98 onto radius using the velocity law ( equation [ eqn : vel ] ) . this optical depth profile is shown as the black dotted line in figure [ fig : fiducial_nvt ] . ] ( equation [ eqn : sobolev ] ) . the @xmath98 profile peaks with @xmath99 at a velocity @xmath100 corresponding to @xmath101 . the optical depth profile for the @xmath102 transition ( not plotted ) is scaled down by the @xmath103 ratio but is otherwise identical . similarly , the optical depth profiles for the @xmath37 transitions are scaled down by @xmath104 and the @xmath86/@xmath87 ratio . for the spatially integrated ( i.e. , 1d ) spectra , the actual dimensions and density of the wind are unimportant provided they scale together to give nearly the same optical depth . therefore , one may consider the choices for @xmath105 , and @xmath106 as largely arbitrary . nevertheless , we adopted values for this fiducial model with some astrophysical motivation , e.g. , values that correspond to galactic dimensions and a normalization that gives @xmath107 . optical depth profiles ( solid and dotted ; black ) for the fiducial wind model ( see table [ tab : fiducial ] for details ) . the density and velocity laws are simple @xmath108 and @xmath109 power - laws ; these curves have been scaled for plotting convenience . the optical depth profile was calculated two ways : ( i ) using the sobolev approximation ( equation [ eqn : sobolev ] ; solid black curve ) and ( ii ) summing the opacity at small and discrete radial intervals in velocity space and then converting to radius with the velocity law ( doted black curve ) . these give very similar results . the wind parameters were set to give an optically thick medium at the inner radius ( @xmath110kpc ) that becomes optically thin at the outer radius ( @xmath111kpc ) . , height=336 ] ccr mgii 2796 & [ @xmath112 & 2.96 + & [ @xmath113&@xmath114 + mgii 2803 & [ @xmath115 & 1.29 + & [ @xmath116&@xmath117 + feii 2586 & [ @xmath118 & 0.59 + & [ @xmath119&@xmath120 + feii 2600 & [ @xmath121 & 1.16 + & [ @xmath122&@xmath123 + feii * 2612 & [ @xmath124&@xmath125 + feii * 2626 & [ @xmath126&@xmath127 + feii * 2632 & [ @xmath128&@xmath129 + using the methodology described in @xmath18 [ sec : method ] , we propagated photons from the source and through the outflow to an ` observer ' at @xmath130 who views the entire wind+source complex . figure [ fig : fiducial_1d ] presents the 1d spectrum that this observer would record , with the unattenuated flux normalized to unit value . the doublet shows the canonical ` p - cygni profile ' that characterizes a continuum source embedded within an outflow . strong absorption is evident at @xmath131 in both transitions ( with equivalent widths @xmath132 and @xmath133 ) and each shows emission at positive velocities . for an isotropic and dust - free model , the total equivalent width of the doublet must be zero , i.e. every photon absorbed eventually escapes the system , typically at lower energy . the wind simply shuffles the photons in frequency space . a simple summation of the absorption and emission equivalent widths ( table [ tab : fiducial_ew ] ) confirms this expectation . focusing further on the absorption , one notes that the profiles lie well above zero intensity and have similar depth even though their @xmath103 values differ by a factor of two . in standard absorption - line analysis , this is the tell - tale signature of a ` cloud ' that has a high optical depth ( i.e.is saturated ) which only partially covers the emitting source ( e.g. * ? ? ? * ; * ? ? ? * ) . our fiducial wind model , however , _ entirely covers the source _ ; the apparent partial covering must be related to a different effect . figure [ fig : noemiss ] further emphasizes this point by comparing the absorption profiles from figure [ fig : fiducial_1d ] against an artificial model where no absorbed photons are re - emitted . as expected from the @xmath98 profile ( figure [ fig : fiducial_nvt ] ) , this ` intrinsic ' model produces a strong @xmath0 doublet that absorbs all photons at @xmath134 , i.e. @xmath135 . the true model , in contrast , has been ` filled in ' at @xmath136 by photons scattered in the wind . an absorption - line analysis that ignores these effects would ( i ) systematically underestimate the true optical depth and/or ( ii ) falsely conclude that the wind partially covers the source . we will find that these are generic results , even for wind models that include dust and are not fully isotropic . turning to the emission profiles of the @xmath0 doublet , one notes that they are quite similar with comparable equivalent widths . this is because the gas is optically thick yielding comparable total absorption . the flux of the @xmath102 transition even exceeds that for @xmath97 because the wind speed is greater than the velocity separation of the doublet , @xmath137 . therefore , the red wing of the @xmath97 emission profile is partially absorbed by @xmath102 and re - emitted at lower frequency . this yields a line ratio that is far below the @xmath138 ratio that one may have naively expected ( e.g. if the line - emission resulted from recombinations ) , and leads us to conclude that the relative strengths of the emission lines are sensitive to both the opacity and velocity extent of the wind . now consider the transitions : the bottom left panel of figure [ fig : fiducial_1d ] covers the majority of the uv1 transitions and several are shown in the velocity plot . the line profile for @xmath139 is very similar to the @xmath0 doublet ; one observes strong absorption to negative velocities and strong emission at @xmath140 producing a characteristic p - cygni profile . splitting the profile at @xmath141 , we measure an equivalent width @xmath142 in absorption and @xmath143 in emission ( table [ tab : fiducial_ew ] ) for a total equivalent width of @xmath144 . in contrast , the @xmath145 resonance line shows much weaker emission and a much higher total equivalent width ( @xmath146 ) , even though the line has a @xmath147 lower @xmath103 value . these differences between the resonance lines ( and between and ) occur because of the complex of non - resonant * transitions that are coupled to the resonance lines ( figure [ fig : energy ] ) . specifically , a resonance photon absorbed at @xmath37 has a finite probability of being re - emitted as a non - resonant photon which then escapes the system without further interaction . the principal effects are to reduce the line emission of @xmath37 and to produce non - resonant line - emission ( e.g.*@xmath148 ) . the reduced @xmath145 emission relative to @xmath139 is related to two factors : ( i ) there is an additional downward transition from the z @xmath20d@xmath23@xmath46 level and ( ii ) the einstein a coefficients of the non - resonant lines coupled to @xmath145 are comparable to and even exceed the einstein a coefficient of the resonant transition . in contrast , the * @xmath149 transition ( associated with @xmath139 ) has an approximately @xmath150 smaller a coefficient than the corresponding resonance line . therefore , the majority of photons absorbed at @xmath1 are re - emitted as @xmath139 photons , whereas the majority of photons absorbed at @xmath151 are re - emitted at longer wavelengths ( * @xmath148 or @xmath152 ) . if we increase @xmath153 ( and especially if we include gas with @xmath154 ) then the @xmath139 emission is significantly suppressed ( e.g. @xmath18 [ sec : ism ] ) . the total equivalent width , however , of the three lines connected to the z @xmath20d@xmath23@xmath46 upper level must still vanish ( photons are conserved in this isotropic , dust - free model ) . transition for the source+wind complex of the fiducial model . the middle panel shows the 1d spectrum with @xmath155 corresponding to @xmath156 and the dotted vertical curves indicate the velocity slices for the emission maps . the source has a size @xmath157kpc , traced by a few pixels at the center of each map . at @xmath158 , the wind has an optical depth of @xmath159 and the source contributes roughly half of the observed flux . at @xmath160 the wind absorbs all photons from the source and the observed emission is entirely due to photons scattered by the wind . amazingly , this emission exceeds the integrated flux at @xmath161 such that the @xmath97 line center is offset from the ( intrinsic ) peak in optical depth . at @xmath162 , both the source and wind contribute to the observed emission . at all velocities , the majority of emission comes from the inner @xmath163kpc . , width=336 ] the preceding discussion emphasizes the filling - in of resonance absorption at @xmath164 and the generation of emission lines at @xmath165 by photons scattered in the wind . we also mapped the emission of the fiducial model to sudy its spatial extent ( see @xmath18 [ sec : monte ] for a description of the algorithm ) . the output is a set of surface - brightness maps in a series of frequency channels yielding a dataset analogous to integral - field - unit ( ifu ) observations . in figure [ fig : fiducial_ifu_mgii ] , we present the output at several velocities relative to the @xmath97transition . at @xmath166 , where the wind has an optical depth @xmath167 ( figure [ fig : fiducial_nvt ] ) , the source contributes roughly half of the observed flux . at @xmath168 , however , the wind absorbs all photons from the source and the observed emission is entirely from photons scattered by the wind . this scattered emission actually exceeds the source+wind emission at @xmath166 such that the absorption profile is negatively offset from the velocity where @xmath98 is maximal ( figure [ fig : fiducial_nvt ] ) . the net result is weaker absorption that peaks blueward of the actual peak in the optical depth profile . table [ tab : line_diag ] reports several kinematic measurements of the absorption and emission features . clearly , these effects complicate estimates for the speed , covering fraction , and total column density of the wind . at @xmath169 , the wind and source have comparable total flux with the latter dominating at higher velocities . transition for the source+wind complex . the results are very similar to those observed for the @xmath0 doublet ( figure [ fig : fiducial_ifu_mgii ] ) . ( lower ) surface - brightness emission maps around the @xmath170 transition for the source+wind complex . in this case , the source is unattenuated yet scattered photons from the wind also make a significant contribution . , width=336 ] similar results are observed for the resonance transitions ( figure [ fig : fiducial_ifu_feii ] ) . for transitions to fine - structure levels of the a @xmath20d@xmath23configuration , the source is unattenuated but there is a significant contribution from photons generated in the wind . at all velocities , the majority of light comes from the inner regions of the wind . the majority of emission occurs within the inner few kpc , e.g. @xmath171 of the light at @xmath172 to @xmath173 comes from @xmath174kpc . the emission is even more centrally concentrated for the transitions . a proper treatment of these distributions is critical for interpreting observations acquired through a slit , i.e. where the aperture has a limited extent in one or more dimensions . a standard longslit on 10m - class telescopes , for example , subtends @xmath175 corresponding to @xmath176kpc for @xmath11 . we return to this issue in @xmath18 [ sec : discuss ] . the results presented in figures [ fig : fiducial_ifu_mgii ] and [ fig : fiducial_ifu_feii ] are sensitive to the radial extent , morphology , density and velocity profiles of this galactic - scale wind . consequently , ifu observations of line emission from low - ion transitions may offer the most direct constraints on galactic - scale wind properties . steradians as viewed from @xmath177 ( source uncovered ) to @xmath178 ( source covered ) . one detects significant emission for all orientations but significant absorption only for @xmath179 . the velocity centroid of the emission shifts from positive to negative velocities as @xmath180 increases and one transitions from viewing the wind as lying behind the source to in front of it . the velocity centroid of emission , therefore , diagnoses the degree of anisotropy for the wind . , width=336 ] in this section , we investigate a series of more complex wind scenarios through modifications to the fiducial model . these include relaxing the assumption of isotropy , introducing dust , adding an ism component within @xmath77 , and varying the normalization of the optical depth profiles . the fiducial model assumes an isotropic wind with only radial variations in velocity and density . angular isotropy is obviously an idealized case , but it is frequently assumed in studies of galactic - scale outflows ( e.g. * ? ? ? there are several reasons , however , to consider anisotropic winds . firstly , galaxies are not spherically symmetric ; the sources driving the wind ( e.g. supernovae , agn ) are very unlikely to be spherically distributed within the galaxy . secondly , the galactic ism frequently has a disk - like morphology which will suppress the wind preferentially at low galactic latitudes , perhaps yielding a bi - conic morphology ( e.g. * ? ? ? * ; * ? ? ? lastly , the galaxy could be surrounded by an aspherical gaseous halo whose interaction would produce an irregular outflow . with these considerations in mind , we reanalyzed the fiducial model with the 3d algorithm after departing from isotropy . it is beyond the scope of this paper to explore a full suite of anisotropic profiles . we consider two simple examples : ( i ) half the fiducial model , where the wind density is set to zero for @xmath181 steradians ( i.e. a hemispherical wind ) . this model is viewed from @xmath177 ( source uncovered ) to @xmath178 ( source covered ) ; and ( ii ) a bi - conical wind which fills @xmath182 both into and out of the plane of the sky and that is viewed along the axis of rotational symmetry ( @xmath183 is defined along this axis ) . is a full wind ) . for modest angles , one observes similar results to the fiducial model but larger angles ( @xmath184 ) the emission is significantly suppressed at all velocities . , width=336 ] the resulting and profiles for the half wind are compared against the fiducial model ( isotropic wind ) in figure [ fig : anisotropic ] . examining the @xmath0 doublet , the @xmath177 model only shows line - emission from photons scattered off the back side . these photons , by definition , have @xmath185 relative to line - center ( a subset have @xmath186 because of turbulent motions in the wind ) . when viewed from the opposite direction ( @xmath178 ) , the absorption lines dominate , but there is still significant line - emission at @xmath187 and at @xmath188 from photons that scatter through the wind which fills in the absorption . the key difference between this and the isotropic wind is the absence of photons scattered to @xmath189 ; this also implies deeper @xmath102 absorption at @xmath190 . the shifts in velocity centroid and asymmetry of the emission lines serve to diagnose the degree of wind isotropy , especially in conjunction with analysis of the absorption profiles . the results are similar for the @xmath37 resonance lines . the @xmath191 line , meanwhile , shows most clearly the offset in velocity between the source unobscured ( @xmath177 ) and source covered ( @xmath178 ) cases . the offset of the * lines is the most significant difference from the fully isotropic wind . we have also analyzed the predicted profiles for a series of bi - conical winds with @xmath192 , each viewed along the axis of rotational asymmetry with the wind covering the source . figure [ fig : biconical ] and table [ tab : line_diag ] summarizes the results for several cases . for @xmath193 , the results are very similar to the fiducial model ; the profiles show similar equivalent widths for absorption and emission . the key quantitative difference is that the * line - emission is modestly suppressed at @xmath194 because the portion of the wind with that projected velocity has been removed . for more highly collimated winds @xmath195 , however , the line - emission is several times weaker than the fiducial model . similarly , the absorption - line profiles more closely track the intrinsic optical depth of the outflow . of course , these same models when viewed ` sideways ' would show no absorption but strong line - emission . such events are sufficiently rare ( rubin et al . , in prep ) that we consider this level of anisotropy to be uncommon . nevertheless , the results for a bi - conic clearly have important implications for the nature of absorption and line - emission in our fiducial model . ) . the primary effect of dust is to suppress the line - emission relative to the continuum . a more subtle but important effect is that the redder photons in the emission lines ( corresponding to positive velocity offsets relative to line - center ) suffer greater extinction . this occurs because the ` redder ' photons that we view have travelled farther to scatter off the backside of the wind . note that the absorption lines are nearly unmodified until @xmath196 , a level of extinction that would preclude observing the source altogether . , width=336 ] as described in @xmath18 [ sec : dust_method ] , one generally expects dust in astrophysical environments that contain cool gas and metals . this dust modifies the observed wind profiles in two ways . first , it is a source of opacity for all of the photons . this suppresses the flux at all wavelengths by @xmath197 but because we re - normalize the profiles , this effect is essentially ignored . second , photons that are scattered by the wind must travel a greater distance to escape and therefore suffer from greater extinction . a photon that is trapped for many scatterings has an increased probability of being absorbed by dust . section [ sec : dust_method ] describes the details of our treatment of dust ; we remind the reader here that we assume a constant dust - to - gas ratio that is normalized by the total optical depth @xmath65 photons would experience if they traveled from the source to infinity without scattering . line - emission from the fiducial wind model , as attenuated by dust with a range of @xmath65values . the black ( solid ) curve shows the model results . the dashed ( green ) curve shows @xmath198 . it is obvious that the equivalent widths ( i.e. , the flux relative to the continuum ) do not follow this scaling . the dotted ( red ) line , meanwhile , plots @xmath199 . this simple approximation is a good representation of the results for our radiative transfer calculations . , height=336 ] in figure [ fig : dust ] , we show the and profiles of the fiducial model ( @xmath200 ) against a series of models with @xmath201 . for the transitions , the dominant effect is the suppression of line emission at @xmath202 . these ` red ' photons have scattered off the backside of the wind and must travel a longer path than other photons . dust leads to a differential reddening that increases with velocity relative to line - center . this is a natural consequence of dust extinction and is most evident in the * @xmath148 emission profile which is symmetrically distributed around @xmath169 in the @xmath203 model . the degree of suppression of the line - emission is relatively modest , however . specifically , we find that the flux is reduced by a factor of the order of ( 1+@xmath65)@xmath19 ( figure [ fig : dust_tau ] ) , instead of the factor exp(@xmath65 ) that one may have naively predicted . in terms of absorption , the profiles are nearly identical for @xmath204 . one requires very high extinction to produce a deepening of the profiles at @xmath190 . we conclude that dust has only a modest influence on this fiducial model and , by inference , on models with moderate peak optical depths and significant velocity gradients with radius ( i.e. scenarios in which the photons scatter only one to a few times before exiting ) . for qualitative changes , one requires an extreme level of extinction ( @xmath196 ) . in this case , the source would be extinguished by 15magnitudes and could never be observed . even @xmath205 is larger than typically inferred for the star - forming galaxies that drive outflows ( e.g. * ? ? ? for the emission lines , the dominant effect is a reduction in the flux with a greater extinction at higher velocities relative to line - center . in these respects , dust extinction crudely mimics the behavior of the anisotropic wind described in section [ sec : anisotropic ] . in the emission lines from @xmath48 for the fiducial model to @xmath206 for the ism+wind model . the profiles , meanwhile , show several qualitative differences . the @xmath37 resonance transitions each exhibit much greater absorption at @xmath48 than the fiducial model . the resonant line - emission is also substantially reduced , implying much higher fluxes for the non - resonant lines ( e.g.*@xmath148,2626 ) . lastly , we note that the @xmath145 absorption profile provides a good ( albeit imperfect ) representation of the wind opacity and therefore offers the best characterization of an ism component . , width=336 ] the fiducial model does not include gas associated with the ism of the galaxy , i.e. material at @xmath207kpc with @xmath208 . this allowed us to focus on results related solely to a wind component . the decision to ignore the ism was also motivated by the general absence of significant absorption at @xmath187 in galaxies that exhibit outflows ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? on the other hand , the stars that comprise the source are very likely embedded within and fueled by gas from the ism . consider , then , a modification to the fiducial model that includes an ism . specifically , we assume the ism component has density @xmath209 for @xmath210 with @xmath211kpc , an average velocity of @xmath212 , and a larger turbulent velocity @xmath213 . the resultant optical depth profile @xmath98 is identical to the fiducial model for @xmath214kpc , has a slightly higher opacity at @xmath215kpc , and has a very large opacity at @xmath216kpc . in figure [ fig : ism_spec ] , the solid curves show the and profiles for the ism+wind and fiducial models . in comparison , the dotted curve shows the intrinsic absorption profile for the ism+wind model corresponding to the ( unphysical ) case where none of the absorbed photons are scattered or re - emitted . focus first on the doublet . as expected , the dotted curve shows strong absorption at @xmath187 and blueward . the full models , in contrast , show non - zero flux at these velocities and even a normalized flux exceeding unity at @xmath187 . in fact , the ism+wind model is nearly identical to the fiducial model ; the only quantitative difference is that the velocity centroids of the emission lines are shifted redward by @xmath217 . we have also examined the spatial distribution of emission from this model and find results qualitatively similar to the fiducial wind model ( figures [ fig : fiducial_ifu_mgii ] and [ fig : fiducial_ifu_feii ] ) . there are , however , several qualitative differences for the transitions . first , the @xmath145 transition in the ism+wind model shows much stronger absorption at @xmath218 to @xmath219 . in contrast to the doublet , the profile is not filled in by scattered photons . instead , the majority of @xmath145 photons that are absorbed are re - emitted as * @xmath220 photons . in fact , the @xmath145 profile very nearly matches the profile without re - emission ( compare to the dotted lines ) ; this transition provides a very good description of the intrinsic ism+wind optical depth profile . we conclude that resonant transitions that are coupled to ( multiple ) non - resonant , electric dipole transitions offer the best diagnosis of ism absorption . the differences in the absorption profiles are reflected in the much higher strengths ( @xmath221 ) of emission from transitions to the excited states of the a @xmath20d@xmath23 configuration . this occurs because : ( 1 ) there is greater absorption by the @xmath37resonance lines ; and ( 2 ) the high opacity of the ism component leads to an enhanced conversion of resonance photons with @xmath222 into * photons . this is especially notable for the @xmath139 transition whose coupled * transition shows an equivalent width nearly @xmath223 stronger than for the fiducial model . the relative strengths of the @xmath139 and * @xmath149 lines provide a direct diagnostic of the degree to which the resonance line photons are trapped , i.e. the peak optical depth of 2600 and the velocity gradient of the wind . because of the high degree of photon trapping within the ism component , this model does suffer more from dust extinction than the fiducial model . we have studied the ism+wind model including dust with @xmath224 ( normalized to include the ism gas ) . all of the emission lines are significantly reduced . the emission is affected most because these lines are resonantly trapped . the * emission is also reduced relative to the dust - free model , but the absolute flux still exceeds the fiducial model ( table [ tab : line_diag ] ) . at large negative velocity offsets from systemic , the two profiles are nearly identical . we conclude that a dusty ism model could show significant absorption at @xmath194 in resonance lines ( including ) with strong line - emission in the * transitions . . as expected , the strength of absorption increases with increasing @xmath106 ; this also results in stronger line - emission . note that the @xmath145 emission is always weak . only its absorption increases with @xmath106 and actually exceeds the depth of @xmath139 for @xmath225 . the depth of the doublet , meanwhile , always falls below a relative flux of 0.3 while the @xmath102 emission rises steadily with @xmath106 . , width=336 ] ccrccccccccccc mgii 2796 + & fiducial&[@xmath226 & 4.78 & 2.83&0.94&@xmath227&@xmath228&[@xmath229&@xmath230 & 2.48&@xmath231&@xmath232 & 215 + & @xmath233&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&[@xmath235&@xmath117 & 2.10&@xmath236&@xmath237 & 483 + & @xmath238&[@xmath112 & 4.78 & 2.98&1.03&@xmath239&@xmath240&[@xmath241&@xmath242 & 1.78&@xmath243&@xmath244 & 107 + & @xmath245&[@xmath112 & 4.78 & 3.55&0.91&@xmath246&@xmath247&[@xmath248&@xmath249 & 1.78&@xmath250&@xmath251 & 215 + & @xmath252&[@xmath253 & 4.78 & 4.79&2.19&@xmath254&@xmath255&[@xmath256&@xmath257 & 1.07&@xmath250&@xmath258 & 54 + & @xmath65=1&[@xmath259 & 4.77 & 2.94&0.95&@xmath227&@xmath260&[@xmath261&@xmath262 & 1.90&@xmath231&@xmath263 & 193 + & @xmath65=3&[@xmath264 & 4.78 & 3.07&1.03&@xmath265&@xmath266&[@xmath267&@xmath268 & 1.48&@xmath250&@xmath269 & 150 + & ism&[@xmath270 & 6.36 & 2.67&0.90&@xmath271&@xmath272&[@xmath273&@xmath274 & 2.36&@xmath275&@xmath276 & 236 + & ism+dust&[@xmath277 & 6.42 & 4.06&1.06&@xmath265&@xmath278&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234 + mgii 2803 + & fiducial&[@xmath279 & 3.29 & 1.19&0.76&@xmath280&@xmath265&[@xmath281&@xmath282 & 2.55&@xmath283&@xmath284 & 449 + & @xmath233&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&[@xmath285&@xmath286 & 1.93&@xmath287&@xmath288 & 482 + & @xmath238&[@xmath289 & 3.29 & 1.82&0.97&@xmath290&@xmath291&[@xmath292&@xmath293 & 1.81&@xmath294&@xmath295 & 107 + & @xmath245&[@xmath296 & 3.29 & 1.33&0.65&@xmath297&@xmath298&[@xmath299&@xmath300 & 2.12&@xmath301&@xmath302 & 535 + & @xmath252&[@xmath289 & 3.29 & 2.60&1.74&@xmath303&@xmath297&[@xmath304&@xmath305 & 1.33&@xmath301&@xmath306 & 562 + & @xmath65=1&[@xmath307 & 3.28 & 1.41&0.83&@xmath290&@xmath308&[@xmath309&@xmath310 & 1.95&@xmath283&@xmath311 & 417 + & @xmath65=3&[@xmath312 & 3.26 & 1.67&0.94&@xmath313&@xmath308&[@xmath314&@xmath315 & 1.50&@xmath283&@xmath316 & 375 + & ism&[@xmath317 & 6.49 & 1.03&0.67&@xmath318&@xmath319&[@xmath320&@xmath321 & 2.57&@xmath322&@xmath323 & 439 + & ism+dust&[@xmath324 & 6.51 & 2.59&1.02&@xmath290&@xmath325&[@xmath326&@xmath327 & 1.03&@xmath328&@xmath328 & 21 + feii 2586 + & fiducial&[@xmath329 & 0.82 & 0.61&1.01&@xmath330&@xmath331&[@xmath332&@xmath120 & 1.11&@xmath333&@xmath334 & 128 + & @xmath233&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&[@xmath335&@xmath120 & 1.06&@xmath236&@xmath336 & 232 + & @xmath238&[@xmath118 & 0.82 & 0.60&1.01&@xmath337&@xmath338&[@xmath339&@xmath340 & 1.06&@xmath341&@xmath342 & 87 + & @xmath245&[@xmath343 & 0.82 & 0.96&1.57&@xmath337&@xmath280&[@xmath119&@xmath120 & 1.09&@xmath336&@xmath344 & 203 + & @xmath252&[@xmath343 & 0.82 & 1.03&1.88&@xmath337&@xmath325&[@xmath345&@xmath346 & 1.03&@xmath336&@xmath347 & 58 + & @xmath65=1&[@xmath329 & 0.82 & 0.61&1.04&@xmath330&@xmath348&[@xmath349&@xmath350 & 1.06&@xmath301&@xmath336 & 116 + & @xmath65=3&[@xmath351 & 0.94 & 0.60&1.06&@xmath330&@xmath352&[@xmath353&@xmath346 & 1.04&@xmath354&@xmath342 & 58 + & ism&[@xmath355 & 1.90 & 1.60&2.75&@xmath356&@xmath357&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234 + & ism+dust&[@xmath358 & 1.84 & 1.63&2.96&@xmath301&@xmath359&[@xmath360&@xmath361 & 1.03&@xmath362&@xmath363 & 12 + feii 2600 + & fiducial&[@xmath364 & 1.87 & 1.18&1.08&@xmath365&@xmath366&[@xmath367&@xmath368 & 1.70&@xmath231&@xmath369 & 312 + & @xmath233&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&@xmath234&[@xmath370&@xmath371 & 1.46&@xmath372&@xmath373 & 346 + & @xmath238&[@xmath374 & 1.87 & 1.18&1.31&@xmath375&@xmath376&[@xmath377&@xmath378 & 1.39&@xmath379&@xmath295 & 87 + & @xmath245&[@xmath380 & 1.87 & 1.72&1.07&@xmath381&@xmath382&[@xmath383&@xmath384 & 1.49&@xmath385&@xmath386 & 404 + & @xmath252&[@xmath380 & 1.87 & 2.12&2.71&@xmath375&@xmath271&[@xmath387&@xmath388 & 1.15&@xmath385&@xmath389 & 346 + & @xmath65=1&[@xmath390 & 1.87 & 1.20&1.14&@xmath365&@xmath391&[@xmath392&@xmath393 & 1.45&@xmath231&@xmath394 & 265 + & @xmath65=3&[@xmath364 & 1.95 & 1.25&1.31&@xmath395&@xmath318&[@xmath396&@xmath397 & 1.22&@xmath398&@xmath251 & 196 + & ism&[@xmath399 & 3.12 & 1.88&1.19&@xmath395&@xmath400&[@xmath401&@xmath402 & 1.15&@xmath403&@xmath404 & 242 + & ism+dust&[@xmath405 & 3.06 & 2.12&1.40&@xmath398&@xmath406&[@xmath407&@xmath361 & 1.02&@xmath408&@xmath409 & 12 + feii * 2612 + & fiducial&&&&&&&[@xmath410&@xmath411 & 1.24&@xmath412&@xmath413 & 241 + & @xmath233&&&&&&&[@xmath414&@xmath129 & 1.22&@xmath415&@xmath372 & 172 + & @xmath238&&&&&&&[@xmath416&@xmath129 & 1.21&@xmath417&@xmath418 & 172 + & @xmath245&&&&&&&[@xmath419&@xmath420 & 1.29&@xmath417&@xmath341 & 316 + & @xmath252&&&&&&&[@xmath421&@xmath422 & 1.10&@xmath423&@xmath424 & 144 + & @xmath65=1&&&&&&&[@xmath425&@xmath397 & 1.21&@xmath417&@xmath426 & 207 + & @xmath65=3&&&&&&&[@xmath427&@xmath428 & 1.16&@xmath417&@xmath429 & 184 + & ism&&&&&&&[@xmath430&@xmath431 & 2.66&@xmath356&@xmath432 & 138 + & ism+dust&&&&&&&[@xmath433&@xmath434 & 1.53&@xmath356&@xmath435 & 161 + feii * 2626 + & fiducial&&&&&&&[@xmath436&@xmath437 & 1.29&@xmath417&@xmath413 & 217 + & @xmath233&&&&&&&[@xmath438&@xmath439 & 1.16&@xmath283&@xmath440 & 171 + & @xmath238&&&&&&&[@xmath441&@xmath439 & 1.16&@xmath442&@xmath443 & 143 + & @xmath245&&&&&&&[@xmath444&@xmath445 & 1.41&@xmath442&@xmath446 & 314 + & @xmath252&&&&&&&[@xmath447&@xmath448 & 1.14&@xmath283&@xmath269 & 114 + & @xmath65=1&&&&&&&[@xmath449&@xmath378 & 1.23&@xmath417&@xmath341 & 194 + & @xmath65=3&&&&&&&[@xmath450&@xmath428 & 1.17&@xmath417&@xmath451 & 194 + & ism&&&&&&&[@xmath452&@xmath453 & 4.10&@xmath356&@xmath356 & 103 + & ism+dust&&&&&&&[@xmath454&@xmath455 & 1.70&@xmath356&@xmath456 & 137 + feii * 2632 + & fiducial&&&&&&&[@xmath457&@xmath129 & 1.12&@xmath458&@xmath459 & 194 + & @xmath233&&&&&&&[@xmath460&@xmath448 & 1.09&@xmath461&@xmath398 & 142 + & @xmath238&&&&&&&[@xmath462&@xmath448 & 1.11&@xmath463&@xmath464 & 142 + & @xmath245&&&&&&&[@xmath465&@xmath397 & 1.14&@xmath458&@xmath466 & 285 + & @xmath252&&&&&&&[@xmath467&@xmath468 & 1.14&@xmath469&@xmath470 & 684 + & @xmath65=1&&&&&&&[@xmath471&@xmath472 & 1.10&@xmath442&@xmath473 & 160 + & @xmath65=3&&&&&&&[@xmath474&@xmath350 & 1.08&@xmath463&@xmath463 & 103 + & ism&&&&&&&[@xmath475&@xmath476 & 1.81&@xmath413&@xmath477 & 137 + & ism+dust&&&&&&&[@xmath478&@xmath479 & 1.30&@xmath413&@xmath459 & 148 + the final modification to the fiducial model considered was a uniform variation of the normalization of the optical depth profiles . specifically , we ran a series of additional models with @xmath106 at 1/3 , 3 , and 10 times the fiducial value of @xmath480 . the resulting and profiles are compared against the fiducial model in figure [ fig : norm ] . inspecting , one notes that the models with 1/3 and @xmath481 behave as expected . higher / lower optical depths lead to greater / weaker @xmath97 absorption and stronger / weaker @xmath102 emission . in the extreme case of @xmath482 , the @xmath102 absorption and @xmath97emission have nearly disappeared and one primarily observes very strong @xmath97 absorption and @xmath102 emission . in essence , this wind has converted all of the photons absorbed by the doublet into @xmath102 emission . similar behavior is observed for the transitions . interestingly , the @xmath145 transitions never show significant emission , only stronger absorption with increasing @xmath106 . in fact , for @xmath483 the peak optical depth of @xmath145 actually exceeds that for @xmath139 because the latter remains significantly filled - in by scattered photons . by the same token , the flux of the * @xmath148emission increases with @xmath106 . table [ tab : line_diag ] presents a series of quantitative measures of the and absorption and emission lines for the fiducial model ( @xmath18 [ sec : fiducial ] ) and a subset of the models presented in this section . listed are the absorption and emission equivalent widths ( @xmath484 ) , the peak optical depth for the absorption @xmath485 , the velocity where the optical depth peaks @xmath486 , the optical depth - weighted velocity centroid @xmath487 / \int dv \ln[i(v)]$ ] , the peak flux @xmath488 in emission , the velocity where the flux peaks @xmath489 , and the flux - weighted velocity centroid of the emission line @xmath490 . we discuss several of these measures in @xmath18 [ sec : discuss ] . this section presents several additional wind scenarios that differ significantly from the fiducial model explored in the previous sections . in each case , we maintain the simple assumption of isotropy . the lyman break galaxies ( lbgs ) , uv color - selected galaxies at @xmath4 , exhibit cool gas outflows in , , etc . transitions with speeds up to 1000(e.g . * ; * ? ? ? researchers have invoked these winds to explain enrichment of the intergalactic medium ( e.g. * ? ? ? * ; * ? ? ? * ) , the origin of the damped systems @xcite , and the formation of ` red and dead ' galaxies ( e.g. * ? ? ? although the presence of these outflows was established over a decade ago , the processes that drive them remain unidentified . similarly , current estimates of the mass and energetics of the outflows suffer from orders of magnitude uncertainty . recently , ( * ? ? ? * hereafter s10 ) introduced a model to explain jointly the average absorption they observed along the sightlines to several hundred lbgs and the average absorption in gas observed transverse to these galaxies . their wind model is defined by two expressions : ( i ) a radial velocity law @xmath491 ; and ( ii ) the covering fraction of optically thick ` clouds ' @xmath492 . for the latter , s10 envision an ensemble of small , optically thick @xmath493 clouds that only partially cover the galaxies . for the velocity law , they adopted the following functional form : @xmath494 with @xmath495 the constant that sets the terminal speed , @xmath77 the inner radius of the wind ( taken to be 1kpc ) , and @xmath6 the power - law exponent that describes how steeply the velocity curve rises . their analysis of the lbg absorption profiles implied a very steeply rising curve with @xmath496 . this velocity expression is shown as a dotted line in figure [ fig : lbg_sobolev]a . the covering fraction of optically thick clouds , meanwhile , was assumed to have the functional form @xmath497 with @xmath498 and @xmath499 the maximum covering fraction . from this expression and the velocity law , one can recover an absorption profile @xmath500 $ ] , written explicitly as @xmath501 the resulting profile for @xmath502 , @xmath503 , @xmath504 , and @xmath505 is displayed in figure [ fig : lbg_sobolev]b . in the following , we consider two methods for analyzing the lbg wind . both approaches assume an isotropic wind and adopt the velocity law given by equation [ eqn : lbg_vlaw ] . in the first model , we treat the cool gas as a diffuse medium with unit covering fraction and a radial density profile determined from the sobolev approximation . we then apply the monte carlo methodology used for the other wind models to predict and line profiles . in the second lbg model , we modify our algorithms to more precisely mimic the concept of an ensemble of optically thick clouds with a partial covering fraction on galactic scales . to highlight the evolution in the quantities at small radii . for example , note how rapidly the velocity rises from @xmath506 to 2kpc . the solid curve shows the sobolev solution for the mg@xmath21 gas density derived from the average absorption profile of lbg galaxies ( see below ) . the density falls off initially as @xmath507 and then steepens to @xmath508 . ( b ) the average profile for lbg cool gas absorption as measured and defined by s10 ( black solid curve ) . overplotted on this curve is a ( red ) dotted line that shows the @xmath97 absorption profile derived from the density ( and velocity ) law shown in the upper panel . , width=336 ] as demonstrated in figure [ fig : lbg_sobolev ] , the wind velocity for the lbg model rises very steeply with increasing radius before flattening at large radii . under these conditions , the sobolev approximation provides an accurate description of the optical depth for the flowing medium . the sobolev line optical depth profile can be derived from the absorption profile ( equation [ eqn : lbg_i ] ) , @xmath509 this simple expression assumes that the source size is small compared the wind dimensions and neglects the effects of light scattered in the wind ( for more general inversion formulae that do not make these assumptions , see @xcite ) . the sobolev approximation ( equation [ eqn : sobolev ] ) provides a simple expression for the optical depth at each radius in terms of the local density and velocity gradient . in general , the velocity gradient must be taken along the direction of propagation of a photon , however in this model the central source is relatively small and the radiation field is therefore primarily radially directed . in this case the velocity gradient in the radial direction , @xmath510 , is the relevant quantity , and we may use equation [ eqn : sobolev ] to invert the optical depth and determine the density profile , @xmath511 the solid curve in figure [ fig : lbg_sobolev]a shows the resultant density profile for mg@xmath21 assuming that the @xmath97 line follows the intensity profile drawn in figure [ fig : lbg_sobolev]b . this is a relatively extreme density profile . from the inner radius of 1kpc to 2kpc , the density drops by over 2 orders of magnitude including nearly one order of magnitude over the first 10pc . beyond 2kpc , the density drops even more rapidly , falling orders of magnitude from 2 to 100kpc . we verified that the sobolev - derived density profile shown in figure [ fig : lbg_sobolev]a reproduces the proper absorption profile by discretizing the wind into a series of layers and calculating the integrated absorption profile . this calculation is shown as a dotted red curve in figure [ fig : lbg_sobolev]b ; it is an excellent match to the desired profile ( black curve ) . to calculate the optical depth profiles for the other and transitions , we assume @xmath512 and also scale @xmath513 by the @xmath103 product . ) , scattered and re - emitted photons significantly modify the absorption profiles ( especially and @xmath139 ) and produce strong emission lines . the result , for especially , is a set of profiles that do not match the average observed lbg absorption profile . , width=336 ] we generated and profiles for this lbg - sobolev wind using the 1d algorithm with no dust extinction ; these are shown as red curves in figure [ fig : lbg_spec ] . for this analysis , one should focus on the @xmath97 transition . the dotted line in the figure shows the intensity profile when one ignores re - emission of absorbed photons . by construction , it is the same profile and for the reduced fe@xmath21 abundance . ] described by equation [ eqn : lbg_i ] and plotted in figure [ fig : lbg_sobolev]b . in comparison , the full model ( solid , red curve ) shows much weaker absorption , especially at @xmath514 to @xmath515 because scattered photons fill in the absorption profile . in this respect , our lbg - sobolev model is an inaccurate description of the observations which show more uniform equivalent widths among differing transitions ( s10 ) . the model also predicts significant emission in the lines and several of the @xmath17 transitions . emission associated with cool gas has been observed for @xmath17 transitions in lbgs @xcite , but the @xmath17 transitions modeled here lie in the near - ir and have not yet been investigated . on the other hand , there have been no reported detections of significant line - emission related to low - ion resonance transitions ( e.g. ) in lbgs , only @xmath516 star - forming galaxies @xcite . the principal result of the lbg - sobolev model is that the scattering and re - emission of absorbed photons significantly alters the predicted absorption profiles for the input model . this is , of course , an unavoidable consequence of an isotropic , dust - free model with unit covering fraction . ) . the remaining panels show the cumulative mass , energy , and momentum of this flow . all of the curves rise very steeply at small radii and then rise steadily to large radii . the principal result is that the majority of mass , energy , and momentum in the wind are contained at large radii . we derive similar results for the lbg - partial covering model if we assume the wind is composed of identical clouds optically thick to strong metal - line transitions ( e.g. 1526 ) . the energy and momentum in the outer regions of the outflow may be very difficult to generate with standard galactic - scale wind scenarios . , height=336 ] because we have explicit velocity and density profiles for the lbg - sobolev model , it is straightforward to calculate the radial distributions of mass , energy , and momentum of this wind . these are shown in cumulative form in figure [ fig : lbg_cumul ] . before discussing the results , we offer two cautionary comments : ( i ) the conversion of @xmath87 to @xmath84 assumes a very poorly constrained scalar factor of @xmath517 . one should give minimal weight to the absolute values for any of the quantities ; ( ii ) the sobolev approximation is not a proper description of the s10 lbg wind model ( see the following sub - section ) . these issues aside , we may inspect figure [ fig : lbg_cumul ] to reveal global properties of this lbg - sobolev wind . one obvious result is that all of the curves rise very steeply at small radii . we find , for example , that @xmath518 of the mass is contained within the inner 2kpc . all of the curves continue to rise , however , such that the majority of energy , mass , and momentum in the wind is transported by its outermost layers ( i.e. , @xmath519kpc ) . this is somewhat surprising given that the density is @xmath520 orders of magnitude lower at these radii than at @xmath521pc . more importantly , we question whether any physical process could produce a wind with such extreme profiles . in the previous sub - section , we described a sobolev inversion that reproduces the average absorption profile of lbgs in cool gas transitions when scattered photons are ignored . a proper analysis that includes scattered photons , however , predicts line - profiles that are qualitatively different from the observations because scattered photons fill - in absorption and generate significant line - emission ( similar to the fiducial wind model ; @xmath18 [ sec : fiducial ] ) . we also argued that the implied mass , energy , and momentum distributions of this wind ( figure [ fig : lbg_cumul ] ) were too extreme . this lbg - sobolev model , however , is not precisely the one introduced by s10 ; those authors proposed an ensemble of optically thick clouds with a partial covering fraction described by equation [ eqn : covering ] . in contrast , the lbg - sobolev model assumes a diffuse medium with a declining density profile but a unit covering fraction . one may question , therefore , whether these differences lead to the failures of the lbg - sobolev model . to more properly model the lbg wind described in s10 , we performed the following monte carlo calculation . first , we propagate a photon from the source until its velocity relative to line - center resonates with the wind ( the photon escapes if this never occurs ) . the photon then has a probability @xmath522 of scattering . if it scatters , we track the photon until it comes into resonance again for a scattered @xmath97 photon . ] or escapes the system . in this model , all of the resonance transitions are assumed to have identical ( infinite ) optical depth at line center . the results of the full calculation ( absorption plus scattering ) are shown as the black curve in figure [ fig : lbg_spec ] . remarkably , the results are essentially identical to the lbg - sobolev calculation for the @xmath97 transition ; scattered photons fill - in the absorption profiles at @xmath48 and yield significant emission lines at @xmath523 . similar results are observed for the other resonance lines and significant emission is observed for the * transitions , centered at @xmath187 . in contrast to the profiles , the @xmath145 line much more closely resembles its intrinsic profile ( dotted curves ) . this occurs because most of these absorbed photons fluoresce into * emission . the equivalent width of @xmath145 even exceeds that for @xmath139 , an inversion that , if observed , would strongly support this model . a robust conclusion of our analysis is that these simple models can not reproduce the observed profiles of resonantly trapped lines ( like the doublet ) because of the emission from scattered photons . in particular , the observations show much greater opacity at velocities within @xmath524 of systemic ( s10 ) . in order to achieve a significant opacity at @xmath187 , one must either suppress the line - emission ( e.g. with severe dust extinction or anisotropic winds ) or include a substantial ism component that absorbs light at @xmath185 ( e.g. @xmath18 [ sec : ism ] ) . absent such corrections ( which may be insufficient ) , we conclude that the lbg model introduced by s10 is not a valid description of the data . also , similar to our lbg - sobolev calculation ( figure [ fig : lbg_sobolev ] ) , this clump model predicts the majority of mass , energy , and momentum are carried in the outer regions of the wind . in fact , for clumps that are optically thick to strong metal - line transitions ( e.g. 1526 ) we estimate because the latter assumes a diffuse medium with unit filling factor . ] that the wind would carry nearly a total mass of @xmath525 , with the majority at large radii . the resonant line - emission predicted for should also arise in other transitions . indeed , we caution that the line - emission observed in lbg spectra ( e.g. * ? ? ? * ) may result from the galactic - scale outflow , not only the stellar winds of massive stars . a proper treatment of these radiative transfer effects is required to quantitatively analyze these features . as discussed in the introduction , the importance of various physical mechanisms in driving galactic outflows remains an open question . several studies advocate a primarily energy - driven wind scenario , in which the hot wind produced by the thermalized energy from sne ejecta entrains cool clouds via ram pressure ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? alternatively , momentum deposition by photons emitted by luminous star clusters onto dust grains in the ism may also play an integral role in driving large - scale outflows ( e.g. * ? ? ? * ; * ? ? ? both of these mechanisms likely contribute more or less significantly in a given galaxy ; however , the latter ( radiation - pressure driving ) has the added advantage that it does not destroy cool gas clouds as it acts . here we explore the line - profiles produced by a radiation - pressure driven wind model as described by @xcite . ) . at these velocities , the absorption is not filled - in by scattered photons and one recovers absorption lines that closely resemble the intrinsic optical depth profile ( dotted curve ) . although the absorption profiles are very different from the fiducial model , the line - emission has similar peak flux and velocity centroids the * emission also has a similar equivalent width , yet the lines are much broader in the radiation - driven model . , width=336 ] these authors start by assuming the galaxy has luminosity @xmath526 and is an isothermal sphere with mass profile @xmath527 , where @xmath528 is the velocity dispersion . the gas mass profile is simply @xmath529 , with @xmath530 a constant gas fraction . the dust has opacity @xmath531 and is optically thin , such that the force per unit mass of wind material due to radiation is @xmath532 . ignoring gas pressure , and assuming a steady state for the flow ( i.e. , the mass outflow rate , @xmath533 , is constant ) , the momentum equation for the wind is @xmath534 substituting @xmath535 for @xmath536 and solving for @xmath537 , @xcite find @xmath538 ( their equ . 26 ) , where @xmath539 at @xmath540 and @xmath541 . the corresponding density profile for the gas is given by @xmath542 with @xmath543 the mass of the hydrogen atom . similar to the fiducial wind model , this wind has a decreasing density law with @xmath84 roughly proportional to @xmath108 . in contrast to the fiducial model , the velocity law is nearly constant with radius before decreasing sharply at large radii . to produce a wind whose @xmath97 optical depth profile peaks at @xmath544 , we set @xmath545 , @xmath546 , and @xmath547 . the density peaks at @xmath548 where @xmath549 . the velocity , meanwhile , peaks at @xmath550 with @xmath551 . this corresponds to a mass flow of @xmath552 . figure [ fig : rad_spec ] presents the and profiles for this radiation - driven wind model compared against the fiducial wind model . in contrast to the fiducial model , the optical depth peaks at much larger velocity ( here @xmath553 ) . at these velocities , the absorption is not filled - in by scattered photons and one recovers absorption lines that more closely follow the intrinsic optical depth profile ( dotted curve ) . the line - emission , meanwhile , has similar peak flux and velocity centroids to the fiducial model ( see also table [ tab : plaw_diag ] ) . the * emission also has a similar equivalent width , yet the lines of the radiation - driven model are much broader . this reflects the fact that the majority of absorption occurs at @xmath554 . the kinematics of * line - emission , therefore , independently diagnose the intrinsic optical depth profile of the flow . we conclude that the radiation - driven model has characteristics that are qualitatively similar to the fiducial model ( e.g. strong blue - shifted absorption and significant line - emission ) with a few quantitative differences that , in principle , could distinguish them with high fidelity observations . the fiducial model assumed power - law descriptions for both the density and velocity laws ( equations [ eqn : density],[eqn : vel ] ) . the power - law exponents were arbitrarily chosen , i.e. with little physical motivation . in this sub - section , we cursorily explore the results for a series of other power - law expressions , also arbitrarily defined . our intention is to illustrate the diversity of and profiles that may result from modifications to the density and velocity laws . we consider three different density laws ( @xmath555 ) and three different velocity laws ( @xmath556 ) for nine wind models ( table [ tab : plaw_parm ] , figure [ fig : plaws]a ) . each of these winds extends over the same inner and outer radii as the fiducial model . the density and velocity normalizations @xmath557 have been modified to yield a @xmath97 optical depth profile that peaks at @xmath558 and then usually decreases to @xmath167 ( figure [ fig : plaws]b ) . this choice of normalization was observationally driven to yield significant absorption lines . all models assume a doppler parameter @xmath559 , full isotropy , and a dust - free environment . lastly , the abundances of mg@xmath21 and fe@xmath21 scale with hydrogen as in the fiducial model ( table [ tab : fiducial ] ) . , dotted @xmath560 , dot - dash @xmath561 ) and velocity laws ( colored ; green @xmath562 , red @xmath563 , blue @xmath564 ) for a series of power - law models ( table [ tab : plaw_parm ] ) . for comparison we also plot the velocity and density laws for the fiducial model ( thin , black curves ) . panel ( b ) plots the optical depth profiles of the @xmath97 line for the nine models built from the density and velocity laws of panel ( a ) . each of these were normalized to have peak optical depths of 10 to 1000 and to be optically thin at small or large velocity . , width=336 ] cccccc a & @xmath565 & 0.4000 & @xmath566 & 2.0 & 0.8 + b & @xmath565 & 0.5000 & @xmath567 & 50.0 & 1.1 + c & @xmath565 & 0.3000 & @xmath568 & 100.0 & 1.8 + d & @xmath569 & 0.0100 & @xmath566 & 2.0 & 2.1 + e & @xmath569 & 0.0100 & @xmath567 & 50.0 & 1.7 + f & @xmath569 & 0.0200 & @xmath568 & 100.0 & 1.6 + g & @xmath570 & 0.0100 & @xmath566 & 2.0 & 4.4 + h & @xmath570 & 0.0010 & @xmath567 & 50.0 & 3.2 + i & @xmath570 & 0.0001 & @xmath568 & 100.0 & 1.8 + figure [ fig : plaws_spec ] presents the and profiles for the full suite of power - law models ( see also table [ tab : plaw_diag ] ) . although these models differ qualitatively from the fiducial model in their density and velocity laws , the resultant profiles share many of the same characteristics . each shows significant absorption at @xmath571 , extending to the velocity where @xmath98 drops below 0.1 ( figure [ fig : plaws]b ) . all of the models also exhibit strong line - emission , primarily at velocities @xmath572 . this emission fills - in the absorption at @xmath573 such that the profiles rarely achieve a relative flux less than @xmath574 at these velocities . similar to the radiation - driven wind ( @xmath18 [ sec : radiative ] ) , the power - law models that have significant opacity at @xmath575 tend to have larger peak optical depths . another commonality is the weak or absent line - emission at @xmath145 ; one instead notes strong * @xmath148 emission that generally exceeds the equivalent width of the fiducial model . one also notes that the width of the * emission is systematically higher for models where the optical depth in absorption peaks at large velocity from the systemic . one of the few obvious distinctions between these models and the fiducial wind is the higher peak optical depth of absorption in the @xmath139transition . this occurs primarily because these models have higher intrinsic optical depths ( i.e. higher @xmath86 values ) . 2796 + & fiducial&[@xmath226 & 4.78 & 2.83&0.94&@xmath227&@xmath228&[@xmath229&@xmath230 & 2.48&@xmath231&@xmath232 & 215 + & lbg sob.&[@xmath576 & 2.62 & 1.35&0.45&@xmath577&@xmath578&[@xmath248&@xmath579 & 1.84&@xmath580&@xmath581 & 193 + & lbg cov.&[@xmath582 & 2.67 & 1.40&0.41&@xmath583&@xmath584&[@xmath585&@xmath586 & 2.23&@xmath356&@xmath587 & 172 + & radiation&[@xmath588 & 2.86 & 1.72&3.00&@xmath589&@xmath590&[@xmath591&@xmath592 & 2.13&@xmath593&@xmath594 & 236 + & a&[@xmath595 & 9.29 & 5.02&2.92&@xmath596&@xmath597&[@xmath598&@xmath599 & 3.30&@xmath580&@xmath600 & 118 + & b&[@xmath601&11.08 & 7.54&3.00&@xmath602&@xmath603&[@xmath604&@xmath605 & 2.89&@xmath356&@xmath606 & 172 + & c&[@xmath607 & 2.54 & 1.82&1.86&@xmath271&@xmath608&[@xmath609&@xmath610 & 2.09&@xmath356&@xmath611 & 247 + & d&[@xmath612 & 4.21 & 2.14&1.11&@xmath243&@xmath613&[@xmath614&@xmath615 & 3.58&@xmath250&@xmath616 & 172 + mgii 2803 + & fiducial&[@xmath279 & 3.29 & 1.19&0.76&@xmath280&@xmath265&[@xmath281&@xmath282 & 2.55&@xmath283&@xmath284 & 449 + & lbg sob.&[@xmath617 & 1.53 & 0.46&0.24&@xmath618&@xmath618&[@xmath619&@xmath620 & 1.58&@xmath301&@xmath621 & 332 + & lbg cov.&[@xmath622 & 2.56 & 0.98&0.37&@xmath623&@xmath624&[@xmath625&@xmath626 & 2.35&@xmath627&@xmath628 & 396 + & radiation&[@xmath629 & 2.32 & 1.29&2.65&@xmath630&@xmath631&[@xmath632&@xmath633 & 1.86&@xmath461&@xmath634 & 246 + & a&[@xmath635 & 7.06 & 1.98&0.78&@xmath636&@xmath637&[@xmath638&@xmath639 & 5.03&@xmath640&@xmath641 & 610 + & b&[@xmath642 & 5.74 & 1.80&1.27&@xmath228&@xmath643&[@xmath644&@xmath645 & 4.20&@xmath627&@xmath646 & 749 + & c&[@xmath647 & 2.17 & 1.49&1.85&@xmath318&@xmath648&[@xmath649&@xmath650 & 1.94&@xmath287&@xmath651 & 246 + & d&[@xmath652 & 3.69 & 0.19&0.77&@xmath432&@xmath435&[@xmath653&@xmath626 & 3.33&@xmath283&@xmath654 & 332 + feii 2586 + & fiducial&[@xmath329 & 0.82 & 0.61&1.01&@xmath330&@xmath331&[@xmath332&@xmath120 & 1.11&@xmath333&@xmath334 & 128 + & lbg sob.&[@xmath655 & 0.18 & 0.04&0.07&@xmath656&@xmath400&[@xmath657&@xmath327 & 1.05&@xmath356&@xmath342 & 35 + & lbg cov.&[@xmath658 & 2.38 & 1.93&0.65&@xmath659&@xmath613&[@xmath660&@xmath411 & 1.33&@xmath342&@xmath661 & 313 + & radiation&[@xmath662 & 1.00 & 0.00&0.02&@xmath356&@xmath356&[@xmath663&@xmath472 & 1.09&@xmath333&@xmath664 & 220 + & a&[@xmath665 & 1.88 & 1.66&0.33&@xmath666&@xmath667&[@xmath668&@xmath669 & 1.12&@xmath301&@xmath670 & 197 + & b&[@xmath671 & 2.20 & 1.97&0.60&@xmath672&@xmath673&[@xmath674&@xmath388 & 1.09&@xmath342&@xmath675 & 429 + & c&[@xmath676 & 0.79 & 0.71&2.51&@xmath313&@xmath677&[@xmath678&@xmath120 & 1.08&@xmath251&@xmath679 & 174 + & d&[@xmath680 & 0.88 & 0.68&3.00&@xmath354&@xmath466&[@xmath681&@xmath327 & 1.03&@xmath679&@xmath682 & 35 + feii 2600 + & fiducial&[@xmath364 & 1.87 & 1.18&1.08&@xmath365&@xmath366&[@xmath367&@xmath368 & 1.70&@xmath231&@xmath369 & 312 + & lbg sob.&[@xmath683 & 0.55 & 0.24&0.11&@xmath684&@xmath685&[@xmath686&@xmath687 & 1.23&@xmath688&@xmath689 & 208 + & lbg cov.&[@xmath690 & 2.43 & 1.38&0.44&@xmath260&@xmath691&[@xmath692&@xmath693 & 2.02&@xmath694&@xmath695 & 381 + & radiation&[@xmath696 & 2.21 & 1.44&3.00&@xmath697&@xmath577&[@xmath698&@xmath699 & 1.59&@xmath398&@xmath394 & 242 + & a&[@xmath700 & 4.61 & 3.51&1.05&@xmath701&@xmath702&[@xmath703&@xmath704 & 2.34&@xmath694&@xmath705 & 612 + & b&[@xmath706 & 5.10 & 4.20&1.88&@xmath707&@xmath708&[@xmath709&@xmath710 & 1.86&@xmath294&@xmath711 & 935 + & c&[@xmath712 & 1.34 & 1.09&2.04&@xmath713&@xmath714&[@xmath715&@xmath716 & 1.43&@xmath717&@xmath718 & 219 + & d&[@xmath719 & 1.68 & 1.13&3.00&@xmath720&@xmath721&[@xmath722&@xmath723 & 1.47&@xmath398&@xmath724 & 219 + feii * 2612 + & fiducial&&&&&&&[@xmath410&@xmath411 & 1.24&@xmath412&@xmath413 & 241 + & lbg sob.&&&&&&&[@xmath725&@xmath257 & 1.07&@xmath356&@xmath459 & 115 + & lbg cov.&&&&&&&[@xmath726&@xmath727 & 1.79&@xmath356&@xmath333 & 563 + & radiation&&&&&&&[@xmath728&@xmath393 & 1.26&@xmath356&@xmath435 & 425 + & a&&&&&&&[@xmath729&@xmath730 & 1.35&@xmath356&@xmath731 & 735 + & b&&&&&&&[@xmath732&@xmath730 & 1.14&@xmath283&@xmath733 & 873 + & c&&&&&&&[@xmath734&@xmath268 & 1.20&@xmath735&@xmath459 & 241 + & d&&&&&&&[@xmath736&@xmath455 & 2.37&@xmath356&@xmath435 & 103 + feii * 2626 + & fiducial&&&&&&&[@xmath436&@xmath437 & 1.29&@xmath417&@xmath413 & 217 + & lbg sob.&&&&&&&[@xmath353&@xmath340 & 1.06&@xmath301&@xmath415 & 57 + & lbg cov.&&&&&&&[@xmath737&@xmath738 & 1.22&@xmath356&@xmath415 & 434 + & radiation&&&&&&&[@xmath739&@xmath740 & 1.42&@xmath356&@xmath243 & 343 + & a&&&&&&&[@xmath741&@xmath274 & 1.71&@xmath356&@xmath742 & 720 + & b&&&&&&&[@xmath743&@xmath744 & 1.22&@xmath356&@xmath278&1039 + & c&&&&&&&[@xmath745&@xmath420 & 1.24&@xmath746&@xmath747 & 217 + & d&&&&&&&[@xmath748&@xmath749 & 3.53&@xmath356&@xmath747 & 103 + feii * 2632 + & fiducial&&&&&&&[@xmath457&@xmath129 & 1.12&@xmath458&@xmath459 & 194 + & lbg sob.&&&&&&&[@xmath750&@xmath327 & 1.07&@xmath751&@xmath751 & 46 + & lbg cov.&&&&&&&[@xmath752&@xmath753 & 1.36&@xmath413&@xmath477 & 479 + & radiation&&&&&&&[@xmath754&@xmath397 & 1.12&@xmath413&@xmath354 & 353 + & a&&&&&&&[@xmath755&@xmath293 & 1.19&@xmath751&@xmath756 & 581 + & b&&&&&&&[@xmath757&@xmath371 & 1.17&@xmath758&@xmath759 & 741 + & c&&&&&&&[@xmath760&@xmath402 & 1.10&@xmath440&@xmath413 & 228 + & d&&&&&&&[@xmath761&@xmath397 & 1.63&@xmath413&@xmath477 & 91 + while there is commonality between models , the specific characteristics of the absorption / emission profiles do exhibit significant diversity . table [ tab : plaw_diag ] compares measures of the absorption and emission profiles against the fiducial model . in detail , the line profiles differ in their peak optical depths , the velocity centroids of absorption / emission , and their equivalent widths . we find that many of the differences are driven by differences in the velocity laws , i.e. the kinematics of the outflow . at the same time , models with very different density / velocity laws can produce rather similar results . for example , profiles that have declining opacity with increasing velocity offset from systemic can be obtained with a radial velocity law that increases ( e.g. the fiducial model ) or decreases ( model d ) . the implication is that absorption profiles alone are likely insufficient to fully characterize the physical characteristics of the outflow . we now discuss the principal results of our analysis and comment on the observational consequences and implications . the previous sections presented idealized wind models for cool gas outflows , and explored the absorption / emission profiles of the @xmath0 doublet and the uv1 multiplet . in addition to the ubiquitous presence of blue - shifted absorption , the wind models also predict strong line - emission in both resonance and non - resonance transitions . for isotropic and dust - free scenarios , this is due to the simple conservation of photons : every photon emitted by the source eventually escapes the system to maintain zero total equivalent width . a principal result of this paper , therefore , is that galaxies with observed cool gas outflows should also generate detectable line - emission . indeed , line - emission from low - ion transitions has been reported from star - forming galaxies that exhibit cool gas outflows . this includes emission related to @xcite , emission @xcite , * emission from lbgs @xcite , and , most recently , significant * emission @xcite . a variety of origins have been proposed for this line - emission including agn activity , recombination in regions , and back - scattering off the galactic - scale wind . our results suggest that the majority of the observed line - emission is from scattered photons in the cool gas outflows of these star - forming galaxies . indeed , line - emission should be generated by all galaxies driving a cool gas outflow . on the other hand , there are many examples of galaxies where blue - shifted absorption is detected yet the authors report no significant line emission . this includes flows @xcite , and absorption in ulirgs @xcite , and the extreme outflows identified by @xcite . similarly , there have been no reports of resonance line - emission from low - ion transitions in the lbgs , and many @xmath13 galaxies show no detectable or * emission despite significant blue - shifted absorption ( rubin et al . , in prep ) . these non - detections appear to contradict a primary conclusion of this paper . we are motivated , therefore , to reassess several of the effects that can reduce the line - emission and consider whether these prevent its detection in many star - forming galaxies . dust is frequently invoked to explain the suppression of line - emission for resonantly trapped transitions ( e.g. ) . indeed , a photon that is trapped for many scatterings within a dusty medium will be preferentially extincted relative to a non - resonant photon . in @xmath18 [ sec : dust ] , we examined the effects of dust extinction on and emission . the general result ( figure [ fig : dust_tau ] ) is a modest reduction in flux that scales as ( 1+@xmath65)@xmath19 instead of exp(@xmath65 ) . although the photons are resonantly trapped , they require only one to a few scatterings to escape the wind thereby limiting the effects of dust . this reflects the moderate opacity of the doublet ( e.g. relative to ) and also the velocity law of the fiducial wind model . in scenarios where the photons are more effectively trapped , dust does suppress the emission ( e.g. in the ism+wind model from @xmath18 [ sec : ism ] ) . in contrast to , the resonance photons may be converted to non - resonant photons that freely escape the wind . therefore , a wind model that traps photons for many scatterings does not similarly trap @xmath37 photons . the effects of dust are much reduced and , by inference , the same holds for any other set of transitions that are coupled to a fine - structure level ( e.g./ * ) . in summary , dust does reduce the line flux relative to the continuum , but it generally has only a modest effect on the predicted emission and a minor effect on * emission . another factor that may reduce line - emission is wind anisotropy . the flux is lower , for example , if one eliminates the backside to the wind ( e.g. the source itself could shadow a significant fraction of the backside ) . similarly , a bi - conical wind can have significantly lower line - emission , at least for the fiducial model ( @xmath18 [ sec : anisotropic ] ) . for the anisotropic winds explored in this paper , the line - emission scales roughly as @xmath762 where @xmath763 is the angular extent of the wind . because we require that the wind points toward us , it is difficult to reduce @xmath763 much below @xmath764 and , therefore , anisotropy reduces the emission by a factor of order unity similarly , an anisotropically emitting source would modify the predicted line - emission . for example , if the backside of the galaxy were brighter / fainter then would one predict brighter / fainter emission relative to the observed continuum . to reduce the emission , the brightest regions of the galaxy would need to be oriented toward earth . although this would not be a generic orientation , most spectroscopic samples are magnitude - limited and biased towards detecting galaxies when observed at their brightest . in principle , this might imply a further reduction of order unity . together , dust and anisotropic models may reduce the line - emission of and by one to a few factors of order unity . for some galaxies , these effects may explain the absence of significant line - emission , but they may not be sufficient to preclude its detection . there is another ( more subtle ) effect that could greatly reduce the observed line - emission : slit - loss . as described in figure [ fig : fiducial_ifu_mgii ] , emission from the wind is spatially extended with a non - zero surface brightness predicted at large radii . this implies a non - negligible luminosity emitted beyond the angular extent of the galaxy . the majority of observations of star - forming galaxies to date have been taken through a spectroscopic slit designed to cover the brightest continuum regions . slits with angular extents of @xmath765 subtend roughly @xmath766kpc for galaxies at @xmath767 . therefore , a @xmath768 slit covering a galaxy with our fiducial wind would cover less than half the wind . the result is reduced line - emission when compared to the galaxy continuum . ) relative to the observed absorption equivalent width ( @xmath769 ) for a series of transitions : ( solid @xmath97 ; dotted @xmath102 ; dashed @xmath139 ) . the @xmath770/@xmath769 ratio is plotted as a function of slit width relative to twice the radius @xmath771 , defined to be where the sobolev optical depth @xmath772 . the black curves correspond to the fiducial wind model ( @xmath18 [ sec : fiducial ] ; @xmath773kpc ) , the red curves are for the lbg - partial covering scenario ( @xmath18 [ sec : covering ] ; @xmath774kpc ) , and the radiation - driven wind ( @xmath18 [ sec : radiative ] ; @xmath775kpc ) has blue curves . for all of the wind models , the @xmath776/@xmath769 ratio rises very steeply with slit width and then plateaus at @xmath777 . therefore , a slit that exceeds @xmath778 will admit nearly all of the photons scattered to our sightline . , height=336 ] we explore the effects of slit - loss as follows . we model the slit as a perfect , infinitely - long rectangle centered on the galaxy . we then tabulate the equivalent width of the line - emission through slits with a range of widths , parameterized by @xmath771 : twice the radius where the wind has a sobolev optical depth of @xmath772 . the line - emission will be weak beyond this radius because the photons have only a low probability of being scattered . for the fiducial wind model , the @xmath97 transition has @xmath773kpc ( figure [ fig : fiducial_nvt ] ) . the predicted equivalent width in emission @xmath770 relative to the absorption equivalent width @xmath769 is presented in figure [ fig : obs_slit ] for a series of transitions for the fiducial wind model ( black curves ) . the @xmath770/@xmath769 curves rise very steeply with increasing slit width and then plateau when the slit width reaches @xmath777 . for the fiducial model , the emission is concentrated toward the source ; this derives from the density and velocity profiles but is also a simple consequence of geometric projection . figure [ fig : obs_slit ] reveals similar results for other wind models . the curves rise so steeply that the effects of slit - loss are minor ( order unity ) unless the slit - width is very small . nevertheless , the results do motivate extended aperture observations , e.g. integral field unit ( ifu ) instrumentation , that would map the wind both spatially and spectrally . although several effects can reduce the line - emission relative to the absorption of the outflow , our analysis indicates that detectable line - emission should occur frequently . furthermore , the line - emission could be suppressed so that it does not exceed the galaxy continuum yet still ( partially ) fills - in the absorption profiles . indeed , dust and anisotropic winds preferentially suppress line - emission at @xmath140 ( figures [ fig : anisotropic],[fig : dust ] ) . the remaining emission would still modify the observed absorption profiles ( e.g.figure [ fig : noemiss ] ) and may complicate conclusions regarding characteristics of the flow . _ an analysis of cool gas outflows that entirely ignores line - emission may incorrectly conclude that the source is partially covered , that the gas has a significantly lower peak optical depth , and/or that a @xmath572 component is absent . _ we now examine several quantitative effects of line - emission . which includes the flux of scattered photons and the ` intrinsic ' equivalent width @xmath779 that ignores photon scattering . the dashed ( dotted ) curves trace a 50% ( 10% ) reduction in @xmath769relative to @xmath779 . one notes a reduction in @xmath769 by @xmath780 for the @xmath97 transition ( the effect is generally larger for @xmath102 ) . the effects of scattered photons are reduced for the transitions because a fraction ( in fact a majority for @xmath145 ) of the absorbed photons fluoresce as * emission at longer wavelengths and do not ` fill - in ' the absorption profiles . the figure shows results for all of the models presented in tables [ tab : line_diag ] and [ tab : plaw_diag ] . , height=336 ] figure [ fig : obs_ew ] demonstrates one observational consequence : reduced measurements for the absorption equivalent width @xmath769 of the flow . in the case of , which has the most strongly affected transitions , @xmath769 is reduced by @xmath781 from the intrinsic equivalent width ( the equivalent width one would measure in the absence of scattered photons ) . in turn , one may derive a systematically lower optical depth or velocity extent for the wind , and therefore a lower total mass and kinetic energy . the effects are most pronounced for wind scenarios where the peak optical depth occurs near @xmath169 . geometric projection limits the majority of scattered photon emission to have @xmath782 ; therefore , the absorption profiles are filled - in primarily at these velocities . $ ] for @xmath97 profiles of the wind models studied in this paper ( tables [ tab : line_diag ] , [ tab : plaw_diag ] ) with @xmath783 the minimum normalized intensity of the absorption profile . cases where @xmath485 exceeds 3 are presented as lower limits . the @xmath485 values are plotted against the velocity @xmath486 where @xmath784 . in all of the models , the true peak optical depth @xmath785 . the much lower ` observed ' @xmath485 values occur because scattered @xmath97photons have filled - in the absorption profiles at velocities @xmath786 . these effects , therefore , are greatest for wind models where the optical depth peaks near @xmath572 because the majority of scattered photons have this relative velocity . indeed , models with @xmath787 all show @xmath788 . , height=336 ] another ( related ) consequence is the reduction of the peak depth of absorption . in figure [ fig : obs_peaktau ] , we plot the observed peak optical depth @xmath485 for @xmath97 versus the velocity where the profile has greatest depth for the various wind models ( i.e.@xmath485 vs. @xmath486 from tables [ tab : line_diag ] and [ tab : plaw_diag ] ) . in every one of the models , the true peak optical depth @xmath789 . for the majority of cases with @xmath790 , one observes @xmath791 and would infer the wind is not even optically thick ! this occurs because photons scattered by the wind have ` filled - in ' the absorption at velocity @xmath187 . in contrast , wind models with @xmath787 all yield @xmath792 . the results are similar for the @xmath102 profile . in fact , one generally measures a similar @xmath485 for each line and may incorrectly conclude that the source is partially covered . in the case of , one may even observe a ( non - physical ) inversion in the apparent optical depths of @xmath145 and @xmath139 ( e.g.figure [ fig : norm ] ) . and @xmath97 absorption profiles have greatest depth . results for all of the models studied in this paper are presented ( tables [ tab : line_diag ] , [ tab : plaw_diag ] ) . it is evident that wind models with a peak optical depth near the systemic velocity have a @xmath97 absorption profile shifted blueward by one to several hundred . analysis of such profiles may lead to the false conclusion that ( i ) the majority of mass in the wind is travelling at a higher velocity ; and ( ii ) there is no gas with @xmath572 . the dashed curve traces the one - to - one line . , height=336 ] these effects are reduced for the absorption profiles , especially for @xmath145 . this is because a significant fraction ( even a majority ) of the emission is florescent * emission at longer wavelengths which does not affect the absorption profiles . therefore , the absorption equivalent widths ( @xmath769 ) more closely follow the intrinsic values , one derives more accurate peak optical depths , and the absorption kinematics more faithfully reflect the motions of the flow . regarding the last point , one also predicts a velocity offset between the and absorption - line centroids ( figure [ fig : obs_vtau ] ) . this affects the analysis of gas related to the ism of the galaxy and also material infalling at modest speeds . figure [ fig : obs_vtau ] also emphasizes that the profiles may misrepresent the kinematics of the bulk of the gas . analysis of these lines , without consideration of line - emission , may lead to incorrect conclusions on the energetics and mass flux of the wind . ) and noise has been added to give a s / n=7 per pixel . note the suppressed peak in line - emission ; one may even infer the flux of @xmath97 exceeds that of @xmath102 . the @xmath97 absorption is also shifted to shorter wavelengths , i.e. to a greater velocity offset from systemic . ( b ) the solid curve shows a stack of 100 spectra of the ism+wind profile , each degraded to a s / n=2 per pixel , and offset from systemic by a normal deviate with @xmath793 to mimic uncertainty in the redshift of the galaxies . this treatment is meant to illustrate the implications of stacking galaxy spectra to study outflows ( e.g. * ? ? ? * ; * ? ? ? the main difference from the single galaxy observation shown in panel ( a ) is the smearing of line - emission and absorption that reduces the height / depth of each . the effects would likely be even more pronounced if one studied spectra with a diversity of profiles . , height=336 ] we emphasize that all of these effects are heightened by the relatively low spectral resolution and s / n characteristic of the data commonly acquired for @xmath794 star - forming galaxies . figure [ fig : obs_lris]a shows one realization of the @xmath0 doublet for the ism+wind model convolved with the line - spread - function of the keck / lris spectrometer ( a gaussian with fwhm=250 ) and an assumed signal - to - noise of s / n=7 per 1 pixel . both the absorption and emission are well detected , but it would be difficult to resolve the issues discussed above ( e.g. partial covering , peak optical depth ) with these data . one also notes several systematic effects of the lower spectral resolution , e.g. reduced peak flux in the line - emission and a systematic shift of @xmath97 absorption to more negative velocity . we have also modelled an observation of the ism+wind model using a stacked galaxy spectrum ( figure [ fig : obs_lris]b ) . specifically , we averaged 100 identical profiles from the fiducial model degraded to a s / n=2pix@xmath19 and shifted by a random velocity offset with @xmath795 . this treatment illustrates the effects of using stacked galaxy spectra to study outflows ( e.g. * ? ? ? * ; * ? ? ? the main difference between this and the single galaxy observation shown in panel ( a ) is the smearing of line - emission and absorption that reduces the height / depth of each . the effects would be even more pronounced if one studied spectra with a diversity of profiles . special care is required , therefore , to interpret properties of the wind ( and ism ) from such spectral analysis . ( solid ) and * @xmath149 ( dotted ) transitions . both profiles peak at small radii and decrease by several orders of magnitude before reaching the outer edge of the wind . the profiles are sufficiently shallow , however , that the azimuthally integrated flux declines by only a factor of @xmath796 from @xmath521kpc to @xmath797kpc . therefore , a sensitive ifu observation could map the emission ( and , in principle , the kinematics ) from @xmath77 to approximately @xmath78 . , height=336 ] the previous few paragraphs sounded a cautionary perspective on the implications for absorption - line analysis of galactic - scale outflows in the presence of ( expected ) significant line - emission . while this is a necessary complication , we emphasize that such analysis remains one of the few observational techniques at our disposal to study outflows . furthermore , direct analysis of the line - emission offers new and unique constraints on the characteristics of the outflow . and , when coupled with the absorption - line data , the two sets of constraints may break various degeneracies in the physical characteristics of the outflow . we now consider a few examples . the most obvious characteristics probed by the line - emission are the size and morphology of the outflow @xcite . line emission is predicted to extend to radii where the sobolev optical depth exceeds a few tenths . the principle challenge is to achieve sufficient sensitivity to detect the predicted , low surface - brightness emission . as figure [ fig : obs_sb ] demonstrates , the surface brightness at the inner wind radius of our fiducial wind exceeds that at the outer radius by several orders of magnitude . nevertheless , an instrument that sampled the entire wind ( e.g. a large format ifu or narrow band imager ) may detect the emission in azimuthally - averaged apertures . for the fiducial wind model , for example , the azimuthally integrated flux falls by only a factor of 10 for projected radii of 1 to 15kpc . of the transitions considered in this paper , emission is preferred for this analysis because ( i ) it has the largest equivalent width in absorption and ( ii ) there are fewer emission channels per absorption line than for . the lines ( especially @xmath102 ) frequently have the highest peak and integrated fluxes ( figure [ fig : obs_sb ] ) . on the other hand , the transitions are more susceptible to the effects of dust extinction and for some galaxies * emission could be dominant . in either case , the study of spatially - extended line - emission from photons scattered by a galactic - scale wind offers a direct means to study the morphology and radial extent of these phenomena . the next generation of large - format optical and infrared integral - field - units are well - suited to this scientific endeavor ( e.g. kcwi on keck , muse on the vlt ) . the kinematic measurements of the line - emission also offer insight into physical characteristics of the wind . in @xmath18 [ sec : dust ] , we emphasized that dust extinction preferentially suppresses photons scattered off the backside of the wind ( e.g. @xmath140 ) so that the line - centroid is shifted to negative velocities . similarly , anisotropic models with reduced emission off the backside yield emission lines that are centered blueward of the galaxy s systemic redshift . these effects are most prominent in the * emission . the centroids of the lines , meanwhile , are sensitive to the optical depth profile of the wind . for example , a wind with flows exceeding @xmath798 will shift the @xmath97 centroid to bluer wavelengths ( e.g. figure [ fig : norm ] ) . although these effects are modest ( tens of ) , they may be resolved by moderate resolution spectroscopy . ) of * @xmath148 emission versus the velocity where the resonance lines achieve peak optical depth @xmath486 in absorption . the @xmath799 values rise steadily with increasing offset of @xmath486 from systemic . the width of the * @xmath148emission line , therefore , offers an independent diagnostic of the wind speed . , height=336 ] the emission - line velocity widths also reveal characteristics of the wind . in particular , the * emission - line widths are sensitive to the optical depth of the wind at large velocity offsets from systemic . this is illustrated in figure [ fig : obs_edelv ] which plots the 90% velocity width @xmath799 of * @xmath148 versus the velocity where the wind optical depth is maximal ( @xmath486 ) . we find that @xmath799 increases with @xmath486 such that a large width for * emission requires a wind profile with large optical depths at @xmath800 . the broadening of * emission occurs because this emission is dominated by single scatterings which trace all components of the wind . lastly , the line flux ratios of pairs of transitions are sensitive to characteristics of the wind . the most obvious example is the relative emission of the @xmath0doublet . specifically , an outflow whose velocity exceeds the doublet spacing ( @xmath32 ) will convert @xmath97 photons into @xmath102 emission resulting in a flux ratio that is inverted relative to the intrinsic optical depth profiles ( e.g.figure [ fig : norm ] ) . another important example is the relative flux of * emission relative to . a dusty medium with velocity near systemic ( e.g. an ism component ) may significantly suppress emission yet have a significant * line flux because the latter is not resonantly trapped . in this paper , we have explored the predicted absorption and emission - line profiles of a set of simple galactic - scale outflow models . this analysis implemented monte carlo radiative transfer techniques to propogate resonant photons through an expanding medium , allowing for their conversion to non - resonant photons ( e.g. * ) . our work focused on the @xmath0 and uv1 multiplet of rest - frame uv transitions , but the results apply to most other lines used to probe cool gas . our primary findings are summarized as follows : 1 . isotropic , dust - free wind models conserve photon flux . therefore , the blue - shifted absorption - line profiles commonly observed in star - forming galaxies are predicted to be accompanied by emission - lines with similar equivalent width . this holds even for non - extreme anisotropic and dust - extincted scenarios . 2 . the line - emission occurs preferentially at the systemic velocity of the galaxy and ` fills - in ' the absorption profiles at velocity offsets @xmath782 from systemic . for transitions that are only coupled to the ground - state ( e.g. , , ) , this implies much lower , absorption - line equivalent widths ( by up to @xmath2 ) and observed absorption profiles that are significantly offset in velocity from the intrinsic optical depth profile . 3 . analysis of cool gas outflows that entirely ignores this line - emission may incorrectly conclude that the source is partially covered , that the gas has a significantly lower peak optical depth , and/or that gas with velocities near systemic ( e.g. from the ism or even an infalling component ) is absent . resonance transitions that are strongly coupled to non - resonant lines ( e.g. , ) produce emission dominated by the optically - thin , fine - structure transitions . as such , these resonance absorption lines offer the best characterization of the opacity of the wind and also of gas with velocities near systemic . 5 . dust extinction modestly affects models where resonance photons are trapped for only a few scatterings . models with high opacity at small radii and at systemic velocity ( e.g. with an optically thick , ism component ) can effectively extinguish resonantly trapped emission ( ) but have weaker effect on non - resonant lines ( * ) . we examined two scenarios designed to mimic the wind model proposed by @xcite for @xmath4 lyman break galaxies . our implementation of this model genreates substantial line - emission from scattered photons that greatly modifies the predicted line - profiles so that this model does not reproduce the observed line - profiles . significant line - emission is a generic prediction of simple wind models , even in the presence of dust , anisotropic flows , and when viewed through finite apertures . we have explored the 2d emission maps ( figure [ fig : fiducial_ifu_mgii ] ) and surface brightness profiles ( figure [ fig : obs_sb ] ) of the winds . sensitive , spatially - extended observations will map the morphology and radial extent of the outflows . these data afford the best opportunity to estimate the energetics and mass - flux of galactic - scale outflows . the kinematics and flux ratios of the emission lines constrain the speed , opacity , dust extinction , and morphology of the wind . when combined with absorption - line analysis , one may develop yet tighter constraints on these characteristics .
|
we analyze the absorption and emission - line profiles produced by a set of simple , cool gas wind models motivated by galactic - scale outflow observations .
we implement monte carlo radiative transfer techniques that track the propagation of scattered and fluorescent photons to generate 1d spectra and 2d spectral images .
we focus on the @xmath0 doublet and uv1 multiplet at @xmath1 , but the results are applicable to other transitions that trace outflows ( e.g. , , ) . by design ,
the resonance transitions show blue - shifted absorption but one also predicts strong resonance and fine - structure line - emission at roughly the systemic velocity .
this line - emission ` fills - in ' the absorption reducing the equivalent width by up to @xmath2 , shift the absorption - lin centroid by tens of , and reduce the effective opacity near systemic .
analysis of cool gas outflows that ignores this line - emission may incorrectly infer that the gas is partially covered , measure a significantly lower peak optical depth , and/or conclude that gas at systemic velocity is absent ( e.g. an interstellar or slowly infalling component ) . because the lines are connected by optically - thin transitions to fine - structure levels , their profiles more closely reproduce the intrinsic opacity of the wind .
together these results naturally explain the absorption and emission - line characteristics observed for star - forming galaxies at @xmath3 .
we also study a scenario promoted to describe the outflows of @xmath4 lyman break galaxies and find prfiles inconsistent with the observations due to scattered photon emission .
although line - emission complicates the analysis of absorption - line profiles , the surface brightness profiles offer a unique means of assessing the morphology and size of galactic - scale winds .
furthermore , the kinematics and line - ratios offer powerful diagnostics of outflows , motivating deep , spatially - extended spectroscopic observations .
# 1 10^#1 5 ly-5 6 ly-6 7 ly-7 @xmath5
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum bits ( qubits ) based on polarization or spatial degrees of freedom of optical modes have several advantages : they are easily manipulated and measured ; they exist in a low noise environment and ; they are easily communicated over comparitively long distances . recently considerable progress has been made on implementing two qubit gates in optics using the measurement induced non - linearities proposed by knill , laflamme and milburn @xcite . non - deterministic experimental demonstrations have been made @xcite and theory has found significant ways to reduce the resource overheads @xcite . nevertheless , the number of photons and gate operations required to implement a near deterministic two qubit gate remains high . a possible solution to this problem is the optical quantum zeno gate suggested by franson et al @xcite , @xcite . this gate uses passive two - photon absorption to suppress gate failure events associated with photon bunching at the linear optical elements , using the quantum zeno effect @xcite . in principle a near deterministic , high fidelity control - sign ( cz ) gate can be implemented between a pair of photonic qubits in this way . however , the slow convergence of the zeno effect to the ideal result , with ensuing loss of fidelity , and the effect of single photon loss raises questions about the practicality of this approach . here we consider a model of the gate that includes the effects of finite two - photon absorption and non - negligible single photon absorption . we obtain analytic expressions for the fidelity of the gate and its probability of success in several scenarios and show how the inclusion of optical distilling elements @xcite can lead to high fidelity operation under non - ideal conditions for tasks such as cluster state construction @xcite . the paper is arranged in the following way . we begin in the next section by introducing our model in an idealized and then more realistic setting and obtain results for a free - standing cz gate . in section 3 we focus on using the gate as a fusion element @xcite for the construction of , for example , optical cluster states . we introduce a distillation protocol that significantly improves the operation of the gate in this scenario . in section 4 we summarize and conclude . franson et al @xcite suggested using a pair of optical fibres weakly evanescently coupled and doped with two - photon absorbing atoms to implement the gate . as the photons in the two fibre modes couple the occurence of two photon state components is suppressed by the presence of the two - photon absorbers via the zeno effect . after a length of fibre corresponding to a complete swap of the two modes a @xmath2 phase difference is produced between the @xmath3 term and the others . if the fibre modes are then swapped back by simply crossing them , a cz gate is achieved . we model this system as a succession of @xmath4 weak beamsplitters followed by 2-photon absorbers as shown in fig . [ fig : ourcsign ] . as @xmath5 the model tends to the continuous coupling limit envisaged for the physical realization . the gate operates on the single - rail encoding @xcite for which @xmath6 and @xmath7 with the kets representing photon fock states . [ fig : cz ] shows how the single rail cz can be converted into a dual rail cz with logical encoding @xmath8 and @xmath9 with @xmath10 a fock state with @xmath11 photons in the horizontal polarization mode and @xmath12 photons in the vertical . + the general symmetric beam splitter matrix has the form : @xmath13\ ] ] according to figure [ fig : ourcsign ] , after the first beam splitter , the four computational photon number states become : @xmath14 to illustrate the operation of the gate we first assume ideal two - photon absorbers , i.e. they completely block the two - photon state components but do not cause any single photon loss . propagation through the first pair of ideal two - photon absorbers gives the mixed state @xmath15 where @xmath16 is the evolved two - mode input state obtained for the case of no two - photon absorption event and @xmath17 is the vacuum state obtained in the case a two - photon absorption event occurs . the individual components of @xmath16 transform as @xmath18 notice that , because we are embedded in a dual rail circuit , we can distinguish between the @xmath19 state that corresponds to input state @xmath20 and the @xmath21 state that results from two - photon absorption of input state @xmath22 . equation ( [ eqn : complete ] ) describes the transformation of each unit , hence repeating the procedure @xmath4 times gives , latexmath:[\[\begin{aligned } describing the transformations giving the evolved input state after @xmath4 units , @xmath24 . there are three conditions to satisfy for building a cz gate . the first condition is @xmath25 " , so that @xmath26 and @xmath27 . the second condition is @xmath28 " ( equivalently , @xmath29 ) , where @xmath30 is any integer , so that @xmath31 and @xmath32 and @xmath33 . phase shifters are needed to correct the sign of the output state of @xmath34 and @xmath35 for odd @xmath30 , but here we simply set @xmath36 . the last condition is @xmath37 " ( i.e. @xmath38 ) , such that a minus sign is induced on @xmath39 . this condition is always true because we are using many weak beam splitters ( i.e. @xmath40 is small ) . let @xmath41 , then swapping the fibres gives the transformations @xmath42 ) on the probability amplitude of the @xmath39 state . if there is some way to herald failure , i.e. two - photon absorption events , then the fidelity of the gate will be @xmath43 , where @xmath44 is the target state , and the probability of success will be @xmath45 . on the other hand if two - photon absorption events are unheralded then the fidelity will be @xmath46 . for simplicity we consider the equally weighted superposition input state @xmath47 . the corresponding @xmath24 after the zeno - cz gate is @xmath48 , to be compared with the target state @xmath49 . the heralded fidelity and probability of success are then @xmath50 and @xmath51 respectively . as @xmath4 becomes very large and hence tends to the continuous limit , @xmath52 tends to one , and so both @xmath53 and @xmath54 approach one . the previous analysis is clearly unrealistic as it assumes infinitely strong two - photon absorption but negligible single photon absorption . we now include the effect of finite two - photon absorption and non - negligible single photon loss . let @xmath55 and @xmath56 be the probability of single photon and two - photon transmission respectively for one absorber . here the parameter @xmath57 , where @xmath58 is the length of the absorber and @xmath59 is the corresponding proportionality constant related to the absorption cross section . furthermore , @xmath0 specifies the relative strength of the two transmissions and relates them by @xmath60 . now each unit of weak beam splitter and absorbers does the following transformation on the computational states of @xmath61 @xmath62 repeating the procedure @xmath4 times with the aforementioned conditions on @xmath40 gives the following latexmath:[\[\begin{aligned } expression for @xmath52 is given by : @xmath64 and we have suppressed the explicit form of the @xmath65 state components as they lie outside the computational basis and so do not explicitly contribute to the fidelity . these expressions can be used to calculate the unheralded fidelity , the heralded fidelity and probability of success . our numerical evaluations are all carried out in the ( near ) continuous limit of large @xmath4 . for a free - standing gate , as depicted in fig.[fig : cz ] , gate failure events are not heralded , thus the unheralded fidelity is appropriate to consider . the fidelity is a function of @xmath66 . as the length of the interaction region is increased ( @xmath66 increased ) the effective strength of the two - photon absorption is increased leading to an improvement in the heralded fidelity , @xmath53 . however , at the same time , the level of single photon absorption is also increasing with the length , acting to decrease the probability of success , @xmath54 . as the unheralded fidelity is @xmath67 , there is a trade - off between these two effects leading to an optimum value for @xmath66 for sufficiently large @xmath0 . an example of the dependence is shown in fig.[fig : fidelity_vs_lambda ] . the fidelity is plotted as a function of @xmath0 with @xmath66 optimized for each point in fig.[fig : fidelity_vs_kappa ] . for large ratios of two - photon absorption to single - photon absorption , @xmath0 , we tend to the ideal case of unit fidelity . however , the conditions required are demanding with absorption ratios of a million to one required for @xmath68 and 100 million to one for @xmath69 . recent estimates suggest @xmath0 s of ten thousand to one may be achievable @xcite , well short of these numbers . in the following we will consider a different scenario in which the gate can be usefully employed with less stringent conditions on @xmath0 . we have seen that the requirements on high fidelity operation for the free - standing gate are quite extreme . we now consider an alternate scenario in which probability of success is traded - off against fidelity by heralding failure events through direct detection . in particular we consider using the zeno gate to implement the fusion technique @xcite . fusion can be used to efficiently construct cluster states @xcite , or re - encode parity states @xcite . we will specifically consider cluster state construction here . essentially , the gate is used to make a bell measurement on a pair of qubits , as depicted in fig.[fig : fusion ] . one of the qubits comes from the cluster we are constructing , whilst the other comes from a resource cluster state , in a known logical state . the bell measurement has the effect of fusing " the resource state onto the existing state . by careful choice of the resource state , large 2-dimensional cluster states , suitable for quantum computation , can be constructed @xcite . because the bell measurement ends with the direct detection of the qubits , the loss of one or both of the photons , or the bunching of two photons in a single qubit mode can immediately be identified in the detection record , and hence failure events will be heralded . effectively we will postselect the density operator @xmath70 , where @xmath71 is the component of the output state which remains in the computational basis and @xmath72 are all the components that do not . the measurement record then allows us to herald the first term of the density operator as successful operation , with fidelity @xmath73 and probability of success of @xmath54 , and the second term as failure . we now consider techniques for improving the heralded fidelity of the gate and then evaluate its performance as a fusion gate . from equation ( [ eqn : incomplete ] ) , we can see that @xmath74 lowers the probability amplitude of the four computational states unevenly as previously discussed by jacobs et al @xcite . by distilling the states with beam splitters and detectors @xcite ( see figure [ fig : cz_distill ] ) , where each beam splitter has a transmission coefficient equal to @xmath75 , the four computational states of @xmath71 become : @xmath76 and @xmath77 respectively . as shown previously , after the cz gate and single photon distillation , the input state @xmath47 becomes @xmath80 . now we require two - photon distillation to renormalise the input state by inducing @xmath52 on the other three computational states as shown in figure [ fig : two_photon_distill ] . to do so , we first apply a bit - flip on the control qubit and then apply a @xmath52-gate ( see figure [ fig : tau_gate ] ) and a single photon distiller on the control qubit with transmission coefficient @xmath81 and another single photon distiller on the target qubit with transmission coefficient @xmath82 and then undo the previous bit - flip by applying another bit - flip on the control qubit . the @xmath52-gate does the same operation as the aforementioned cz gate ( excluding the single photon distillation ) except that no minus sign is induced on the output of @xmath83 . the construction of a @xmath52-gate is described in the next subsection . in summary the two - photon distillation circuit does the following : @xmath84 becomes @xmath85 . now the state can be renormalised to achieve unit fidelity _ independent _ of @xmath66 . the explicit expression for the probability of success is @xmath86 figure [ fig : two_photon_distill_psuccess ] shows the probability of success of this gate for different values of @xmath0 . + we can construct a @xmath52-gate with two 50 - 50 beam splitters , a pair of two - photon absorbers , and some phase shifters , as shown in figure [ fig : tau_gate ] . the first beam splitter performs @xmath87 , @xmath88 and @xmath89 . the pair of two - photon absorbers then induce @xmath81 on both @xmath34 and @xmath35 due to single photon loss , and induce @xmath90 on @xmath91 due to both single photon and two - photon loss . the second beam splitter undoes the operation of the first beam splitter . then with some phase shifters to correct the relative phase between the terms and having @xmath92 , we have a @xmath52-gate that does the following operation : the fusion approach is important because it is the most efficient method known for performing quantum computation using only linear optics . linear optics allows a partial bell measurement to be made with a probability of success of 50% ( assuming ideal detectors ) . in addition the failure mode measures the qubits in the computational basis , which does not affect the state of the remaining qubits in the cluster or parity state . thus a failure event only sacrifices a single qubit from the cluster being constructed and the probability of destroying @xmath94 qubits in the process of achieving a successful fusion is @xmath95 . in contrast , many of the failure events for the zeno gate will simply erase the photon giving no knowledge about its state . for simplicity , and to be conservative , we will assume all events lead to complete erasure of the photon state . in order to recover from this situation the adjoining qubit in the cluster must be measured in the logical basis , thus removing the affect of the erasure @xcite . this means that every failure event sacrifices two qubits from the cluster being constructed and the probability of destroying @xmath94 qubits in the process of achieving a successful fusion is @xmath96 . requiring @xmath97 we estimate that the zeno gate must have @xmath98 to offer an advantage over linear optics . + we can make one final improvement to the set - up by relocating the distillation process for the resource qubit to offline ( see fig.5 ) , which boosts the probability of success . the probability of success is then given by @xmath99 . the plot for the probability of success versus @xmath0 and optimal @xmath66 versus @xmath0 are shown in figure [ fig : psuccessvsnoffline ] and figure [ fig : optimalvsnoffline ] respectively . + the break even point between linear optics and the zeno gate is when @xmath100 , such that the probability of success is about @xmath101 . when @xmath102 the probability of success is about @xmath103 . thus we conclude that an absorption ratio of ten thousand to one or more would produce a zeno gate with significant advantage over linear fusion techniques . + * conclusion * + in this paper , we have modelled franson et al s cz gate with a succession of @xmath4 weak beam - splitters followed by two - photon absorbers , in the ( near ) continuous limit of large @xmath4 . we analysed this cz gate for both the ideal two - photon absorption case and the imcomplete two - photon absorption with single photon loss case , giving analytical and numerical results for the fidelity and probability of success . the result shows that for a free - standing gate we need an absorption ratio @xmath0 of a million to one to achieve @xmath104 and 100 million to one to achieve @xmath105 , where recent estimate only suggests that @xmath106 may be achievable . we therefore employ this gate for qubit fusion , where the requirement for @xmath0 is less restrictive . with the help of partially offline one - photon and two - photon distillations , we can achieve a cz gate with unity fidelity and with probability of success is about 0.87 for @xmath102 . we conclude that when employed as a fusion gate , the zeno gate could offer significant advantages over linear techniques for reasonable parameters . + * acknowledgement * + we thank w.j.munro , a.gilchrist and c.myers for useful discussions . this work was supported by the australian research council and the dto - funded u.s . army research office contract no . w911nf-05 - 0397 . + 99 e. knill , r. laflamme , and g.j . milburn , _ nature _ * 409 * , 46 - 52 ( 2001 ) . j.l.obrien , g.j.pryde , a.g.white , t.c.ralph , d.branning , nature * 426 * , 264(2003 ) . pittman , m.j . fitch , b.c . jacobs , and j.d . franson , phys . a * 68 * , 032316 ( 2003 ) . s.gasparoni , j .- w.pan , p.walther , t.rudolph , and a.zeilinger phys . lett . * 93 * , 020504 ( 2004 ) . n.yoran and b.reznik , phys . lett . * 91 * , 037903 ( 2003 ) . nielsen , phys . rev . lett . * 93 * , 040503 ( 2004 ) . a.j.f.hayes , a.gilchrist , c.r.myers and t.c.ralph , j.opt.b * 6 * , 533 ( 2004 ) . d.e.browne and t.rudolph , phys . lett . * 95 * , 010501 ( 2005 ) . franson , b.c . jacobs , and t.b . pittman , _ pra _ * 70 * , 062302 ( 2004 ) b.c . jacobs , t.b . pittman , and j.d . franson , _ pra _ * 74 * , 010303(r ) ( 2006 ) w. kaiser and c. garrett , _ prl _ * 7 * 229 ( 1961 ) r.t . thew and w.j . munro , _ pra _ * 63 * , 030302(r ) ( 2001 ) a.p . lund , t.c . ralph , _ pra _ , * 66 * , 032307 ( 2002 ) . j.d . franson and s.m . quant - ph _ 0603044 ( 2006 ) r. raussendorf and h.j . briegel , _ prl _ * 86 * 5188 ( 2001 ) t.c . ralph , a.j.f . hayes and a. gilchrist , _ prl _ * 95 * 100501 ( 2005 ) c.m . dawson , h.l . haselgrove , and m.a . nielsen , _ prl _ 96 , 020501 ( 2006 ) l.m . duan and r. raussendorf , _ prl _ * 95 * , 080503 ( 2005 ) y.l . lim , s.d . barrett , a. beige , p. kok , l.c . kwek , _ pra _ * 73 * , 012304 ( 2006 )
|
we have modelled the zeno effect control - sign gate of franson et al ( pra 70 , 062302 , 2004 ) and shown that high two - photon to one - photon absorption ratios , @xmath0 , are needed for high fidelity free standing operation .
hence we instead employ this gate for cluster state fusion , where the requirement for @xmath0 is less restrictive . with the help of partially offline one - photon and two - photon distillations , we can achieve a fusion gate with unity fidelity but non - unit probability of success .
we conclude that for @xmath1 , the zeno fusion gate will out perform the equivalent linear optics gate .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the microscopic calculation of ground state energy and particle density of medium and heavy nuclei based on realistic nucleon - nucleon interaction requires the solution of a formidable many - body problem . for this reason effective nucleon - nucleon interactions have been introduced , like the skyrme @xcite and gogny @xcite forces . they simplify enormously the problem since , by construction , they must be used at the mean field level , and the calculation of the mean single particle potential and of the ground state energy becomes easily manageable . the number of parameters which enter these effective forces is typically around ten and they are adjusted to reproduce finite nuclei and some equilibrium nuclear matter properties . however , recently also data from a theoretically determined neutron matter equation of state ( eos ) @xcite have been used as input ( see also an older attempt in this direction in @xcite ) . generally these forces give rise to an effective nucleon mass @xmath0 with typically @xmath1 in the non - relativistic framework . with these ingredients nuclear mean field theories are very successful to describe nuclear properties as , e.g. binding energies , radii . their use can be extended to the evaluation of more realistic single level schemes , of the nuclear excitation spectra , in particular giant resonances , of the fission barriers , of the nucleon - nucleus optical potential , and so on . the price to be payed is that the connection with the bare nucleon - nucleon interaction is not apparent . the first attempt to connect the effective nn interaction and the underlying bare interaction was the density matrix expansion of negele and vautherin @xcite , based on the assumption that the effective nn interaction can be identified essentially with the brueckner g - matrix calculated in nuclear matter and on the gradient expansion of the non - local one - body density matrix . see also another attempts in the same direction in ref . this approach was recently generalized to include also the vector ( spin ) part of the density matrix at the same level of accuracy as the scalar part @xcite . the challenging program of deriving the effective interaction to be used in nuclear structure studies from the bare nn interaction has still to be completed . an approach similar to the skyrme - like one is the energy density functional ( edf ) approach , where the basic quantity to start with is directly the functional that expresses the energy in terms of the matter density and its gradients . the functional must be minimized to obtain the actual ground state energy and matter density profile . following the method of kohn and sham @xcite , developed in atomic , molecular and solid state physics , the minimization procedure can be performed by introducing a set of auxiliary single particle states and taking for the kinetic energy the slater form in this basis . in this way the minimization procedure gives hartree - like equations for the single particle states , where the interaction part includes in an effective way the overall exchange and correlation contributions . the latter , in condensed matter physics , is taken from accurate calculations for the homogeneous electron system or in a purely phenomenological fashion . this is not the usual way of proceeding in nuclear physics , although it exists , to our knowledge , an earlier attempt in this direction , see ref . @xcite . in this paper we will discuss to what extent the edf approach in nuclear structure can be based on microscopic many - body results on nuclear matter equation of state ( eos ) and which accuracy can be reached . to this purpose we will rely on recent achievements along this line of research and we will analyze the main open problems that hinder the development of the microscopic many - body theory of the nuclear edf . despite the fact that the applicability of the ks - dft approach to self - bound systems , as nuclei , is not obvious @xcite , it is commonly believed that the basis of ks - dft lies in the hohenberg - kohn ( hk ) theorem @xcite , which states that for a fermi system , with a non - degenerate ground state , the total energy can be expressed as a functional of the density @xmath2 only . such a functional reaches its variational minimum when evaluated with the exact ground state density . furthermore , in the standard ks - dft method one introduces an auxiliary set of @xmath3 orthonormal single particle wave functions @xmath4 , where @xmath3 is the number of particles , and the density is assumed to be given by @xmath5 where @xmath6 and @xmath7 stand for spin and iso - spin indices . the variational procedure to minimize the functional is performed in terms of the orbitals instead of the density . as in condensed matter and atomic physics the hk functional @xmath8 is split into two parts : @xmath9+w[\rho]$ ] @xcite . the first piece @xmath10 corresponds to the uncorrelated part of the kinetic energy and within the ks method it is written as @xmath11 the other piece @xmath12 $ ] contains the potential energy as well as the correlated part of the kinetic energy . then , upon variation , one gets a closed set of @xmath3 hartree - like equations with an effective potential , the functional derivative of @xmath12 $ ] with respect to the local density @xmath2 . since the latter depends on the density , and therefore on the @xmath13 s , a self - consistent procedure is necessary . the equations are exact but they only can be of some use if a reliable approximation is found for the otherwise unknown density functional @xmath12 $ ] . it has to be stressed that in the ks - dft formalism the exact ground state wave function is actually not known , the density being the basic quantity . in nuclear physics , contrary to the situation in condensed matter and atomic physics , the contribution of the spin - orbit interaction to the energy functional is very important . non - local contributions have been included in dft in several ways already long ago ( see @xcite for a recent review of this topic ) . consequently , the spin - orbit part also can be split in an uncorrelated part @xmath14 plus a remainder . the form of the uncorrelated spin - orbit part is taken exactly as in the skyrme @xcite or gogny forces @xcite . we thus write for the functional in the nuclear case @xmath15 , where we explicitly split off the coulomb energy @xmath16 because it is a quite distinct part in the hamiltonian . it shall be treated , as usual , at lowest order , i.e. the direct term plus the exchange contribution in the slater approximation , that is @xmath17 , and @xmath18 with @xmath19 and @xmath20 the proton / neutron density . let us now discuss the nuclear energy functional contribution @xmath21 $ ] which contains the nuclear potential energy as well as additional correlations . we shall split it in a finite range term @xmath22 $ ] to account for correct surface properties and a bulk correlation part @xmath23 $ ] that we take from a microscopic infinite nuclear matter calculation @xcite . thus our final ks - dft -like functional reads : @xmath24 for the finite range term we make the simplest phenomenological ansatz possible @xmath25 & = & \frac{1}{2}\sum_{t , t'}\int \int d^3r d^3r'\rho_{t}({\bf r } ) v_{t , t'}({\bf r}-{\bf r'})\rho_{t'}({\bf r ' } ) \nonumber \\ & - & \frac{1}{2}\sum_{t , t ' } \gamma_{t , t'}\int d^3r { \rho_{t}({\bf r } ) } \rho_{t'}({\bf r } ) \label{eq : eq7}\end{aligned}\ ] ] with @xmath26 proton / neutron and @xmath27 the volume integral of @xmath28 . the substraction in ( [ eq : eq7 ] ) is made in order not to contaminate the bulk part , determined from the microscopic infinite matter calculation . finite range terms have already been used earlier , generalizing usual skyrme functionals ( see e.g @xcite ) . in this study , for the finite range form factor @xmath28 we make a simple gaussian ansatz : @xmath29 . we choose a minimum of three open parameters : @xmath30 , and @xmath31 . the only undetermined and most important piece in ( [ eq : eq6 ] ) is then the bulk contribution @xmath32 . as already mentioned , we obtain @xmath32 from microscopic infinite matter calculations , using a realistic bare force , together with a converged hole - line expansion @xcite . we first reproduce by interpolating functions the _ correlation _ part of the ground state energy per particle of symmetric and pure neutron matters , and then make a quadratic interpolation for asymmetric matter . finally the total correlation contribution to the energy functional in local density approximation reads : @xmath33 = \int d^3r \big [ p_s(\rho ) ( 1 - \beta^2 ) + p_n ( \rho)\beta^2 \big ] \rho \label{eq : eq4a}\ ] ] where @xmath34 and @xmath35 are two interpolating polynomials for symmetric and pure neutron matter , respectively , at the density @xmath36 , and @xmath37 is the asymmetry parameter . the interpolating polynomial for symmetric matter has been constrained to allow a minimum exactly at the energy @xmath38 = - 16.00 mev and fermi momentum @xmath39 = 1.36 @xmath40 , i.e. @xmath41= 0.16 @xmath42 . this is within the uncertainty of the numerical microscopic calculations of the eos . it has to be stressed that the use of a polynomial in density is just for practical reasons . the constrained fit was performed by keeping the eos as smooth as possible , thus allowing for some very small deviations from the microscopic calculations below saturation density . an interpolating fit which goes exactly through the calculated eos , as performed in @xcite , gives a not good enough saturation point ( typically e / a = -15.6 mev , @xmath39=1.38 @xmath40 ) . as discussed in @xcite , the low density behavior of the nuclear matter eos is quite intricate and usually not reproduced by skyrme and gogny functionals ( see also ref . @xcite ) , missing quite a substantial part of binding . we show our eos for nuclear and neutron matter in fig . since we want to construct the edf , as much as possible , on the basis of the microscopic calculations , the bulk part @xmath32 of the functional , directly related to bare nn and tbf ( three - body forces ) , is determined once and for all and we will use it in ( 3 ) together with lda . [ fig : eos ] the only open parameters are , therefore , the ones contained in the finite range surface part , eq.([eq : eq7 ] ) , and the strength of the spin - orbit contribution , that we fit to reproduce finite nuclei properties . we , thus , follow _ exactly _ the strategy employed in condensed matter . on the contrary , in nuclear physics , almost exclusively a different strategy is usually adopted ( see , however , ref.@xcite with some ingredients similar as in the present approach ): functionals like the one of ( [ eq : eq6 ] ) , i.e. bulk , surface , etc . , were globally parametrized with typically of the order of ten parameters which , then , were determined fitting _ simultaneously _ some equilibrium nuclear matter ( binding energy per particle , saturation density , incompressibiliy , etc . ) and finite nuclei properties . however , in this way , bulk and surface are not properly separated and early attempts used to miss important infinite nuclear matter properties , as , e.g. stability of neutron matter at high density @xcite and other stability criteria . modern skyrme forces , like the saclay - lyon ( sly ) ones , explicitly use the high density part ( @xmath43 ) of microscopic neutron matter calculations for the eos in the fitting procedure @xcite and thus avoid collapse . therefore , modern functionals usually reproduce reasonably well microscopically determined eos for neutron and nuclear matter @xcite . examples are , among others , the sly - forces @xcite ( see fig.1 in @xcite ) and the fayans functional df3 ( see e.g. fig . 3 in @xcite ) . in this presentation we follow the just discussed alternative approach , that is different on a qualitative level from the usual and allows , via the fit , to reproduce very accurately the microscopic infinite matter results in the whole range of densities considered . this may be important for surface properties and neutron skins in exotic nuclei , what shall be investigated in the future . for open shell nuclei , we still have to add pairing . the formal generalization of the rigorous hk theorem to paired system has been given in ref . @xcite . in the present work our main objective is to discuss the ks - dft scheme for the non pairing part , thus we add pairing in a very simple way within the bcs approach . for this we simply take the density dependent delta force defined in ref @xcite for @xmath44 with the same parameters and in particular with the same cutoff . as far as this amounts to a cutoff of @xmath45 10 mev into the continuum for finite nuclei , we have to deal with single - particle energy levels lying in the continuum . we have simulated it by taking in the pairing window all the quasi - bound levels , i.e. the levels retained by the centrifugal ( neutrons ) and centrifugal plus coulomb ( protons ) barriers . this treatment of the continuum works properly , at least for nuclei not far from the stability valley as it has been extensively shown in @xcite . in this way we obtain two - neutron ( @xmath46 ) and two - proton ( @xmath47 ) energy separations for magic proton and neutron numbers in quite good agreement with the experiment ( see also below ) . in our calculations the two - body center of mass correction has been included in the self - consistent calculation using the pocket formula , based on the harmonic oscillator , derived in ref.@xcite which nicely reproduces the exact correction as it has been shown in @xcite . our functional is now fully defined and , henceforth , we call it bcp - functional . preliminary results based on this functional have been presented in ref . in this section we discuss the fitting protocol to determine the open parameters of our functional and the accuracy that can be reached following the scheme described in the previous section . to this purpose we make a comparison between our recent results and the ones that can be obtained with the gogny functional , which is one of the most accurate phenomenological functional . in this paper we follow a novel strategy , recently proposed @xcite , for the fitting procedure to obtain the set of parameters for the surface and spin - orbit part of the functional , that we call the bcp09 functional . contrary to our previous works we no longer use the binding energies of spherical nuclei for the fit but , following the suggestion of ref . @xcite , we use a set of deformed nuclei carefully chosen in the rare earth , actinide and super - heavy regions instead . the underlying philosophy behind this choice @xcite is that for spherical nuclei the fluctuations in the two most relevant collective variables , namely , the quadrupole deformation and the pairing correlations are large ( i.e. the minimum in those variables is broad in some sense ) and therefore correlations beyond mean field are important and difficult to evaluate . on the other hand , for deformed nuclei the minimum as a function of the quadrupole degree of freedom is stiffer than its counterpart in spherical nuclei and the additional correlations ( rotational energy correction mainly ) are not so difficult to compute in a mean field model . in order to fit the three free parameters of the surface part ( @xmath48 , @xmath49 and @xmath31 ) of bcp and the spin - orbit strength @xmath50 we have taken as in @xcite 84 well deformed nuclei in the rare earth , actinide and super - heavy regions where the experimental binding energies are known . the four parameters are determined by minimizing the mean square root deviation of the binding energy @xmath51 . the theoretical binding energy has been computed as the hartree- fock- bogoliubov ( hfb ) mean field energy plus a rotational energy correction , which differs from the standard one @xcite by a phenomenological factor to account for the approximate calculation of the yoccoz moment of inertia . we also consider an additional correction factor ( not relevant in the present calculation ) to deal with the weak deformation regime ( see , for instance , @xcite ) . also a correction to the binding energy ensuing from the finite size of the harmonic oscillator basis used , as estimated in @xcite , has been included . axial symmetry has been preserved in the hfb calculation as the nuclei chosen are not expected to develop triaxiality . on the other hand reflection symmetry is allowed to be broken in the solution of the hfb equation , to allow for octupole deformation in the ground state which is relevant for a few of the actinides considered . the calculation of the 84 ground states for a set of parameters can be carried out in a powerful personal computer in around half an hour . as a consequence of this figure , an unconstrained and blind search of the four parameters leading to the absolute minimum of @xmath52 is a task out of the scope of the present exploratory considerations . fortunately , the expression for the binding energy depends linearly on three out of the four free parameters ( @xmath48 , @xmath49 and @xmath50 ) and the method suggested by bertsch et al . @xcite in those situation should work well . we have implemented bertsch s procedure for the three parameters @xmath48 , @xmath49 and @xmath50 and performed systematic calculations for as a function of the other parameter @xmath31 . usually , for a given @xmath31 value and with a reasonable choice of starting parameters , the linearized bertsch s method leads to a local minimum of @xmath51 in a couple of iterations . the reason is that the correlation matrix of bertsch s method has in our case two eigenvalues which are two to three orders of magnitude smaller than the remaining one . this fact points to the existence of only one free parameter ( a linear combination of @xmath48 , @xmath49 and @xmath50 with weights corresponding to the components of the eigenvector of the largest eigenvalue ) . at this point it has to be mentioned that the fact that the expression of the binding energy is linear in @xmath48 , @xmath49 and @xmath50 by no means imply the value of binding energy to be globally a linear function of those quantities . the reason is that the binding energy is obtained after a self - consistent hfb calculation and the self - consistency breaks the linear dependence of the wave function on the parameters mentioned above . as a consequence , bertsch method is only valid locally what implies that the procedure will end up in different minimum depending upon the values of the initial parameters . a careful analysis of the results obtained so far indicates that the spin - orbit strength is the most sensitive starting parameter and therefore a search on this degree of freedom should be carried out in addition to the search on @xmath31 . using this new protocol we have performed a fit using for the bulk density dependent part of the functional new interpolating polynomials for the nuclear matter equation of state in a wider range of density with respect to the previous ones ( bcp1 and bcp2 in refs . the new fit has still a binding energy per nucleon @xmath53 mev at saturation density @xmath54 fm @xmath55 ( for a compressibility of 220 mev ) . for the spin - orbit strength we choose to stay around the @xmath56 mev value and perform a minimization of @xmath51 for different values of @xmath31 from 0.85 fm up to 1.00 fm . the minimum value of @xmath51 was obtained for @xmath57 fm . the values of the other parameters are @xmath58 mev , @xmath59 mev and @xmath60 mev @xmath61 for a value of @xmath51 of 0.545 mev . in order to check for the suitability of the spin - orbit strength value used we performed another two minimizations of @xmath51 fixing @xmath31 at 0.9 fm but taking @xmath62 mev @xmath61 and 102.51 mev @xmath61 . the values of @xmath51 obtained are 0.569 mev and 0.566 mev respectively , supporting the assignment of @xmath63 mev @xmath61 as the value leading to the lowest @xmath51 ( this check is incomplete as a more extensive search in the two parameters should be performed ; work in this direction is in progress and will be reported in a near future ) . 16 mev , @xmath57 fm that yield @xmath64 mev are plotted as full ( blue ) dots , whereas results for @xmath6516 mev , @xmath66 fm that yield @xmath67 mev are plotted as open ( green ) circles . the group of points to the left corresponds to the rare earth isotopes whereas the one to the right are for the actinides and superheavy.[fig : sigmabcp09 ] ] in fig . [ fig : sigmabcp09 ] we show the individual differences in binding energy ( @xmath68 ) between the theoretical results and the experimental ones for the 84 nuclei considered . the quantity @xmath69 has been plotted as a function of the number of neutrons @xmath70 and the values corresponding to the same isotope are joined with lines . two sets of parameters have been considered , namely one with @xmath57 fm producing the lowest value of @xmath51 ( full dots ) and other with a slightly higher value @xmath67 mev and obtained fixing @xmath66 fm ( circles ) . it is worth pointing out that the absolute value of @xmath71 never exceeds 1.2 mev . we also observe that the change in parameters only produce minor changes in the @xmath71 s . finally , we notice that the actinides and superheavies look a little bit too underbound as compared to the rare earth isotopes . finding the origin of such a relative underbinding will help to improve the quality of the fit reducing the value of @xmath51 . although the value of @xmath51 was obtained by considering an optimal set of deformed nuclei ( those with a well established deformation ) it is very likely that the @xmath51 value for a complete set of nuclei from proton drip line to neutron drip line will be promising . in order to compare the results obtained with bcp 09 with others of other interactions we have performed calculations with the gogny force and the most recent parametrization d1 m that was made up with the idea of producing a mass table . unfortunately , the details of how the theoretical binding energy was obtained are not detailed enough as to allow a reproduction of them ( the main uncertainty is in the quadrupole motion zero point energy correction ) . we have decided to use the same protocol as with bcp 09 , that is , the rotational energy correction and the finite size of the basis effect are included , and we have left out the zero point energy correction mentioned above . as a consequence , our values for @xmath51 of d1 m are too high and in order to make a more fair comparison we just shifted all the binding energies computed by a constant quantity ( 2.8 mev ) obtained as to minimize @xmath51 . with this readjustment we obtain for d1 m and the 84 nuclei considered the value @xmath72 mev which is slightly better than our value but , on the other hand , of similar quality to the one of the bcp 09 functional . in fig . [ fig : sigmabcp09d1 m ] we have plotted the individual values of @xmath71 both for bcp 09 and gogny d1 m . we notice how the behavior of @xmath71 for the gogny d1 m as a function of @xmath70 for each isotopic chain is different from the one of bcp 09 ; decreasing with increasing @xmath70 in the case of gogny d1 m whereas it is increasing with increasing @xmath70 for bcp 09 . it is also worth to point out that gogny d1 m force produces results in the actinide and super - heavy nuclei which are more spread out than the ones of the bcp 09 functional . by comparing both results we can conclude that , at least for the 84 deformed nuclei considered , the performance of bcp 09 is as good as the one of gogny d1 m and at a fraction of the computational cost ( the absence of exchange terms and non local pairing in bcp makes the numerical evaluation of the quantities entering the hfb equation much faster than for the gogny d1 m case ) . this is an encouraging starting point for a more detailed study of the viability of bcp 09 in the description of low energy nuclear structure . but this time the bcp results with @xmath6516 mev , @xmath57 fm that yield @xmath64 mev are compared to those of the gogny d1 m force . see text for further details . [ fig : sigmabcp09d1 m ] ] next we want to explore the ability of our proposed bcp09 energy density functional to describe nuclear ground - state properties of other nuclei not included in the fitting protocol used to obtain the free parameters that describe the surface and spin - orbit contributions to the functional . to this end we have computed the ground - state energies of 161 even - even and 306 odd spherical nuclei . these nuclei are chosen to be spherical according to the deformation properties of the compilation of mller and nix @xcite . to deal with odd nuclei , we have used the blocking approach on top of the bcs calculation . the differences @xmath73 , where @xmath74 are the theoretical predictions of the bcp09 functional and @xmath75 the experimental values taken from @xcite , are displayed in fig . [ fig : diffbsph ] . the agreement found between the theoretical prediction and the experimental values is fairly good finding an energy rms @xmath76 1.3 mev . it should be mention that both , even - even and odd nuclei , give basically the same rms when they are considered separately . from the figure we can see that the theoretical ground - state energies are scattered , rather symmetrically , around @xmath77 within a window of about @xmath78 3 mev with the only exception of the nucleus @xmath79er . combining these values for 467 spherical nuclei with the results obtained for the 84 deformed nuclei used in the fitting protocol , we find a global rms @xmath801.2 mev , a value that can be improved if the analysis of the ground - state energies include more deformed nuclei . in fig . [ fig : diffrsph ] we show the differences between the calculated and experimental charge radii @xmath81 . in this figure we have displayed the differences @xmath82 for 88 even - even and 111 odd spherical nuclei for which the experimental charge radii are known @xcite . from this figure we see that the bcp09 energy density functional predicts , on the average , smaller charge radii than the experimental values . with few exceptions , the theoretical charge radii are scattered within a window of 0.04 fm around an average values @xmath82 = -0.02 fm . the global quality of this fit of the charge radii is given by rms @xmath83=0.030 fm and again the rms of the even - even and odd nuclei are basically the same . this global rms can be compared with one provided by the hfb-8 model @xcite which is @xmath83=0.0275 fm , however computed using 782 experimental data . as compared with the results provided by the earlier bcp2 functional , that included some experimental charge radii in the fit , the @xmath83 values are roughly the same , but in the case of the bcp2 functional the theoretical charge radii are rather overestimated , at least for light nuclei . after having defined a performing functional in the realm of nuclear masses one has to check its suitability in describing other nuclear properties like deformations . a detailed study of deformations with bcp 09 is underway and will be reported in the future , but in order to give a taste of the results already obtained we show in fig . [ fig : mfprop240pu ] the typical outcome of a fission barrier calculation for the nucleus @xmath84pu that has also been discussed in ref . @xcite for early versions of the bcp functional . in fission studies the energy of the system obtained in the hfb framework is obtained as a function of the quadrupole moment by constraining the hfb solution . in this way a potential energy curve ( pec ) is obtained that in the present case is displayed in the lower panel of fig . [ fig : mfprop240pu ] for the bcp1 and bcp09 functionals and the gogny d1s force . the bcp1 and bcp09 curves have been shifted by 0.65 mev and 4.5 mev respectively as to make the first minimum at @xmath85 b ( @xmath86 0.26 ) coincide in energy . after the first ( ground state ) minimum there is a barrier ( first fission barrier ) that is higher for d1s than for the bcp functionals . the height of this barrier is known to be very sensitive to triaxial effects which are still not accounted for in the present calculation . after the first fission barrier there is an additional excited local minimum at @xmath87 b ( @xmath860.73 ) that corresponds to the fission isomer . the excitation energy of the fission isomer obtained with the two bcp functionals is more or less the same with a value of 3.5 mev . for the gogny d1s the result is slightly higher ( around 4.5 mev ) . the following maximum corresponds to the second fission barrier which is definitely much higher for gogny d1s than for the bcp functional . after the second fission barrier the typical decrease of the energy consequence of the coulomb repulsion between the nascent fragments ( the real shape of the nucleus is depicted at the corresponding @xmath88 value as a contour plot of the real density at a value of the contour of 0.08 @xmath42 ) . as the two fragments are still connected by the neck the nuclear part of the interaction still has an effect , explaining the differences observed between the three calculations . finally , in the upper panel of fig . [ fig : mfprop240pu ] the octupole ( @xmath89 ) and hexadecapole ( @xmath90 ) moments of the mass distribution obtained with the three functionals / interactions are shown . the values lie on top of each others indicating ( as the multipole moments considered are the most relevant ) that the shapes of the nucleus along the whole fission path are essentially the same for the three calculations . for the two bcp functionals and gogny force used in the calculations is shown . in the upper panel , the octupole and hexadecapole moments are displayed again for the three interactions / force used . the lines lie on top of each other and are hardly distinguisable . see text for further details . [ fig : mfprop240pu ] ] the problems to be clarified in the edf approach can be in general classified in two categories . on one hand , the use of the nuclear matter eos must be taken with care since the microscopic calculations can not reach the accuracy usually required in nuclear structure calculations and contains intrinsic and unavoidable uncertainties discussed below . on the other hand , finite size effects must be included in the functional with a minimal set of parameters . the procedure of doing that is not straightforward and requires extensive analysis . following the ks scheme , the nuclear eos calculated microscopically should be included in the edf mainly by an analytic fit that reproduces accurately both the nuclear matter and neutron matter results . in this way the saturation point , as well the whole density dependence of the binding energy , can be incorporated in the edf . however , the microscopic calculations can not be considered without any uncertainty . besides the point of numerical accuracy , it is well known that it is necessary to introduce three - body forces in the microscopic calculations in order to reproduce the correct saturation point , which however contains necessarily a phenomenological uncertainty . therefore the tbf can be adjusted to reproduce different saturation points . furthermore the microscopic data do not fix uniquely the fitted eos . we found that the quality of the nuclear data fitting is very sensitive to the saturation point of the interpolated eos . the optimal value of the energy per particle at saturation seems to be close to -16 mev , but even a shift as small as 0.1 mev of this quantity can deteriorate appreciably the quality of the nuclear data fit . this should be a general feature , independent of the particular edf that is used . less sensitivity seems to be present for the saturation density . for the eos obtained from the bhf theory , the tbf contribution below saturation is quite small and the bhf is expected to be accurate at low density , so that the main problem can be considered as how to fix the saturation point . of course the just discussed sensitivity on the saturation point could be considered a not serious problem , in view of the phenomenological uncertainty that affects its position . all these problems appear absent for the neutron matter eos , for which the tbf contribution is smaller and correlation energy is substantially reduced . a separate discussion is needed for the type and strength of the tbf that are used in microscopic nuclear matter calculations . the numerical calculations of the binding energy of nuclear few - body systems , like triton and alpha particle , can be performed numerically very accurately . if realistic two - body forces , like the argonne v@xmath91 potential , are used the binding energies turn out systematically underestimated . to remedy to this drawback usually one adds a semi - phenomenological tbf , like the urbana model @xcite , that is adjusted to reproduce the experimental binding energies . the unpleasant discovery is that the use of the same tbf in microscopic calculations for nuclear matter does not produce a good saturation point . the tbf appear too repulsive . indeed , in general semi - phenomenological tbf s contain a repulsive and attractive part , and the standard procedure is to reduce the repulsive component in order to get a good saturation point . at phenomenological level this means that one needs at least a four - body force in a non - relativistic many - body scheme . it has to be mentioned in this respect that non - local two - body forces @xcite which are able to reproduce the binding energy of few - nucleons systems fail to reproduce the correct saturation point , that then turns out to be at too high density and too low energy @xcite . recently @xcite it has been shown that one can get a reasonable saturation point by keeping the tbf obtained by fitting few - body systems if for the effective two - body force the so - called v@xmath92 interaction is used . the latter is obtained by projecting out the high momenta of the bare nn interaction and renormalizing accordingly the interaction at low momenta . the resulting interaction is then phase - equivalent to the original bare nn interaction up to the momentum cutoff , but it is much softer , and therefore it can be treated perturbatively . nuclear matter calculations with v@xmath92 are not saturating , most probably because they misses the effects of the pauli principle and of the self - consistent single particle potential , which are distint features of the g - matrix . the final saturation point is a consequence of the compensation between the too large attraction of v@xmath92 and the too large repulsion of the tbf . all these uncertainties are of course embodied in the final eos and affect the detailed properties of the resulting edf . unfortunately there is a more basic problem for the low density eos . at low enough density symmetric matter is expected to be unstable towards cluster formation . the picture of an homogeneous matter can not be kept at very low density . the real eos is therefore altered by the appearance of clusters of different sizes . however the eos which includes cluster formation can not be used in the edf , since such a proliferation of light nuclei can not be present in the low density region of finite nuclei . this is a general problem for any edf . the absence of clusters at the nuclear surface is due to the small value of the diffusion of the density profile , which prevents any long range correlation to dominate . this fact offers a partial justification of taking the homogeneous matter eos at the bhf level . indeed , short range correlations are well taken into account by the bhf procedure , and therefore the corresponding eos should be able to describe the properties of the bulk contribution to the nuclear edf even in the surface region . this problem still needs further analysis before it can be considered satisfactorily solved . in the standard ks procedure the kinetic energy contribution is taken at the independent particle level with an effective mass equal to the bare one . this is in agreement with the bhf scheme , where the kinetic energy part is also kept at the free value and the whole correlation contribution is included in the interaction energy part coming from the g - matrix . this means that also the correlated part of the kinetic energy is included in the g - matrix contribution . indeed the momentum dependence of the g - matrix is clearly the origin of the deviation of the effective mass value from the bare one . it would not be difficult to separate this effect from the g - matrix contribution , so that the kinetic energy would include a ( density dependent ) effective mass , while leaving the rest of correlations as a genuine interaction energy . this leaves some freedom to the way the nuclear matter eos is actually included in the edf , which allows to go beyond the usual ks scheme . it can be expected that different procedures are not necessarily equivalent . this problem has still to be analyzed in detail . the density matrix expansion of negele and vautherin @xcite suggests that the surface contribution to the binding energy of nuclei may also be extracted from nuclear matter calculations . this would be very welcome , since this would further reduce the number of adjustable parameters and link the functional even more to the microscopic approach . essentially only the spin - orbit and pairing contributions would then need some phenomenological adjustments . however , the g - matrix also has spin - spin interaction terms and , in principle , the spin orbit also could be extracted if the vector part of the one - body density matrix is properly taken into account @xcite . to what extent such a goal can be achieved remains an open problem . let us , however , point to some possible improvements over the past procedures . in first place we want to point out that it is our believe that neither the dme of negele- vautherin , nor the one of campi - bouissy @xcite are the optimal procedures . we have long standing experience with the semiclassical dme of wigner and kirkwood which is based on a systematic expansion in powers of @xmath93 . most accurate results are obtained , at least for the scalar part of the one- body density matrix , with this method @xcite . a second problem stems from the way how the g - matrix of an infinite matter calculation is used for a finite nucleus . the standard procedure is the lda , that is one replaces the @xmath39 dependence by a density dependence via the standard relation @xmath94 . the @xmath39 dependence enters mostly into the pauli operator @xmath95 . however , dependence also can be in the self consistent single particle energies . we want to point out that just replacing @xmath39 by @xmath96 is not necessarily the best procedure . our argument goes as follows . the two particle propagator entering the g - matrix equation @xmath97 [ e - \hat h_1 - \hat h_2 -i\eta]^{-1}\ ] ] has an obvious semiclassical , i.e. @xmath98 limit , namely we just replace the single particle hamiltonians @xmath99 by their classical counterparts @xmath100 with @xmath101 the phase space variables of position and momenta . with this approximation the two particles move according to their respective classical hamiltonians , each . supposing for simplicity a local mean field @xmath102 ( the argument can be generalized to nonlocal mean field potentials ) , we see that in the semiclassical two body propagator two local chemical potentials appear : @xmath103 and @xmath104 ( @xmath105 ) and , therefore , two fermi momenta and , thus , two densities at positions @xmath106 appear in the semiclassical propagator and correspondingly in the g - matrix . since the effective range of the interaction is over 2 fm wide , this non locality in the densities may be of quite some importance . one may call this the non local density approximation ( nlda ) . the standard lda is recovered in putting @xmath107 . this nlda effect may be more important , or at least of equal importance , than to keep gradient expansions of the density itself . it could be interesting to test this . since finite size effects are surface effects , one may argue that only relatively low densities will be involved . it has recently been shown , that at least in neutron matter for low densities a separable force works very well for the g - matrix @xcite , reducing very much the numerical effort . so eventually one could combine the nlda approach with a separable force to get to a surface term . of course , one will not imagine that with such a procedure one will hit the good result right on the spot . but even if the result is only semi quantitatively correct , it still may lead to a reduction of the open parameters and to more insight into the underlying physics . this may be then a first step in the direction to get everything from the microscopic input , i.e. from the underlying bare forces , also for finite nuclei . we have presented a formulation of the energy functional method that enables to keep , to a certain extent , the connection with the nuclear matter equation of state as calculated microscopically from many - body theory and realistic nucleon - nucleon interactions . the functional is constructed following closely the kohn and sham method , where the bulk part of the functional is taken directly from the microscopic calculations and kept fixed in the fitting procedure of the adjustable parameters of the functional . the latter include the surface part of the functional and the spin - orbit strength , ending up with a total of four free parameters . the pairing interaction and the corresponding strength has been taken from the simplest standard choices . despite the analysis was not carried out up to the optimal level , leaving space for refinements , the accuracy of the results can compete with the one obtained with the best purely phenomenological functional like gogny . these promising results open the possibility of building an energy density functional closely connected with the bare nucleon - nucleon forces . developments in this direction could be obtained along the density matrix expansion method of ref . @xcite or its refinements , which could further reduce the number of adjustable parameters . however , we pointed out the various open problems that must be solved before this project can be completed . the incorporation of the nuclear eos in the density functional can be done in different ways , and this leaves some ambiguity in the direction of a universal nuclear functional . furthermore the very low density part of the eos can not be taken literarily from microscopic calculations of homogeneous nuclear matter . in fact cluster formation is expected to dominate this region of the eos . this appears a serious problem for any density functional theory which is constructed in such a way to be compatible with the nuclear matter case . further studies and analysis are needed to proceed further in this project on the nuclear density functional . work supported in part by micinn ( spain ) grants fpa2007 - 66069 and fpa2008 - 03865-e / in2p3 , and by the consolider - ingenio 2010 program ( spain ) cpan , csd2007 - 00042 . x. v. also acknowledges the support from fis2008 - 01661 ( spain and feder ) and 2009sgr-1289 ( spain ) . this work was supported by compstar , a research networking programme of the european science foundation .
|
in recent years impressive progress has been made in the development of highly accurate energy density functionals , which allow to treat medium - heavy nuclei . in this approach one
tries to describe not only the ground state but also the first relevant excited states . in general
, higher accuracy requires a larger set of parameters , which must be carefully chosen to avoid redundancy . following this line of development , it is unavoidable that the connection of the functional with the bare nucleon - nucleon interaction becomes more and more elusive . in principle
, the construction of a density functional from a density matrix expansion based on the effective nucleon - nucleon interaction is possible , and indeed the approach has been followed by few authors .
however , to what extent a density functional based on such a microscopic approach can reach the accuracy of the fully phenomenological ones remains an open question .
a related question is to establish which part of a functional can be actually derived by a microscopic approach and which part , on the contrary , must be left as purely phenomenological . in this paper
we discuss the main problems that are encountered when the microscopic approach is followed . to this purpose
we will use the method we have recently introduced to illustrate the different aspects of these problems .
in particular we will discuss the possible connection of the density functional with the nuclear matter equation of state and the distinct features of finite size effects proper of nuclei .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the dynamic of a small quantum system interacting weakly with uncontrollable degrees of freedom is well understood when a markovian approximation applies . in this situation , after tracing - out the environment , the system density matrix evolution can be well approximated by a lindblad equation @xcite . besides that the applicability of the markovian approximation range over many areas of physics @xcite there exist several real systems whose dynamics present strong departures from it . the main signature of this departure is the presence of strong non - exponential decay behaviors , such as power law and stretched exponential . some examples are nanocrystal quantum dots under laser radiation schlegel , brokmann , grigolini , superconducting qubits makhlinreport , makhlin , falci , spin environments @xcite , dephasing in atomic and molecular physics @xcite , electron transfer and exciton dynamics in proteins @xcite , and molecular systems maintained in a glassy environment @xcite , to name but a few . these and another specific experimental situations rise up the necessity of finding formalism and effective evolutions able to describe the corresponding non - markovian dynamics . when the environment is modeled as an infinite set of normal modes , departure from a markov approximation can be related to the corresponding spectral density function . this situation was extensively studied for the spin - boson and boson - boson models , where exacts solutions are available leggett , grifoni , lang , hanggi , milena . exact solutions can also be formulated for more general systems . nevertheless , due to the huge analytical and numerical efforts needed for getting the non - markovian system dynamics , alternative numerical methods based in a special decomposition of the spectral density function were formulated @xcite . anomalous system dynamics also arise from a random matrix modeling of the system - environment interaction @xcite . this approach naturally applies when describing environments characterized by a complex dynamics as for example chaotic ones . outside from a microscopic point of view , there exist an increasing interest in describing non - markovian effects in open quantum systems by introducing memory contributions in lindblad evolutions wilkie , budini , cresser , lidar . this procedure provide easy manageable dynamics . while most of these models are phenomenological , in this paper we will relate the presence of strong memory effects in the standard lindblad theory @xcite with the microscopic interaction of a system with a complex structured environment . we will base our considerations in a splitting of the full hilbert space of the bath in a direct sum of sub - reservoirs , constructed in such a way that each one guarantees the conditions for the applicability of a markov approximation . our motivation for formulating this splitting comes from systems embedded in glassy environments , where the underlying disorder produce localized bath states , inducing a natural shell structure of modes , each set having a different coupling strength with the system @xcite . thus , we associate a different markovian sub - bath to each set of states . as we will demonstrate , the splitting assumption allows us to generalize the usual born - markov approximation , which in a natural way leads to the formulation of lindblad equations characterized by a random dissipative rate . as is well known from a classical context , master equations with random rates @xcite are useful for describing strong non - markovian effects @xcite . here we will demonstrate that the same scheme can also be applied in a quantum context under the previous conditions , i.e. , a complex environment under the splitting condition . as an example , we will study the anomalous dissipative dynamics of a quantum tunneling system described in a two level approximation @xcite . strong non - exponential behaviors , such as stretched exponential and power law , arise from the interplay between the unitary hopping dynamics and the memory effects induced by the environment . the conditions under which our modeling can be mapped to the spin - boson model and stochastic dynamics are established . in this context , the differences between our framework and other approaches @xcite introduced to deal with non - markovian environments are established . in general , the evolution of a system interacting with a complex environment can not be described in a markovian approximation . while a general treatment is clearly not possible , when the splitting condition applies , for weak system - bath interactions , the reduced dynamics can be described through a generalization of the born - markov approximation . we start by assuming a full microscopic hamiltonian description@xmath0where @xmath1 correspond to the hamiltonian of a system s , and @xmath2 correspond to the hamiltonian of the bath b. the term @xmath3 describes their mutual interaction , with the operators @xmath4 and @xmath5 acting on the system and bath hilbert spaces respectively . in an interaction representation with respect to @xmath6 , the total density matrix @xmath7 evolves as@xmath8,\]]where @xmath9 is the interaction hamiltonian in the heisenberg representation . integrating formally this equation , and substituting the solution in it , the evolution of the reduced system density matrix @xmath10 can be written as@xmath11\ } , \label{exact}\]]where , as usual , a first order contribution was discarded after assuming @xmath12=0 $ ] . from this evolution , the well known born - markov approximation can be deduced blum , cohen , carmichael , weiss . the born approximation consists into assume , at all times , an uncorrelated structure for the total density matrix@xmath13where @xmath14 define a stationary state of the bath . this assumption is consistent up to second order in the interaction hamiltonian . when the decay of the bath correlation defines the small time - scale of the problem , after introducing eq . ( [ born ] ) in eq . ( [ exact ] ) , the markovian approximation leads to a closed local in time density matrix evolution . we remark that the born - markov approximation does not rely in any specific model of bath dynamics @xcite , such as an infinite set of harmonic oscillators . in fact , its master equation is independent of model assumptions used in its derivation @xcite . now we consider a complex environment for which the previous approximations are not valid . as is usual when dealing with complex environments kuznezov , dcohen , gaspard , kottos , instead of defining the bath hamiltonian @xmath15 as an infinite set of normal modes , here we specify it through its eigenstates basis @xmath16 , which in a weak interaction limit , results unmodified by the interaction with the system . as a central hypothesis , we will assume that , while the full action of the environment can not be described in a markov approximation , it is possible to split the full hilbert space of the bath as a direct sum of subspaces , in such a way that each one defines a sub - reservoir able to induce by itself a markovian system evolution @xcite . these hypotheses are the main assumptions that allow us to formulate our results . in conformity with the splitting condition , we write the interaction hamiltonian as a direct sum of sub - hamiltonians@xmath17where @xmath18 here , each operator @xmath19 defines the interaction between the system and each sub - reservoir @xmath20 in order to describe the joint action of all sub - reservoirs over the system , instead of the uncorrelated form eq . ( [ born ] ) , we introduce the _ _ generalized born - approximation__@xmath21where @xmath22=1,$ ] and we have defined@xmath23@xmath24 is the base of eigenvectors that span the subspace corresponding to each sub - reservoir . therefore , each contribution in eq . ( [ rhotot ] ) consists in the external product between a system state @xmath25 and the projection of the stationary bath state @xmath14 over each subspace @xmath26 . in physical terms , each state @xmath27 takes in account the dissipative effects induced by each sub - reservoir . after introducing eq . ( [ rhotot ] ) in eq . ( [ exact ] ) , we get the approximated evolution@xmath28\ } , \notag \end{aligned}\]]where @xmath29 means a trace operation with the states @xmath24 corresponding to each subspace . furthermore , we have introduced the sub - bath density states @xmath30 where@xmath31the normalization condition @xmath32=1 $ ] implies the relation @xmath33 . thus , the set @xmath34 can be seen as a set of probabilities defined by the weight of each subspace in the total stationary bath state . from eq . ( [ rhotot ] ) we can write@xmath35\approx \sum_{r}p_{r}\rho _ { r}(t ) . \label{sum}\]]then , the evolution eq . ( [ borngeneral ] ) is in fact a linear combination of the evolutions corresponding to the set @xmath36 , each one participating with weight @xmath37 . after introducing the _ markovian approximation _ @xcite to the evolution of each state @xmath25 , in a schrdinger representation , we get@xmath38-\left ( \frac{1}{\hbar } \right ) ^{2}\int_{0}^{\infty } dt^{\prime } \label{markov } \\ & & \mathrm{tr}_{b_{r}}\{[h_{i_{r}},[h_{i_{r}}(-t^{\prime } ) , \rho _ { r}(t)\otimes \rho _ { b_{r } } ] ] . \notag \end{aligned}\]]this evolution correspond to the usual born - markov approximation when considering a bath consisting only of the subset of states @xmath39 and characterized by the stationary state @xmath40 . the system density matrix is defined by the linear combination eq . ( [ sum ] ) . the evolution eq . ( [ markov ] ) , disregarding transients of the order of the sub - bath hamiltonian correlation time , can be always well approximated by a lindblad equation @xcite@xmath41+\gamma _ { r}\mathcal{l}% [ \rho _ { r}(t ) ] , \label{lindblad}\]]where @xmath42=(-i/\hbar ) [ h_{s},\bullet ] $ ] is the system liouville superoperator and the lindblad superoperator is defined by@xmath43=\sum_{\alpha } \frac{1}{2}([v_{\alpha } , \bullet v_{\alpha } ^{\dagger } ] + [ v_{\alpha } \bullet , v_{\alpha } ^{\dagger } ] ) .\]]as the underlying microscopic interaction between the system and the environment is the same in each subspace , the set of operators @xmath44 does not depend on index @xmath26 . nevertheless , each subspace has associated a different characteristic dissipative rate @xmath45 . as this rate arises from the interaction of the system with the manifold of states @xmath24 , consistently with the fermi golden rule @xcite , it is proportional to the characteristic interaction strength of each subspace , denoted as @xmath46 multiplied by the corresponding density of states @xmath47 evaluated in a characteristic frequency @xmath48 of the system , i.e. , @xmath49 . correspondingly , the system state follows from the average@xmath50random rate equations were extensively used to model classical anomalous diffusion processes in disordered media klafter , bernasconi , odagaki , miguel . here , we have derived a similar structure for a different physical situation , i.e. , quantum systems embedded in a complex structured environment . as in a classical context , while the density matrixes @xmath25 follows a markovian evolution , the average state @xmath51 evolve with a non - markovian evolution . this evolution can be easily obtained in a laplace domain , where the average eq . ( [ average ] ) takes the form@xmath52with @xmath53 being the laplace variable , and we have used the solutions @xmath54\rho _ { s}(0).$ ] consistently with the uncorrelated initial condition @xmath55 , there not exist any statistical correlation between @xmath56 and the random rate set . thus , the average evolution can be obtained without appealing to a projector technique klafter , bernasconi , odagaki , miguel . in fact , after introducing in eq . ( rholaplace ) the identity in the form @xmath57\rangle ^{-1}$ ] , it is immediate to get the deterministic , closed , non - markovian evolution equation@xmath58+\int_{0}^{t}d\tau \,% \mathbb{l}(t-\tau ) [ \rho _ { s}(\tau ) ] , \label{nomarkov}\]]where the superoperator @xmath59 is defined in the laplace domain by the equation@xmath60=\langle g_{r}(u)\rangle \mathbb{l}(u)[\bullet ] . \label{memory}\]]depending on the set @xmath61 , eq . ( [ nomarkov ] ) may lead to the presence of strong non - markovian decay behaviors in the system dynamics . this characteristic originates from the entanglement of the system which each sub - reservoir , situation explicitly introduced by eq . ( rhotot ) . an example of complex structured environment where the generalized born - markov applies straightforwardly is a bath hamiltonian whose eigenvectors can be labelled with two indexes @xmath62 . the index @xmath63 is continuous , and for each @xmath26 the corresponding sub - manifold of states is able to induce a different system markovian - decay . the difference between the markovian dynamics may originates in the coupling strength of each sub - manifold with the system . on the other hand , it may originates due to strong variations of the bath density of states with index @xmath26 . the system dynamics follows as a superposition of markovian dynamics whose weights are taken in account through the generalized born approximation eq . ( [ rhotot ] ) . if the decay induced by each sub - manifold is the same , the generalized born approximation reduces to the usual one , and then a markovian evolution is obtained . further examples can be established in the context of random band - matrix models @xcite , where the markovian sub - baths , for example , may be associated to subspaces with a different characteristic bandwidth . classical master equations with random rates are characterized by equations similar to those obtained previously . nevertheless , as in general the underlying numbers of states is infinite , some kind of approximation is necessary in order to obtain the operator @xmath59 , as for example an effective medium approximation @xcite . here , we introduce a similar approximation in order get a general characterization of the dynamics . thus , in eq . ( [ memory ] ) we discard the dependence introduced by the lindblad superoperator @xmath64 in the propagator @xmath65 , i.e. , @xmath66 . then , we get the approximated solution @xmath67 from where it follows the evolution@xmath68+\int_{0}^{t}d\tau k(t-\tau ) e^{(t-\tau ) \mathcal{l}_{h}}\mathcal{l}% [ \rho _ { s}(\tau ) ] . \label{aproximada}\]]in this approximation all information about the random rate is introduced through the kernel function@xmath69as in a classical context , this kernel can be associated with a waiting time distribution @xmath70 and a survival probability @xmath71 defined by@xmath72 in classical master equations , these objects define a continuous time random walk @xcite . in the quantum case , a similar stochastic dynamics can be constructed @xcite . it consists in the application at random times of the superoperator @xmath73 , implying the transformation @xmath74 $ ] , while during the intervals between these disruptive actions the system evolves with its unitary dynamics , @xmath75 $ ] . the intervals between the successive applications of @xmath76 follows from the waiting time distribution @xmath70 . the function @xmath77 defines the corresponding survival probability , @xmath78 . thus , the average over different realizations of the random times can be written as@xmath79 . \label{estocastica } \end{aligned}\]]from here , in a laplace domain , it is straightforward to recuperate the evolution eq . ( [ aproximada ] ) . when @xmath80i , with @xmath81 a completely positive superoperator @xcite , a similar stochastic dynamics can be formulated after introducing a limit procedure @xcite . we remark that the stochastic interpretation [ eq . ( [ estocastica ] ) ] was constructed after associating to the kernel @xmath82 a waiting time distribution and a survival probability , eq . ( [ survival ] ) . this association does not rely in the generalized born - markov approximation , neither it was deduced from a conditional continuous time measurement theory @xcite . therefore , it is not clear if one can associate to the stochastic dynamics a random signal of a measurement apparatus . if this is the case , contradictions between environmental decoherence and wave - function collapse may arise @xcite . as an example of our formalism , in this section we will characterize the dissipative dynamics of a quantum tunneling system described in a two level approximation @xcite and driven by a complex environment . then , the system hamiltonian can be written as @xmath83 the first term , proportional to the @xmath84-pauli matrix @xmath85 define the energy of the effective levels , and the second one , proportional to the @xmath86-pauli matrix @xmath87 , introduce the reversible hopping between the two effective states . the complex environment will be represented by the lindblad superoperator@xmath43=\frac{1}{2}([\sigma _ { z}\bullet , \sigma _ { z}]+[\sigma _ { z},\bullet \sigma _ { z } ] ) , \label{lindblad}\]]and an arbitrary set @xmath88 of random rates and weights . for fixed rate , this superoperator induces a dynamics equivalent to a thermal environment in a high temperature limit @xcite . the evolution of the system density matrix is defined by eqs . ( [ nomarkov ] ) and ( [ memory ] ) . here , we write the evolution in terms of the components of the bloch vector , which are defined by the mean value of the pauli matrixes , @xmath89 , with @xmath90 @xmath91 and @xmath84 . we get [ bloch ] @xmath92thus , the system evolution is completely characterized by three memory kernels @xmath93 , @xmath94 , and @xmath95 . in appendix a , we give the exact expressions of these kernels for arbitrary random rates , joint with the kernels that arise from the effective approximation eq . ( [ aproximada ] ) . from eq . ( [ bloch ] ) it is straightforward to write the system density matrix evolution [ eq . ( [ nomarkov ] ) ] as a sum of lindblad superoperators , each one characterized by a different kernel . when the hopping frequency is zero , @xmath96 , the dynamics reduce to a dispersive one . thus , the coherences decay continuously while the population of each effective level remains constant . in this limit , from appendix a , for arbitrary set @xmath61 we get the exact kernels [ kerneldispersivo ] @xmath97 , \\ \gamma _ { y}(t ) & = & k(t)\cos [ \omega _ { a}t ] , \\ \upsilon ( t ) & = & k(t)\sin [ \omega _ { a}t ] . \end{aligned}\]]where @xmath82 is defined in the laplace domain by eq . ( [ k(u ) ] ) . we note that these kernels also arise from the effective approximation eq . ( aproximada ) , indicating that for @xmath96 , both evolutions coincide . from eqs . ( [ bloch ] ) , the exact solution of the bloch vector is given by [ disperso ] @xmath98s_{x}(0)-\sin [ \omega _ { a}t]s_{y}(0)\},\ \ \ \ \ \ \ \ \\ s_{y}(t ) & = & p_{0}(t)\{\sin [ \omega _ { a}t]s_{x}(0)+\cos [ \omega _ { a}t]s_{y}(0)\},\ \ \ \ \ \ \ \ \\ s_{z}(t ) & = & s_{z}(0 ) , \end{aligned}\]]where @xmath71 is the survival probability defined by its laplace transform eq . ( [ survival ] ) , which in the time domain reads @xmath99.$ ] consistently , we note that the exact solutions eqs . ( [ disperso ] ) correspond to an average over markovian solutions , each one characterized by a rate @xmath45 and participating with weight @xmath37 . depending on the distribution of the dissipation rate , arbitrary forms of the decay can be obtained from this average over exponential functions . hence the non - markovian behavior can be observed in the relaxation of the density matrix to the stationary state . the form of the set @xmath61 depends on the specific structure of the complex environment . here , we will determine this set in a phenomenological way as a function of the system decay behavior . we will be interested in obtaining anomalous decay dynamics such as _ power law_. a possible set consistent with this decay is @xmath100,\;\;\;\;\;\;p_{r}=(1-e^{-a})\exp [ -ar ] , \label{expo}\ ] ] where @xmath101,$ ] @xmath102 scale the random rates , and the constants @xmath103 and @xmath104 measure the exponential decay of the random rates and their corresponding weights . with these definitions , it is simple to demonstrate that after a transient of order @xmath102 , the waiting time distribution and its associated survival probability , eq . ( [ survival ] ) , present a power law decay behavior @xcite , @xmath105 , and @xmath106 where @xmath107 . clearly , this behavior is reflected in the system dynamics . , @xmath108 , @xmath109 , @xmath110 , @xmath111 , and the markovian limit @xmath112 . in all cases we take @xmath113.,height=272 ] when @xmath114 , the kernel @xmath82 corresponding to the set eq . ( expo ) can be well approximated by the expression@xmath115with the definitions@xmath116 the scaling of these parameters can be motivated by considering a two dimensional set of random rates @xcite . from eq . ( [ k(u ) ] ) and ( [ survival ] ) , the waiting time distribution and its associated survival probability can be obtained as @xmath117from here , is it is simple to proof that @xmath118 is a completely monotone function @xcite , which implies that @xmath71 decays in a monotonous way or equivalently , @xmath119 . in figure 1 we plot the survival probability @xmath71 by assuming the kernel eq . ( [ kernel ] ) for different values of @xmath120 . we note that in a short time regime , the decay is an exponential one , while in an asymptotic regime a power law behavior is present@xmath121,\ \ \ \ \ \ \ \ \ \ p_{0}(t)\simeq \frac{% \beta ^{1-\alpha } } { \gamma \gamma ( 1-\alpha ) } \frac{1}{t^{\alpha } } , \]]where @xmath122 is the gamma function . these asymptotic behaviors follows immediately from eq . ( [ sur ] ) . when the dispersion of the random rate @xmath123 is zero @xmath124 , consistently the dynamics reduce to a markovian one , @xmath125 , which implies the pure exponential decay @xmath126 $ ] and @xmath127 $ ] . in the next subsection we will characterize the tunneling dynamics by assuming a complex environment characterized by the random rate set eq . ( expo ) or equivalently by the kernel eq . ( [ kernel ] ) . here we will analyze the tunneling dynamics for a symmetric case @xmath128 , which arise when the two effective levels have the same energy . from appendix a , the exact kernels read [ kernelhoping ] @xmath129as before , the kernel @xmath130 is defined by eq . ( [ k(u ) ] ) . the exact solution of the bloch vector can be obtained in a laplace domain . we get [ hoping ] @xmath131s_{z}(0)+\delta s_{y}(0)\},\ \ \ \ \ \ \end{aligned}\]]where we have defined @xmath132 which can also be expressed as @xmath133 . in this case it is not possible to find in the time domain a general exact solution for arbitrary memory kernels . a simple analytical solution is only available in a markovian case [ @xmath125 ] [ markovian ] @xmath135 \notag \\ & & -\lambda ^{-1}[(\gamma /2)s_{y}(0)+\delta s_{z}(0)]\sinh [ \lambda t],\ \ \ \ \ \\ s_{z}(t ) & = & e^{-\gamma t/2}\{s_{z}(0)\cosh [ \lambda t ] \notag \\ & & + \lambda ^{-1}[(\gamma /2)s_{z}(0)+\delta s_{y}(0)]\sinh [ \lambda t],\ \ \ \ \ \end{aligned}\]]where @xmath136 , and @xmath137 defines the unique dissipative rate . notice that in the limit of null dissipation , a periodic hopping between the effective levels is obtained . considering as initial condition the upper eigenstate of @xmath138 . the envelopes are given by @xmath139 . from top to bottom , the parameters are @xmath140 , @xmath141 , and @xmath142 . in all cases we take @xmath143 @xmath144 , and @xmath145,height=257 ] for arbitrary random rates @xmath61 , the dynamics can be characterized in different regimes . first , in the case @xmath146 , from eqs . ( [ hoping ] ) it is possible to get the approximated solutions [ aproxhoping ] @xmath147s_{y}(0)-\sin [ \delta t]s_{z}(0)\},\ \ \ \ \ \ \ \\ s_{z}(t ) & \simeq & p_{0}(t/2)\{\sin [ \delta t]s_{y}(0)+\cos [ \delta t]s_{z}(0)\}.\ \ \ \ \ \ \ \end{aligned}\]]thus , the dynamics consist in a periodic tunneling between the two effective states , and whose decay can be written in terms of the survival probability . as in the previous case , this solution correspond to an average over the corresponding markovian solutions , i.e. , eq . ( [ markovian ] ) written in the limit of small decay rate when compared to the tunneling frequency @xmath148 . in figure 2 we plot the average of the @xmath84-pauli matrix which follows from eq . ( [ hoping ] ) with the kernel eq . ( [ kernel ] ) . as initial condition we take the upper eigenstate of @xmath85 . we verified that the exact solutions are well described by the approximation eq . ( [ aproxhoping ] ) for parameters values satisfying @xmath149 . as the envelope decay is given by @xmath150 , by increasing the average rate @xmath151 , the dynamics decay in a faster way . this dependence is broken when the average rate is much greater than the hopping frequency . in the limit @xmath152 , the dissipative dynamics dominates over the tunneling one . in figure 3 we plot @xmath153 [ eq . ( [ hoping ] ) ] for different values of the characteristic parameters of the kernel eq . ( [ kernel ] ) . we note that by increasing the average rate @xmath137 , a slower decay is obtained . thus , the dynamics develops a zeno - like effect @xcite . from the exact solution eq . ( hoping ) , the characteristic decay of the bloch vector can be approximated by the expressions @xmath154 where we have introduced @xmath155 and the function @xmath156^{-1}\simeq u^{-1}p_{0}(\delta ^{2}/u)$ ] . for the kernel defined by eq . ( [ kernel ] ) , the characteristic decay @xmath157 results @xmath158 as can be seen in figure 3 ( dotted line ) , besides the oscillatory behavior , after the transient @xmath159 , this function provides an excellent fitting of the decay dynamics . considering as initial condition the upper eigenstate of @xmath138 . the fitting decay curves ( dotted lines ) are given by eq . ( [ zenodecay ] ) . from top to bottom , the parameters are @xmath160 , @xmath161 , @xmath162 , @xmath163 , @xmath164 , and @xmath165 . in all cases we take @xmath143 @xmath166 and @xmath145 in the inset we show the same graphic in a log - log scale.,height=279 ] the function @xmath167 is characterized by a reach variety of behaviors . first , we note that in the markovian limit , @xmath168 , we get an exponential decay with rate @xmath169 , which clearly diminish by increasing @xmath137 . in the non - markovian case , in a short time regime , we can approximate@xmath170\},\]]while in an asymptotic long time limit we get@xmath171thus , the dispersion of the random rate ( measured by @xmath172 ) induce , at short times , an extra stretched exponential decay , while in the asymptotic regime it scales a power law behavior [ @xmath173 . the characteristic rates of both regimes arise from a competence between the unitary and dissipative dynamics . we notice that by increasing the dispersion rate @xmath172 , the characteristic rate of the stretched exponential decay is increased , while the rate for the power law regime is decreased . the dependence in the hopping frequency @xmath174 is the inverse one . the zeno - like effect can be _ qualitatively _ understood in terms of the stochastic evolution corresponding to the effective approximation eq . ( [ aproximada ] ) . this stochastic process develops in the system hilbert space and consists in the application at random times of the superoperator @xmath73 , which in view of eq . ( [ lindblad ] ) reads @xmath175=\sigma _ { z}\bullet \sigma _ { z}$ ] , while in the intermediates times the system evolves with its unitary evolution @xmath176.$ ] the superoperator @xmath81 implies the disruptive transformations @xmath177 , @xmath178 , @xmath179 , while the unitary dynamics is equivalent to a rotation around the @xmath180-direction . in the limit of vanishing hopping frequency @xmath174 , the continuous applications of the superoperator @xmath76 kill the @xmath180-@xmath181 components and frozen the dynamics in the initial condition @xmath182 . thus , a pure zeno effect is recuperated . for @xmath183 the decay dynamics is determined from the competence between the transformations induced by @xmath81 and @xmath184 , defining the zeno - like regime . this interpretation is exact in a markovian limit and always valid for the effective master equation eq . ( aproximada ) . in obtaining the previous results we have assumed an infinite set of random rates , eq . ( [ expo ] ) , whose effects can be approximated by the kernel eq . ( [ kernel ] ) . while this election guarantees the presence of an asymptotic power law decay , strong non - exponential behaviors can be obtained in an _ intermediate regime _ by considering only a finite set , @xmath185 , of random rates @xmath61 . on the other hand , for a finite set , the asymptotic system dynamics is always markovian and characterized by the inverse rate @xmath186 . this result follows from @xmath187 in figure 4 we show the decay dynamics induced by an environment characterized by a finite set of random rates @xmath45 ( @xmath188 ) with equal weights , @xmath189 . each curve follows from a superposition of markovian solutions , eq . ( [ markovian ] ) with @xmath190 . the set of rates @xmath191 of each plot differ in a multiplicative factor , in such a way that the relation between the average rate @xmath192 and the corresponding fluctuation rate @xmath193/\langle \gamma _ { r}\rangle $ ] remains constant in all curves . for the case @xmath194 the random rates are @xmath195 , @xmath142 , @xmath196 , @xmath197 , @xmath198 , @xmath199 and @xmath200 . considering as initial condition the upper eigenstate of @xmath201 and a finite set of random rates with equal weights . from top to bottom , the parameters are @xmath202 , @xmath163 , @xmath203 , @xmath204 , @xmath205 , and @xmath206 . in all cases we take @xmath207 , @xmath208 and @xmath145 for @xmath209 the dotted lines correspond to the fitting @xmath210 $ ] , while for @xmath211 they corresponds to @xmath212^{-\protect\delta } $ ] ( see text).,height=264 ] in order to enlighten the intermediate non - exponential regime , we have plotted the shifted average @xmath213 , with @xmath214 . in the deep zeno - like regime [ @xmath215 , @xmath216 can be well approximated by an stretched exponential behavior @xmath217 $ ] , with @xmath218 and @xmath219 . for @xmath220 , a power law fitting is more adequate @xmath221^{-\delta } , $ ] with @xmath222 and @xmath223 . we note that a similar non - exponential fitting was found in ref . @xcite by considering the action of a finite bath , which can be associated with a glassy environment . on the other hand , the oscillatory effects in the decay of @xmath216 arise from the markovian solutions , eq . ( markovian ) , corresponding to the rates satisfying @xmath224 . in fact , for the markovian solution , this condition delimits the change between a monotonous and an oscillatory decay behavior . consistently , we notice that by increasing the average rate , the amplitude of the oscillations are smaller . a similar effect can be seen in figure 3 . our formalism relies on the applicability of the generalized born - markov approximation . here we explore the possibility of mapping its dynamics with other models that also induce anomalous decay behaviors . _ spin - boson model _ : the spin - boson model is defined by the total hamiltonian @xmath225where the bath hamiltonian @xmath226 $ ] corresponds to a set of harmonic oscillators , and @xmath227 . the bath is characterized by the spectral density function@xmath228and assumed to be in equilibrium at temperature @xmath229 . as is well known , the reduced system dynamics can be obtained in an exact way leggett , grifoni , lang , hanggi , milena . it reads [ spin - boson ] @xmath230\ \ \ \ \ \ \ \\ & & + y_{b}^{(s)}(t)s_{x}(0)+y_{b}^{(a)}(t)s_{y}(0 ) , \notag \\ s_{y}(t ) & = & \frac{1}{\delta } \frac{ds_{z}(t)}{dt } , \\ \frac{ds_{z}(t)}{dt } & = & \int_{0}^{t}d\tau \lbrack k_{a}^{(a)}(t-\tau ) -k_{a}^{(s)}(t-\tau ) s_{z}(\tau ) ] \ \ \ \ \ \ \ \notag \\ & & + k_{b}^{(a)}(t)s_{x}(0)+k_{b}^{(s)}(t)s_{y}(0)\ } , \end{aligned}\]]where the corresponding kernels can be written as functions of @xmath231 . on the other hand , it is possible to write the exact averaged evolution eq . ( [ bloch ] ) in the form eqs . ( [ spin - boson ] ) . in appendix b we present the kernels corresponding to each dynamics . from these expressions , it is simple to proof that to first order in @xmath174 , after disregarding a phase factor , both set of kernels can be mapped under the condition @xmath232=\exp [ -q^{\prime } ( t ) ] , \label{map}\ ] ] where@xmath233,\]]define the real part of the thermal bath correlation . we remark that the mapping eq . ( [ map ] ) is only valid in a high temperature limit , condition consistent with the lindblad structure eq . ( [ lindblad ] ) . in this context , from eq . ( [ map ] ) , it is possible to enlighten the difference between the present approach and that developed in refs . tannor , ulrich . in our approach , which relies in splitting the hilbert space of the bath as a _ direct sum _ of subspaces , @xmath234 $ ] is written as a sum of exponential functions , each one associated to each markovian sub - reservoir . instead , in refs . @xcite , @xmath235 is expressed as a sum of exponential functions . this representation relies in an artificial discomposing of the spectral density function @xmath231 as a sum of individuals terms . thus , the hilbert space of the bath is effectively split in an _ external product _ of subspaces , each one associated to a non - markovian sub - reservoir . as in our approach , the system density matrix can be written in terms of a set of auxiliary sub - density - matrixes . nevertheless , their evolution involves coupling among them all . _ stochastic hamiltonian _ : decoherence in small quantum systems is also modeled by introducing stochastic elements in the system evolution . this situation arises naturally in many physical systems makhlinreport , makhlin , falci , adrian . consistently with the spin - boson model we consider the stochastic hamiltonian@xmath236\sigma _ { z}+\delta \sigma _ { x}\},\]]where @xmath237 is a classical non - white noise term . by assuming @xmath238 , where @xmath239 means an average over realizations of the noise , in the limit of vanishing @xmath174 it is simple to solve the stochastic dynamics and obtain the average of the pauli matrixes . the final evolution is the same as in eq . ( [ disperso ] ) after replacing @xmath71 with the average dephasing factor @xmath240\rangle \rangle _ { \xi } $ ] . thus , the generalized born - markov approximation can be mapped to the stochastic hamiltonian evolution under the condition @xmath241 , which explicitly reads @xmath232=\big\langle\big\langle\exp [ i\int_{0}^{t}d\tau \xi ( \tau ) ] \big\rangle\big\rangle_{\xi } .\]]this condition can be consistently satisfied if the dephasing factor @xmath242 decays in a monotonous way . we have presented a theoretical approach intended to describe the dynamic of small quantum systems interacting with a complex structured environment . our formalism is based in an extension of the well known born - markov approximation , which relies in the possibility of splitting the environment as a direct sum of sub - reservoirs , each one being able to induce by itself a markovian system dynamics . then , we have demonstrated that the full action of the complex environment can be described through a random lindblad master equation . the set of random rates follows from a fermi golden rule . thus , they are proportional to the characteristic coupling strength of each subspace multiplied by the corresponding sub - density of states evaluated in a characteristic frequency of the system . the associated probabilities are defined by the weight of each subspace in the stationary state of the bath . from a phenomenological point of view , the set of random rates and weights can be determined in a consistent way in function of the system decay . in fact , the system dynamics is characterized by a non - markovian master equation that in function of the random rate set can develop strong non - exponential decays . as an example we worked out the dissipative dynamic of a quantum tunneling system in a two level approximation . we have introduced a set of random rates that lead to the presence of asymptotic power law decay . in the limit of small hopping frequency , when compared with the average rate , we have showed that a zeno - like phenomenon arises , which is characterized by a stretched exponential and a power law decay . these behaviors follow from the interplay between the unitary dynamics and the entanglement - memory - effects induced by the reservoir . for the tunneling dynamics , we have also demonstrated that non - exponential decays arise even by considering a small set of random rates . furthermore , we have established the conditions under which the random lindblad evolution can be mapped to a spin - boson model and a stochastic hamiltonian evolution . finally , we want to emphasize that the present results define a new framework for describing anomalous quantum system dynamics , which consists in taking the characteristic rate of a lindblad equation as a random distributed variable . we remark that this approach was not derived from an ensemble of identical systems whose local interactions with the environment can be approximated by different markovian evolutions . in fact , the underlying microscopic physics can be related to a _ single _ quantum system coupled to an environment with a complex structured spectral density function and whose dynamical influence over the system can be approximated by a direct sum of markovian sub - reservoirs . thus , our approach may be relevant for the description of anomalous decay processes in individual mesoscopic systems embedded in a condensed phase environment schlegel , brokmann , grigolini . a natural example for which the generalized born - markov approximation may applies are glassy reservoirs , where the underlying configurational disorder produce a hierarchical distribution of coupling strength between the single system and the corresponding localized eigenstates of the reservoir @xcite . here we present the exact expressions for the kernels @xmath243 , @xmath244 , and @xmath245 that define the evolution of the pauli operators average , eqs . ( [ bloch ] ) . for arbitrary rates @xmath246 , the kernels read [ exactkernels ] @xmath247(u+b)+u\omega _ { a}^{2}\},\ \ \ \ \ \\ \gamma _ { y}(u ) & = & d\{[u(u+b)+\delta ^{2}](u+c)+u\omega _ { a}^{2}\},\ \ \ \ \ \\ \upsilon ( u ) & = & d(b - c)u\omega _ { a},\ \ \ \ \ \end{aligned}\]]where @xmath248 denotes the function @xmath249+\delta ^{2}\}[u+b(u)]+u\omega _ { a}^{2}}.\ ] ] the extra functions @xmath250 and @xmath251 are defined by@xmath252where we have introduced@xmath253(u+\gamma _ { r})+u\omega _ { a}^{2}}% .\]]using that the laplace transform of @xmath254 is given by @xmath255 , in the case @xmath96 it is possible to recuperate the expressions of section iii - a , eqs . ( [ kerneldispersivo ] ) . on the other hand , taking @xmath256 it is straightforward to get the results of section iii - c , eqs . ( [ kernelhoping ] ) . in an effective approximation , ( [ aproximada ] ) , the corresponding kernels read @xmath257\ } , \\ \gamma _ { y}(t ) & = & k(t)\cos [ \varphi t ] , \\ \upsilon ( t ) & = & k(t)\frac{\omega _ { a}}{\varphi } \sin [ \varphi t ] , \\ \phi _ { x}(t ) & = & k(t)\frac{\omega _ { a}\delta } { \varphi ^{2}}\{1-\cos [ \varphi t]\ } , \\ \phi _ { y}(t ) & = & k(t)\frac{\delta } { \varphi } \sin [ \varphi t ] , \end{aligned}\ ] ] where @xmath258 the extra kernels @xmath259 and @xmath260 couples the derivative of @xmath153 to the averages @xmath261 and @xmath262 respectively , i.e. , @xmath263 for the exact evolution , these kernels vanish . the kernels of the spin - boson model eqs . ( [ spin - boson ] ) , in lowest order in @xmath174 read @xcite [ kernelspinboson ] @xmath264 , \\ k_{a}^{(a)}(t ) & \simeq & \delta ^{2}y_{b}^{(a)}(t)\sin [ q^{\prime \prime } ( t ) ] , \\ k_{a}^{(s)}(t ) & \simeq & \delta ^{2}y_{b}^{(s)}(t)\cos [ q^{\prime \prime } ( t ) ] , \\ y_{a}^{(a)}(t ) & \simeq & -\delta y_{b}^{(a)}(t)\cos [ q^{\prime \prime } ( t ) ] , \\ k_{b}^{(s)}(t ) & \simeq & \delta y_{b}^{(s)}(t ) , \\ k_{b}^{(a)}(t ) & \simeq & -\delta y_{b}^{(a)}(t ) , \\ y_{b}^{(s)}(t ) & \simeq & \cos [ \omega _ { a}t]e^{-q^{\prime } ( t ) } , \\ y_{b}^{(a)}(t ) & \simeq & -\sin [ \omega _ { a}t]e^{-q^{\prime } ( t ) } , \end{aligned}\]]where @xmath265 and @xmath266 are defined by [ kerneldisorder ] @xmath268 @xmath269 , \\ y_{a}^{(a)}(u ) & = & t(u)\delta \lbrack \omega _ { a}-\upsilon ( u ) ] , \\ y_{b}^{(s)}(u ) & = & t(u)\delta \lbrack u+\gamma _ { x}(u ) ] , \\ k_{b}^{(a)}(u ) & = & t(u)\delta \lbrack \omega _ { a}-\upsilon ( u ) ] , \\ y_{b}^{(s)}(u ) & = & t(u)[u+\gamma _ { y}(u ) ] , \\ y_{b}^{(a)}(u ) & = & -t(u)[\omega _ { a}-\upsilon ( u ) ] , \end{aligned}\]]where we have introduced the structure of these kernels is the same as those of the spin - boson model in the limit of vanishing @xmath174 , which implies that @xmath174 only appears through the unitary evolution . in fact , in this limit we can approximate eqs . ( [ exactkernels ] ) by @xmath271/2 $ ] and @xmath272/2i$ ] . after introducing these expressions in eqs . ( [ kerneldisorder ] ) , it is simple to get @xmath273p_{0}(t)$ ] , and @xmath274p_{0}(t)$ ] . then , disregarding in eqs . ( [ kernelspinboson ] ) the phase contribution proportional to @xmath275 , which is valid in a high temperature limit wilkie , a mapping with eqs . ( [ kerneldisorder ] ) can be done after imposing the equality @xmath276 . the stochastic dynamics can be formally extended to the case @xmath80i @xcite after introducing the superoperator @xmath277=\{$]i@xmath278i@xmath279\}\rho $ ] , where @xmath280 must be intended as a control parameter . then , the evolution eq . ( [ aproximada ] ) is recuperated in the limit in which simultaneously @xmath281 and the number of events by unit of time go to infinite , the last limit being controlled by the sprinkling distribution @xmath282 @xcite . this object is defined as the probability density for an event at time @xmath283 , disregarding the possibility of extra events in @xmath284 . therefore , it is given by @xmath285 where @xmath286 is the step function [ here defined as @xmath287 for @xmath288 , and @xmath289 for @xmath290 , which implies @xmath291.$ ] in the laplace domain it reads @xmath292 $ ] , which allows to write @xmath293 . when the random rate assumes only two different values @xmath294 , with probabilities @xmath295 , the exact kernel reads @xmath296 $ ] where @xmath297 $ ] . the rate @xmath172 is defined in terms of the fluctuations @xmath193/\langle \gamma _ { r}\rangle .$ ] the extra rates are @xmath298 , and @xmath299
|
in this paper we demonstrate that lindblad equations characterized by a random rate variable arise after tracing out a complex structured reservoir .
our results follows from a generalization of the born - markov approximation , which relies in the possibility of splitting the complex environment in a direct sum of sub - reservoirs , each one being able to induce by itself a markovian system evolution .
strong non - markovian effects , which microscopically originate from the entanglement with the different sub - reservoirs , characterize the average system decay dynamics . as an example
, we study the anomalous irreversible behavior of a quantum tunneling system described in an effective two level approximation . stretched exponential and power law decay behaviors arise from the interplay between the dissipative and unitary hopping dynamics .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
hierarchical models and observations suggest that galaxy mergers and interactions play a key role in galaxy assembly and star formation , but to what extent is still unclear . studies of gas - rich mergers in the local universe ( e.g. , antennae ; see * ? ? ? * ) and n - body simulations @xcite have revealed fundamental signatures of the galaxy merger process , including tidal tails , multiple nuclei , and violent bursts of star formation . while interaction - induced star formation is thought to be primarily responsible for ultra luminous infrared galaxies ( ulirgs , which have @xmath13 ) both locally and at high redshift @xcite , luminous infrared galaxies ( lirgs , @xmath14 ) appear to have multiple driving mechanisms , merger - induced star formation being only one . luminous infrared ( ir ) galaxies are thought to be the dominant producers of the cosmic infrared background ( cirb ) , and major contributors to the evolution of the cosmic star formation rate ( csfr ) of galaxies , especially at @xmath15 @xcite . the rapid decline from @xmath8 of the csfr density has been linked to a decline in the merger rate . however , recent close pair studies have suggested that the merger rate has remained fairly constant from @xmath8 @xcite , and at @xmath15 the ir population is dominated by morphologically normal galaxies @xcite . the combination of these two results suggest that the bulk of star formation at @xmath8 is not driven by major mergers . however it must be noted that different merger selection criteria probe different stages of the merger process . quantitative measurements of galaxy asymmetry ( @xcite ) are more likely to probe later stages , while early stage mergers can be identified by carefully searching for close companions . there should be some overlap between these techniques if galaxy pairs are close enough to have induced strong tidal interactions , but galaxies in pairs could also have normal morphologies , hence if early stage mergers are not considered , the impact interactions / merging have will be underestimated . traditionally , close pair studies have been carried out in the optical / near - ir @xcite . however recent investigations have begun to explore the mid - ir properties ( star formation ) of galaxy pairs , finding a mid - ir enhancement in pairs separated by less then ten s of kpc s @xcite . the amount of ir luminosity stemming from individual processes ( star formation or fueling an agn ) in interacting pairs and mergers still remains open . to investigate this question we have conducted a study of the frequency of mips 24 detected , and undetected close optical galaxy pairs and morphologically defined mergers in the _ spitzer _ first look survey ( fls ) . we find that the fraction of 24 detected , optically selected close pairs and mergers increases with redshift , and are important contributors to the ir luminosity and star formation rate density at @xmath8 . in the discussion that follows , any calculation requiring cosmology assumes @xmath16=0.3 , @xmath17=0.70 , and h@xmath18=70kms@xmath19mpc@xmath19 . the _ spitzer _ extragalatic component of the fls is a 3.7 @xmath20 region centered around r.a.=@xmath21 , decl.=@xmath22 . observations of this field were taken using all four infrared array camera ( irac ) channels ( fazio et al . 2004 ) and three multiband imaging photometer ( mips ) bands ( rieke et al . 2004 ) . additional ground base images in u*,g from cfht s megacam @xcite , g , i data from palomar 200 " lfc and noao 4-m r and k band ( fadda et al . 2004 ; glassman et al . 2006 in prep ) have also been obtained . this work focuses on the 0.12 @xmath20 acs - hst f814w imaging of the verification strip , which has 3@xmath23 depths in mips 24 of 0.1mjy . object detection and photometry were performed using sextractor ( bertin & arnouts 1996 ) . particular care was taken to ensure accurate de - blending of galaxies in close proximity to one another , while avoiding detections of substructure within a single galaxy , consistent with other reductions of hst imaging with close galaxy pairs in mind ( patton et al . 2005 ) . there were @xmath359,000 sources extracted within the @xmath24 band ( hereafter extracted magnitudes referred to as @xmath25 ) . we compared our number counts to those from the hubble deep field ( hdf ) north and south and determined a limiting magnitude of @xmath2627.4 . using the full mips catalog from the fls we selected 24sources within the area covered by the acs imaging ( @xmath30.12 @xmath20 ) . in order to correlate the mips objects with those identified in the optical we first cross - identified sources from the mips 24 sample to the irac catalog using a tolerance radius of 2.0@xmath27 this choice was primarily motivated by the fwhm of the mips 24 ( psf@xmath3@xmath28 ) and confirmed by visual inspection . we then cross - correlated the irac / mips catalog to the acs sample which we band merged with u * , g and @xmath29 requiring a positional agreement of @xmath30@xmath31 . when multiple counterparts were identified , we selected the closest object . ultimately we found 1155 acs sources also detected by irac and mips at 24 . the redshifts used in this study were determined exclusively from optical spectroscopy . they were obtained by cross - correlating the acs sample , limited to @xmath3226.5 ( @xmath33@xmath329,000 ) with various fls spectroscopic datasets . the vast majority of the included redshifts ( @xmath3497% ) were obtained with the deep imaging multi - object spectrograph ( deimos ) on the w.m . keck ii 10-m telescope ; however , the final sample also included a few redshifts based on sloan digitized sky survey ( sdss ) and wiyn hydra / mos ( marleau et al . 2006 in prep ) spectra . galaxies in the fls verification region were targeted for spectroscopic follow - up during two deimos campaigns that bracketed spitzer s launch . the selection criteria for these campaigns are summarized below . for the 2003 pre - launch campaign , targets were selected based on nir ( @xmath35 ) and optical ( @xmath36 ) colors . the primary sample included sources with @[email protected] , @xmath3819.0 and a @xmath36 color selection that restricted the numbers of low redshift ( @[email protected] ) sources . for the 2004 post - launch campaign , a purely 24 selected sample ( @xmath40@xmath41120@xmath42jy ) was targeted for follow - up . the combined @xmath25 distribution of targeted and detected sources is shown in figure [ fig : magdist ] ( _ top _ ) along with the cumulative redshift identification efficiency ( _ bottom _ ) . the overall spectroscopic completeness ( defined here as the fraction of targeted sources with high quality redshifts ) is @xmath370% for the full sample and @xmath380% for sources with @xmath4325.0 . for a more detailed description of the observing strategy , primary selection criteria and the overall flux and redshift distributions see @xcite . since we were exploring the mid - ir properties of galaxies no optical limit was imposed , instead an ir luminosity cut ( @xmath44 or @xmath45 ) was used , so that a fair comparison could be made at different redshifts . the absolute b magnitude ( @xmath46 ) distribution of the mips spectroscopic sample between 0.5@xmath37@[email protected] probes -21@xmath30@xmath46@xmath30 - 19 fairly uniformly and we are not strongly biased at higher redshifts . cross - correlation of the band - merged photometric catalogs with the redshift samples results in a data set of 476 sources with @xmath47 and @xmath0@xmath39@xmath1 . of those , 245 ( 51% ) are mips 24-detected with a measured @xmath48 . the remaining 231 ( 49% ) are non - detected at 24 . despite the fact that the mips and non - mips galaxies were selected slightly differently , the resultant colors of objects with spectroscopic redshifts have uniform color properties ( figure [ fig : colors ] ) . to properly constrain the role interactions and mergers play in galaxy evolution , all stages of the process must be considered . typically , merger history analyses utilize either pair or structural methods . galaxies in pairs are pre - mergers , or systems undergoing interactions , while morphological or structural methods find galaxies that have already undergone a merger and are dynamically relaxing . when discussing the pair fraction , and inferred merger rates ( as defined in section 4 ) , it must be noted that these measurements are highly dependent on the techniques and selection criteria used to identify ongoing mergers , especially for galaxies at high redshifts . the first method , which we will call the close pair method " , is to count the number of galaxy pairs within some projected separation , @xmath49 , and magnitude difference ( @xmath50 ) . if we assume that these systems will merge within a given time - scale due to dynamical friction , we can determine the merger rate . although not all pairs will merge , they can potentially trigger star formation through gravitational interactions . an alternative is to select merging systems based on morphological indicators either by overall appearance @xcite or computational measurements such as asymmetry ( a ) , and clumpiness ( s ) of a system @xcite , or gini coefficient ( g ) , and @xmath51 parameters @xcite . due to the comparatively limited spatial resolution of _ spitzer _ ( compared with optical imaging ) , seaching for close galaxy pairs or morphological signatures of interaction at mid - ir wavelengths is currently restricted to the nearby universe . however , we can correlate optically - selected pairs / mergers with global mid - ir properties and investigate the ir activity in these systems out to high redshifts . we applied the close pairs technique to identify the average number of close companions per galaxy , hereafter @xmath52 . this measurement is similar in nature to the pair fraction when there are infrequent triples or higher order n - tuples . since this is the case here , @xmath52 will be occasionally referred to as the pair fraction . companions were selected using a standard operational close pair definition of 5@xmath53kpc@xmath54kpc , and an optical magnitude difference ( @xmath50 ) @xmath55 compared to the host galaxy , to select nearly equal mass major mergers . the term host " or primary " galaxy are both used to reference the pair member with a measured redshift . the inner radius is applied to avoid detection of substructure within a galaxy , while the outer @xmath56kpc limit represents the radius within which satellites are expected to strongly interact with the halo of the host and merge within 0.5 - 0.9 gyrs ( patton et al . 1997 ; conselice et al 2003 ) . we find 87 close pairs out of 476 galaxies which fulfill these criteria ( see table 1 ) . to study the fraction of ir - bright galaxies in pairs , we split the pair sample into two sub - sets : those which were detected and those undetected with mips at 24 down to the flux limits of our survey ( 0.1 mjy ) . figure [ fig : stamps ] shows a subset of close pairs ( both detected and undetected at 24 ) with mips contours . due to the small separations of close pairs ( 20@xmath53kpc corresponds to 3.6 at @xmath8 ) relative to the beam of the mips 24 images ( fwhm @xmath36 ) , there are a few instances ( 5 ) where only a single 24 detection is found centered between the pair members ( see middle left image in figure [ fig : stamps ] ) . in these cases we assume all 24 flux is coming from the primary galaxy . since we have redshift information for only the primary galaxy and not the companions we need to consider what fraction of these close pairs are a result of random projection effects . a field correction was determined using two separate methods to account for these close pairs . the first assumed the same optical magnitude and redshift distributions independently for both the detected and undetected 24 samples , while the positions were randomized . the close pair algorithm was applied to 50 realizations of these mock catalogs and the average @xmath52 for each redshift bin was taken to be the pair fraction expected from random . this assumes the absence of clustering . we investigated the environments of 24 detected and undetected objects on scales of @xmath57@xmath5820@xmath53kpc , and found them to be comparable , confirming that the increase in pair - fraction of the 24 detected pairs is not because they preferentially lie in clusters . on the other hand , there is a weak indication that galaxies detected at 24 are more likely to lie in small groups . since such groups may , in some cases , be physical associations , we count such cases as separate pairs . however , the number of these cases is small , and does not influence our results in any significant way . the second method utilizes the @xmath25 magnitude distribution of the full photometric catalog ( @xmath359,000 sources ) , and determines the average number of companions , within 1.5 mags ( @xmath25 ) , normalized to the area covered by 5@xmath53kpc@xmath54kpc . the results obtained from the two field correction methods agreed within @xmath59 , which is negligible compared to the uncertainly in @xmath52 . the average of the two methods was taken to be the final field correction . both the pair catalog and randomly generated catalogs were visually inspected for false pairs due to single galaxies being broken up into multiple components in the source extraction phase , or contaminating stars in the photometric catalog , and were removed . the field - corrected optical pair fractions for the 24 detected and undetected sub - samples are presented in figure [ fig : pfrac ] and table 1 . errors are computed using the jackknife technique @xcite , e.g. given a sample of @xmath33 galaxies the variance is given by [ ( @xmath60@xmath61 . the partial standard deviations , @xmath62 , are computed for each object by taking the difference between @xmath52 , the quantity being measured and the same quantity with the ith galaxy removed , @xmath63 , such that @xmath64 . to allow a more direct comparison to be made between the generally lower - luminosity low - z pairs , and those at higher redshift , we derived pair fractions for mips detected galaxies with an @xmath65 ( approximately the ir luminosity of the famous antenna galaxies ) . in this way we ensure that the sub - luminous galaxies do not strongly influence the pair fractions in the lowest redshift bin . the derived @xmath52 for 24 detected close pairs is @xmath66 at @xmath67 and increases to @xmath68 at z@xmath31 . in contrast , close pairs with no 24@xmath42 m detection show no increase with redshift and have pair fractions consistent with zero at all redshifts . the higher pair fraction of mips bright sources is marginally significant due to the small number of sources in the highest redshift bin , more mir selected samples between @xmath691 - 1.5 are required to strengthen our findings . cccccc & & 24 detected & & + 0.2 - 0.5 & 32 & 0.188 ( 6 ) & 0.078 ( 2.5 ) & 0.110 @xmath70 & 0.83 + 0.5 - 0.80 & 82 & 0.171 ( 14 ) & 0.057 ( 4.7 ) & 0.114 @xmath71 & 0.93 + 0.80 - 1.0 & 82 & 0.122 ( 10 ) & 0.029 ( 2.4 ) & 0.093 @xmath72 & 0.90 + 1.0 - 1.3 & 49 & 0.429 ( 21 ) & 0.182 ( 8.9 ) & 0.247 @xmath73 & 0.67 + & & 24 undetected & & + 0.2 - 0.5 & 44 & 0.136 ( 6 ) & 0.102 ( 4.5 ) & 0.034@xmath74 & 1.00 + 0.5 - 0.80 & 76 & 0.132 ( 10 ) & 0.134 ( 10.2 ) & 0@xmath75 & 1.00 + 0.80 - 1.0 & 56 & 0.214 ( 12 ) & 0.193 ( 10.8 ) & 0.021@xmath76 & 0.83 + 1.0 - 1.3 & 55 & 0.145 ( 8) & 0.180 ( 9.9 ) & 0 @xmath77 & 0.75 + we would like to be able to rule out the possibility that @xmath52 is biased by the brightest ir sources at @xmath78 , since merger fractions change as a function of luminosity and mass @xcite . to address this we placed a higher ir luminosity limit ( @xmath79 ) on the sample , so that at @xmath15 the same populations were being probed ( optically we are probing -22@xmath58@xmath46@xmath58 - 19 ) . we still find an increase in @xmath52 from the lower ( @xmath80 ) to the higher ( @xmath78 ) redshift bins of similar magnitude compared to when the lower ir limit ( @xmath44 ) was used . therefore the increase in @xmath52 found at @xmath78 is likely not a result of merely probing brighter ir systems but rather due to a physical increase in the merge rate for the 24 population , however deeper 24 imaging and spectroscopy are required to confirm this . when we consider the averaged pair fraction over @xmath81 for the 24 detected sample we find that galaxies above a flux limit of @xmath82mjy are five times more likely to be in a close galaxy pair , than those below this limit . to explore the structural components of galaxies in our sample we used the cas ( concentration , asymmetry , clumpiness ) quantitative classification system @xcite , and visual classifications . to measure the merger fraction using structural classifications we visually inspected the full 24 detected spectroscopic catalog with the following groupings : early type ( e , s0 ) , mid - types ( sa - sb ) , late - types ( sc - irr ) , compact systems , disturbed disks , and mergers . the methodology for carrying out this classification is described in detail in @xcite . basically , each galaxy was viewed on a computer screen and classified into one of our types . overall we find that @xmath83 of 24 detected galaxies are disks , which is consistent with @xcite , while @xmath84 are merging systems and @xmath85 were classified as disturbed disks and are possible minor mergers . a fraction of the disk - dominated objects do show some visual signs of a morphological disturbance , or are in a pair , as we will discuss later in this paper . galaxies undergoing a major merger event can also generally be identified by their large asymmetries in the rest frame optical @xcite . we defined a major merger as a galaxy having an asymmetry ( a ) @xmath86 and @xmath87 ( see figure [ fig : stamps2 ] for examples ) . this limit has been shown to be a clean way to find galaxy mergers , without significant contamination from non - merging galaxies @xcite . figure [ fig : mfract ] shows how the merger fraction for cas defined mergers evolves as a function of redshift for both 24 detected objects ( top panel ) and lirg / ulirg galaxies ( bottom panel ) . as with the 24 detected close pair sample there is an elevated merger fraction compared to other works @xcite in which no 24 flux limit was imposed , and a slight indication of evolution with redshift , but it is statistically consistent with @xmath88 ( dashed line ) , where @xmath89 is the slope of a power - law of form @xmath90 later discussed in @xmath91 . we also performed a cas analysis of our close pairs sample which revealed that 24 detected pairs are notably more asymmetric than the undetected - mips close pairs ( figure [ fig : cas ] ) , suggesting that interactions and collisions may play a role in their ir activity . if the 24 detected close pairs were generally of a different morphological classification than those pairs undetected at 24 the discrepancy in the asymmetries could be explained . to address this issue each close pair was visually inspected and classified by four of the authors to be either disk or bulge - dominated . we find that 81% of the 24 pairs have disk morphologies while @xmath92 of the undetected 24 hosts were also disk dominated , hence the discrepancy between the asymmetries of the two groups is not caused by classification differences , but rather is a physical effect . one of the goals of studying mergers and interactions is to determine how the galaxy merger rate evolves with redshift . most studies of galaxy mergers involve determining the merger fraction , yet the merger rate , which is defined as the number of galaxies merging per unit time per unit volume , is a more physical quantity that can be used to determine the full merger history . the rate in which galaxies merge also affects the mass function of galaxies , and is likely linked to the cosmic star formation rate . since we are considering a very broad range in the merger process , from early - stage or pre - mergers selected via close galaxy pairs , and later - stage mergers chosen based on morphological criteria , we must be careful when determining their respective merger rates , as the time - scales for these processes are all different . there are two variations of the merger rate definition . the first is the number of mergers that a galaxy will undergo per unit time ( @xmath93 ) , and the second is the total number of mergers taking place per unit time per unit co - moving volume ( @xmath94 ) . since we are primarily interested in mergers which are also mid - ir bright systems we will have to restrict ourselves to measuring @xmath93 because the evolution of the 24 luminosity function with redshift is currently not well constrained , and our redshifts are not complete enough to reconstruct this evolution . in order to determine @xmath93 we need to identify systems which are destined to merge . we have approached this measurement from three different perspectives , close pairs to select pre - mergers or interactions , visual inspection to select interactions after the first passage , and late stage mergers , as well as cas criteria which quantitatively selects for later stage mergers . by combining information about the number of ongoing mergers ( @xmath95 ) and the time - scales , ( @xmath96 ) on which they will undergo said merger , one can estimate an overall merger rate @xmath97 . each method of identifying mergers / interactions is capturing a different snapshot of the merger process , each with different merger timescales . the value of @xmath52 is directly proportional to the number of mergers per galaxy ( @xmath95 ) , such that @xmath98 ( @xmath99 is a constant relating to the number of mergers per galaxy ) . hence , the merger rate detemined using close galaxy pairs is given by @xmath100 . the value of @xmath99 depends on the nature of the merging systems under consideration . if one were to identify a pure set of galaxy pairs each consisting of one companion undergoing a merger , then @xmath101 . in our case it exclusively accounts for close pairs which are in doubles and perhaps higher order n - tuples . our definition of @xmath99 differs by a factor two from @xcite which in this instance would have @xmath102 since they have redshifts for both pair members and one merger is made up of two companions . we have redshift information for only one pair member , therefore one merger is made up of a primary and one companion . the merger rate equation for merging galaxies selected by visual classification and cas parameters is simply @xmath103 , where @xmath104 is the galaxy merger fraction . before the merger fraction can be used to calculate the merger rate we need to understand the time - scale in which a merger occurs . each technique of identifying mergers has a different time - scale since each is sensitive to a different interval of the merger process . there are two main methods that have been used to estimate the time - scale of a merger : dynamical friction arguments , and n - body models . the details of these methods are beyond the scope of this paper but see @xcite and @xcite for a review . we take the average merger time - scale for a set of close companions of roughly equal mass to merge as @xmath30.5 gyr@xmath105 , derived from dynamical arguments @xcite . @xcite showed through n - body simulations that visual classification selects on - going mergers over a longer time - scale ( 1.0 gyr@xmath105 ) since the human eye detects both early and later stage mergers , while the asymmetry of a galaxy is sensitive to 0.41 gyr@xmath106 @xcite of the merger sequence . within the past two decades numerous studies have been performed to estimate the evolution of the galaxy merger fraction , using both the close pair technique @xcite and morphological parameters @xcite . evolution in the galaxy merger fraction is often parameterized by a power - law of form @xmath90 , and has yielded a wide range of results , spanning 0@xmath107@xmath89@xmath1075 . the large spread in values is in part due to the different selection criteria used to identify merging systems and biases from optical contamination or redshift completeness . @xcite considered these biases and demonstrated that most results to that date were consistent with their estimate of @xmath108 . recently , optical and near - ir close pair studies @xcite have derived merger fractions with little redshift evolution ( @xmath89@xmath31 ) , as have some morphological studies using ( @xmath109 ) , and @xmath51 @xcite . when we consider all the close pairs identified in our sample , both those detected at 24 and not , we find a merger fraction and rate consistent with recent studies showing little redshift evolution . however , when we separate the pair sample into systems with a 24 detection above 0.1 mjy , and those below it , we do see a stronger evolution of @xmath52 with redshift , ( recall that @xmath110 ) and therefore also in the merger fraction and rate ( figure [ fig : mr ] ) . similarly , visually classified mergers and those identified via asymmetry levels ( @xmath111 ) using the cas parameters , also show redshift evolution in the merger fraction and rate . the merger fraction computed using the different methods are in good agreement when normalized by their respective time - scales , reinforcing the idea that we are probing different phases of the merger process . considering all three merger selection techniques we find the best fit of the merger rate parameterized by @xmath112 to be @xmath113 , 2.12 @xmath114 , with a reduced @xmath115=0.39 . this result suggests that when one considers a sample of close galaxy pairs solely on their optical fluxes , brighter than @xmath116 - 19 , little evolution of the merger rate with redshift is found . however close pairs emitting 24 flux exhibit an increase in the merger rate with redshift . the infrared luminosity limit ( @xmath117 ) imposed on the close pairs and mergers allows us to primarily probe systems in a lirg / ulirg phase at @xmath118 ( see next section for details ) . the increase of the merger fraction and rate of this population of galaxies coupled with the fact that lirg / ulirg galaxies dominate the sfr density at @xmath15 @xcite suggests that merging does in fact play an increasingly important role in star formation out to @xmath8 . one way to quantify the role merging galaxies play in triggering star formation is to investigate their contribution to ir luminosity densities . infrared luminosities ( 8 - 1000 ) were calculated utilizing the 24 fluxes and two different template methods : @xcite in a similar manner as @xcite for the full mips 24 spectroscopic sample ( figure [ fig : lirall ] ) . mips pairs and mergers share a similar luminosity distribution to 24 bright field galaxies , although red - agn seem generally more luminous which is in part due to template mismatches @xcite . the @xmath119 of a galaxy is a combined measure of the reprocessed uv photons intercepted by dust from massive young stars and agn . therefore to investigate the contribution an interacting or merging galaxy makes towards the total @xmath119 density from star formation alone we must first remove agn from our sample . due to the nonuniform rest - frame spectral coverage of our sample we rely on the four - band irac color selection used by @xcite to identify and remove agn candidates ( figure [ fig : agn ] ) . over the modest redshift range of our sample , this method is still effective at separating ir - warm agn from starburst systems . we find an agn contamination rate of @xmath120 for the full 24 sample , while @xmath121 of the hosts in a pair or merger were characterized as agn . with agn candidate objects removed we can infer the contribution to the @xmath119 density from star formation coming from 24 galaxies in a interaction / merger as a function of redshift . we derive the number of statistically real " galaxy pairs from our pair fraction result at each redshift interval and determine the total @xmath119 density from close pairs which is in turn divided by the @xmath119 density from the whole sample . we find that paired galaxies ( @xmath117 ) are responsible for @xmath122 of the ir background stemming from star formation at @xmath8 . since we only know the redshift of the host galaxy we select real " close pairs in a statistical sense , and derive error bars for the close pairs contribution by the spread of 50 realizations of the @xmath119 density from different combinations of 24 galaxy pairs . we also applied this analysis to cas and visually classified mergers , which make up an additional @xmath312% , and @xmath322% of the ir luminosity density respectively . naturally there is a some overlap in mergers identified through close pair criteria and morphological parameters , since interacting pairs can exhibit tidal tails and asymmetric structures , causing them to also be identified morphologically as mergers . we found that @xmath123 of cas defined mergers were also in a close pair , and 31% of visually identified mergers were also classified by cas as merging . in cases where a merging system was identified using multiple techniques it s contribution was only counted once . for example , if a merger identified morphologically ( either through cas or visual inspection ) is also in a close pair it is removed from the morphological merger catalog , or if a cas merger is also identified visually the merger is removed from the visual merger catalog . this insures that no close pair or merger is counted more than once when deriving the contribution from interactions and mergers to the ir luminosity density . the combination of these three merger selection techniques identifies a large range in the merger process , from pre - merger to late stage mergers , implying that @xmath360% of the infrared luminosity density at @xmath39@xmath31 can be attributed to galaxies involved in some stage of a major merger ( figure [ fig : lumden ] ) . the remaining @xmath340% of the ir background from lirg / ulirgs is likely to predominately come from active , isolated gas - rich star - forming spirals , with some contribution from minor mergers . if we exclude visually classified mergers the close pair / merger contribution to the ir density is @xmath338% , in good agreement with @xcite who estimate a moderate contribution from interacting and merging systems of @xmath5836% . it must be noted however that neither @xcite or this work have considered the contribution from minor mergers and are therefore lower limits . an important and highly debated question is : how important are galaxy mergers in understanding the dramatic decline of the cosmic sfr density from @xmath8 to the present day ? it has been well established that mergers and interactions can induce violent bursts of star formation @xcite . so to investigate this contribution we derived the sfr for our 24 detected close pairs and mergers , using their @xmath119 . the infrared luminosity of a galaxy is a star formation rate tracer which is unaffected by the extinction of dust . the dominant heat sources of most dusty , high - opacity systems such as lirgs and starbursts is stellar radiation from young stars . in these types of systems the @xmath119 can be converted into a sfr using the calibration of @xcite , @xmath124 , where @xmath119 is the integrated luminosity from 8 - 1000 as determined in section 6.0 . we estimated the contribution mergers and interactions above @xmath117 make to the sfr density at @xmath51 in two ways . the first is simply to consider their contribution to the @xmath119 density which is a star formation tracer . section 6.0 determined that mergers and interactions at z@xmath31 ( above @xmath117 ) are responsible for 40 - 60% of the ir luminosity density . using the results of @xcite which showed that @xmath1250.7 lirgs produce @xmath126 of the star formation rate density , we can infer that mergers and interactions in lirg / ulirg phase would be responsible for @xmath127 ( @xmath128 ) of the sfr density at @xmath51 , since ir activity traces dusty star formation . the second more detailed approach utilizes the sfr density directly arising from our sample of mergers and interactions . at 1.0@xmath30@[email protected] we find that 59% ( 12 close pairs , and 17 later stage mergers ) of galaxies detected at 24 are involved in some stage of an interaction or merger . in this same redshift range our sample is insensitive to galaxies with ir luminosities @xmath30@xmath129 . to correct for this we derived a scaling factor ( @xmath37 ) simply by comparing the number of observed objects of a given @xmath119 in a specific redshift range to the number expected from models @xcite . however to go any further we must assume that our spectroscopic sample is representative of this population at @xmath31 , and by all accounts this appears to be true . using the derived pair fraction we can then infer the total number of major mergers and interactions occurring ( fulfilling our criteria ) in a given volume and @xmath119 limit . the lower limit of the sfr density at @xmath51 from merging and interacting galaxies is found to be 0.066 @xmath130 . using the extinction corrected lilly - madau " plot ( 0.1585 @xmath130 at @xmath691 ) @xcite we find that mergers and interactions are responsible for at least @xmath131 of the sfr density at @xmath51 ( assuming mergers contribute 60% of the ir density ) . both approaches are in good agreement , and are only a lower limit , since objects flagged as agn were not considered even though some of their @xmath119 is a result of star formation , and minor mergers which have been shown to also induce bursts of star formation were not included . these results have interesting implications for the physical mechanisms that drive the decline in the cosmic sfr ( csfr ) density from @xmath8 to present day . they suggest that when all stages of the merger process are considered ( pre - merger to later stage merger ) major interactions and mergers contribute close to half of the @xmath8 sfr density , and the decline in the number of 24 detected mergers / inteactions is a significant , but perhaps not the primary driver for the decline in the cosmic sfr . this conclusion differs in interpretation from @xcite , which generally suggest that the evolution of the merger rate is not a significant underlying cause of the decline in the cosmic sfr , but rather a strong decrease in the sfr of morphologically undisturbed spiral galaxies is the dominant mechanism . their results do not preclude the possibility that their star forming ( undisturbed ) disks " could be in widely separated pairs , and when we only consider quantitatively defined morphological mergers our results are consistent with theirs stressing the importance of considering the merger process in its entirety . it must also be mentioned that we are probing to higher redshifts than @xcite , which found that major galaxy mergers account for @xmath3030% of the ir luminosity density at @[email protected] , consistent with our findings of 35% at that redshift . our results also agree that at @xmath50.7 isolated undisturbed spiral galaxies are a primary contributor , however , the influence shifts to interactions and mergers at @[email protected] . our findings point to an increased importance of mips bright interactions and mergers to the ir luminosity density and sfr density at @[email protected] . this conclusion is not hampered by the small statistics of the @xmath121 bin . figure [ fig : lumden ] shows the ir luminosity density contribution from interactions / mergers at @[email protected] to be @xmath337% and 52% at @[email protected] , reinforcing this increasing trend . using a spectroscopic sample of field galaxies from the acs component of the fls and dividing it into two subsets , those with a 24 detection ( above @xmath82mjy ) and those without ( or below ) we identified optically merging / interacting systems via close pair statistics and morphological methods . we find that roughly 25% of galaxies emitting at 24 have a close companion at @xmath51 while at @xmath50.5 only @xmath132 are in pairs . in contrast , those undetected at mips 24 showed a pair fraction consistent with zero at all redshifts ( 0.2@xmath30@[email protected] ) . on average mips 24 galaxies are five times more likely than non - mips sources to have a close companion over 0.2@xmath30@[email protected] . when the samples are combined ( regardless of 24 flux ) we find pair fractions consistent with previous studies @xcite showing little evolution with redshift . an important and open question is the cause of star formation in lirg galaxies at high - z . some morphological studies have suggested that since at least half of the lirg galaxies exhibit disk dominated morphologies @xcite at @xmath1250.7 and low non - evolving merger fractions @xcite that the driver of ir activity in high - z lirgs is from on - going star - formation from isolated gas - rich spirals and not merger or interaction induced . one bias of @xmath133 morphological studies involving the identification of merging / intereacting systems is the limitation of detecting low surface brightness features such as tidal tails caused by close interactions , which can lead to an underestimate of the importance of mergers in the evolution of galaxies at @xmath1341 . ultimately both close pair and morphological techniques must be applied and considered , to obtain a complete major merger timeline . our analysis is the first to probe merger rate evolution combining close pairs and later stage mergers while considering the ir activity of these systems . we find that close pair statistics , visually classified mergers , and those identified via quantitative cas parameters all showed similar evolution in their merger rates . fitting the merger rate evolution function @xmath135 for 24 detected mergers above @xmath82mjy , we find @xmath136 . this result agrees with previous claims of an increase ( @xmath137 ) of the merger rate out to @xmath8 @xcite . however this evolution is not seen when ir faint ( @xmath138mjy ) mergers are included , suggesting that it is the lirg - merger population that is evolving with redshift . the mid - ir emission of lirgs is indicative of dust enshrouded star formation ( and some agn activity ) , and at @xmath15 they dominate the ir luminosity density and in turn the volume - averaged star formation rate density at @xmath8 . we estimate that close galaxy pairs are responsible for @xmath139 of the ir luminosity density resulting from star formation at @xmath8 , while later stage mergers contribute @xmath140 . this implies that 40 - 60% of the infrared luminosity density at @xmath8 can be attributed to galaxies involved in some stage of a major merger , indicating that merger - driven star formation is responsible for 30 - 40% of the star formation density at @xmath8 . this value is a lower limit since minor mergers and interactions / mergers with an agn were not considered . ultimately , our findings suggest that interactions and mergers of lirg phase galaxies play an increasingly important role in both the ir luminosity and sfr density from @xmath15 out to @xmath141 , and are vital to our understanding of the evolution and mass assembly of luminous ir galaxies . we would like to thank h. shim , m. i m , r. chary , and c. borys for their contributions to this work , v. charmandaris for useful suggestions , and the anonymous referee for valuable comments that improved the clarity of the paper . this work is based on observations made with the _ spitzer observatory _ , which is operated by the jet propulsion laboratory , california institute of technology , under nasa contract 107 . support for this work was provided in part by the _ spitzer graduate student fellowship _ program and an ontario graduate scholarship in science and technology . the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . abraham , r. g. , van den bergh , s. , glazebrook , k. , ellis , r. s. , santiago , b. x. , surma , p. , & griffiths , r. e. 1996 , , 107 , 1 abraham , r. g. , tanvir , n. r. , santiago , b. x. , ellis , r. s. , glazebrook , k. , & van den bergh , s. 1996 , , 279 , l47 abraham , r. g. , van den bergh , s. , & nair , p. 2003 , , 588 , 218 barnes , j. e. 2004 , , 350 , 798 barton , e. j. , geller , m. j. , & kenyon , s. j. 2000 , , 530 , 660 bell , e. f. , et al . 2005 , , 625 , 23 bertin , e. , & arnouts , s. 1996 , , 117 , 393 bundy , k. , fukugita , m. , ellis , r. s. , kodama , t. , & conselice , c. j. 2004 , , 601 , l123 burkey , j. m. , keel , w. c. , windhorst , r. a. , & franklin , b. e. 1994 , , 429 , l13 carlberg , r. g. , pritchet , c. j. , & infante , l. 1994 , , 435 , 540 carlberg , r. g. , et al . 2000 , , 532 , l1 cassata , p. , et al . 2005 , , 357 , 903 chary , r. , & elbaz , d. 2001 , , 556 , 562 choi , p. i. , et al . 2006 , , 637 , 227 conselice , c. j. 1997 , , 109 , 1251 conselice , c. j. , bershady , m. a. , & jangren , a. 2000 , , 529 , 886 conselice , c. j. , gallagher , j. s. , iii , & wyse , r. f. g. 2002 , , 123 , 2246 conselice , c. j. 2003 , , 147 , 1 conselice , c. j. , bershady , m. a. , dickinson , m. , & papovich , c. 2003 , , 126 , 1183 conselice , c. j. , chapman , s. c. , & windhorst , r. a. 2003 , , 596 , l5 conselice , c. j. , blackburne , j.a . , & papovich , c. 2005 , apj , 620 , 564 conselice , c. j. 2006 , , 638 , 686 cox , t.j . , et al . 2006 , , submitted , astro - ph/0503201 dale , d. a. , helou , g. , contursi , a. , silbermann , n. a. , & kolhatkar , s. 2001 , , 549 , 215 dasyra , k. m. , et al . 2006 , , 638 , 745 efron , b. 1981 , biometrika,68 , 589 efron , b. , & tibshirani , r. 1986 , stat . , 1,54 elbaz , d. , cesarsky , c. j. , chanial , p. , aussel , h. , franceschini , a. , fadda , d. , & chary , r. r. 2002 , , 384 , 848 fadda , d. , jannuzi , b. t. , ford , a. , & storrie - lombardi , l. j. 2004 , , 128 , 1 fadda , d. , et al . 2006 , , 131 , 2859 fazio , g. g. , et al . 2004 , , 154 , 10 kennicutt , r. c. , jr . 1998 , , 36 , 189 lacy , m. , et al . 2004 , , 154 , 166 lacy , m. , et al . 2005 , , 161 , 41 lagache , g. , et al . 2004 , , 154 , 112 lavery , r. j. , remijan , a. , charmandaris , v. , hayes , r. d. , & ring , a. a. 2004 , , 612 , 679 le fvre , o. , et al . 2000 , , 311 , 565 le floch , e. , et al . 2005 , , 632 , 169 lilly , s. j. , le fevre , o. , hammer , f. , & crampton , d. 1996 , , 460 , l1 lin , l. , et al . 2004 , , 617 , l9 lin , l. , et al . 2006 , , accepted , astro - ph/0607272 lotz , j , m. , davis , m. , faber , s. m. , guhathakurta , p. , gwyn , s. , huang , j. , koo , d. c. , le floch , e. et al . 2006 , , submitted . madau , p. , pozzetti , l. , & dickinson , m. 1998 , , 498 , 106 melbourne , j. , koo , d. c. , & le floch , e. 2005 , , 632 , l65 mihos , j. c. , & hernquist , l. 1996 , , 464 , 641 patton , d. r. , pritchet , c. j. , yee , h. k. c. , ellingson , e. , & carlberg , r. g. 1997 , , 475 , 29 patton , d. r. , carlberg , r. g. , marzke , r. o. , pritchet , c. j. , da costa , l. n. , & pellegrini , p. s. 2000 , , 536 , 153 patton , d. r. , et al . 2002 , , 565 , 208 patton , d. r. , grant , j. k. , simard , l. , pritchet , c. j. , carlberg , r. g. , & borne , k. d. 2005 , , 130 , 2043 rieke , g. h. , et al . 2004 , , 154 , 25 sanders , d. b. , soifer , b. t. , elias , j. h. , madore , b. f. , matthews , k. , neugebauer , g. , & scoville , n. z. 1988 , , 325 , 74 schweizer , f. 1982 , , 252 , 455 thompson , r. i. , weymann , r. j. , & storrie - lombardi , l. j. 2001 , , 546 , 694 shim , h. , i m , m. , pak , s. , choi , p. , fadda , d. , helou , g. , & storrie - lombardi , l. 2006 , , 164 , 435 wolf , c. , et al . 2005 , , 630 , 771 xu , c. k. , sun , y. c. , & he , x. t. 2004 , , 603 , l73 yee , h. k. c. , & ellingson , e. 1995 , , 445 , 37 zepf , s. e. , & koo , d. c. 1989 , , 337 , 34
|
by combining the 0.12 square degree f814w _ hubble space telescope _ ( _ hst _ ) and _ spitzer _ mips 24 imaging in the first look survey ( fls ) , we investigate the properties of interacting and merging mid - infrared bright and faint sources at @xmath0z@xmath1 .
we find a marginally significant increase in the pair fraction for mips 24 detected , optically selected close pairs , pair fraction@xmath2 at z@xmath31 , in contrast to @xmath4 at @xmath50.4 , while galaxies below our 24 mips detection limit show a pair fraction consistent with zero at all redshifts . additionally , 24 detected galaxies with fluxes @xmath6mjy are on average five times more likely to be in a close galaxy pair between @xmath0z@xmath1 than galaxies below this flux limit . using the 24 flux to derive the total far - ir luminosity we find that paired galaxies ( early stage mergers ) are responsible for @xmath7 of the ir luminosity density resulting from star formation at @xmath8 while morphologically classified ( late stage ) mergers make up @xmath9
this implies that @xmath10 of the infrared luminosity density and in turn @xmath11 of the star formation rate density at @xmath8 can be attributed to galaxies at some stage of a major merger or interaction .
we argue that , close pairs / mergers in a lirg / ulirg phase become increasingly important contributers to the ir luminosity and star formation rate density of the universe at @xmath120.7 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the motivation for this project is to design a novel optical system for quasi - real time alignment of tracker detector elements used in high energy physics ( hep ) experiments . fox - murphy _ et.al . _ from oxford university reported their design of a frequency scanned interferometer ( fsi ) for precise alignment of the atlas inner detector @xcite . given the demonstrated need for improvements in detector performance , we plan to design and prototype an enhanced fsi system to be used for the alignment of tracker elements in the next generation of electron - positron linear collider detectors . current plans for future detectors require a spatial resolution for signals from a tracker detector , such as a silicon microstrip or silicon drift detector , to be approximately 7 - 10 @xmath0@xcite . to achieve this required spatial resolution , the measurement precision of absolute distance changes of tracker elements in one dimension should be on the order of 1 @xmath0 . simultaneous measurements from hundreds of interferometers will be used to determine the 3-dimensional positions of the tracker elements . in this paper , we describe ongoing r&d in frequency scanned interferometry ( fsi ) to be applied to alignment monitoring of a detector s charged particle tracking system , in addition to its beam pipe and final - focus quadrupole magnets . the university of michigan group has constructed several demonstration fsis with the laser light transported by air or single - mode optical fiber , using single - fiber and dual - laser scanning techniques , and dual - laser with dual - channel for initial feasibility studies . absolute distance was determined by counting the interference fringes produced while scanning the laser frequency . the main goal of the demonstration systems was to determine the potential accuracy of absolute distance measurements that could be achieved under both controlled and realistic conditions . secondary goals included estimating the effects of vibrations and studying error sources crucial to the absolute distance accuracy . two multiple - distance - measurement analysis techniques were developed to improve distance precision and to extract the amplitude and frequency of vibrations . under laboratory conditions , a measurement precision of @xmath1 50 nm was achieved for absolute distances ranging from 0.1 meters to 0.7 meters by using the first multiple - distance - measurement technique . the second analysis technique has the capability to measure vibration frequencies ranging from 0.1 hz to 100 hz with amplitude as small as a few nanometers , without a _ priori _ knowledge@xcite . the multiple - distance - measurement analysis techniques are well suited for reducing vibration effects and uncertainties from fringe & frequency determination , but do not handle well the drift errors , such as from thermal effects . we describe a dual - laser system intended to reduce the drift errors and show some results under realistic conditions . the dual - channel fsi is used to make sanity checks of the displacement of the detector simultaneously . dual lasers with oppositely scanned frequency directions permit cancellation of many systematic errors , making the alignment robust against vibrations and environmental disturbances . we also report on progress using a dual - channel dual - laser fsi with prototype . under realistic environmental conditions , a precision of about 0.2 - 0.3 microns was achieved for a distance of about 57 cm for the prototype . the intensity @xmath2 of any two - beam interferometer can be expressed as @xmath3 , where @xmath4 and @xmath5 are the intensities of the two combined beams , and @xmath6 and @xmath7 are the phases . assuming the optical path lengths of the two beams are @xmath8 and @xmath9 , the phase difference is @xmath10 , where @xmath11 is the optical frequency of the light , and c is the speed of light . for a fixed path interferometer , as the frequency of the laser is continuously scanned , the optical beams will constructively and destructively interfere , causing `` fringes '' . the number of fringes @xmath12 is @xmath13 , where @xmath14 is the optical path difference between the two beams , and @xmath15 is the scanned frequency range . the optical path difference ( opd for absolute distance between beamsplitter and retroreflector ) can be determined by counting interference fringes while scanning the laser frequency . schematic of an optical fiber fsi system.,width=302 ] if small vibration and drift errors @xmath16 occur during the laser scanning , then @xmath17 , @xmath18/2\pi = d_{true}\delta\nu / c + [ \epsilon(t)\nu(t)/c - \epsilon(t0)\nu(t0)/c]$ ] , assuming @xmath19 , @xmath20 , @xmath21 , the measured distance can be written as , @xmath22 a schematic of the fsi system with a pair of optical fibers is shown in figure [ fsi ] the light source is a new focus velocity 6308 tunable laser ( 665.1 nm @xmath23 675.2 nm ) . a high - finesse ( @xmath24 ) thorlabs sa200 f - p is used to measure the frequency range scanned by the laser . the free spectral range ( fsr ) of two adjacent f - p peaks is 1.5 ghz , which corresponds to 0.002 nm . a faraday isolator was used to reject light reflected back into the lasing cavity . the laser beam was coupled into a single - mode optical fiber with a fiber coupler . data acquisition is based on a national instruments daq card capable of simultaneously sampling 8 channels at a rate of 250 ks / s / ch with a precision of 12-bits . omega thermistors with a tolerance of 0.02 k and a precision of 0.01 @xmath25 are used to monitor temperature . the apparatus is supported on a damped newport optical table . in order to reduce air flow and temperature fluctuations , a transparent plastic box was constructed on top of the optical table . pvc pipes were installed to shield the volume of air surrounding the laser beam . inside the pvc pipes , the typical standard deviation of 20 temperature measurements was about @xmath26 . temperature fluctuations were suppressed by a factor of approximately 100 by employing the plastic box and pvc pipes . detectors for hep experiments must usually be operated remotely for safety reasons because of intensive radiation , high voltage or strong magnetic fields . in addition , precise tracking elements are typically surrounded by other detector components , making access difficult . for practical hep application of fsi , optical fibers for light delivery and return are therefore necessary . the beam intensity coupled into the return optical fiber is very weak , requiring ultra - sensitive photodetectors for detection . considering the limited laser beam intensity used here and the need to split into many beams to serve a set of interferometers , it is vital to increase the geometrical efficiency . to this end , a collimator is attached to the optical fiber , the density of the outgoing beam from the optical fiber is increased significantly . the return beams are received by another optical fiber and amplified by a si femtowatt photoreceiver with a gain of @xmath27 . for a fsi system , drifts and vibrations occurring along the optical path during the scan will be magnified by a factor of @xmath28 , where @xmath11 is the average optical frequency of the laser beam and @xmath29 is the scanned frequency range . for the full scan of our laser , @xmath30 . small vibrations and drift errors that have negligible effects for many optical applications may have a significant impact on a fsi system . a single - frequency vibration may be expressed as @xmath31 , where @xmath32 , @xmath33 and @xmath34 are the amplitude , frequency and phase of the vibration , respectively . if @xmath35 is the start time of the scan , eq . [ eq : rdist ] can be re - written as @xmath36/c \label{eq : rdist}\end{aligned}\ ] ] if we approximate @xmath37 , the measured optical path difference @xmath38 may be expressed as @xmath39 \times \sin[\pi f_{vib}(t+t_0)+\phi_{vib } ] \label{eq : rdist_vib}\end{aligned}\ ] ] where @xmath40 is the true optical path difference in the absence of vibrations . if the path - averaged refractive index of ambient air @xmath41 is known , the measured distance is @xmath42 . if the measurement window size @xmath43 is fixed and the window used to measure a set of @xmath44 is sequentially shifted , the effects of the vibration will be evident . we use a set of distance measurements in one scan by successively shifting the fixed - length measurement window one f - p peak forward each time . the arithmetic average of all measured @xmath44 values in one scan is taken to be the measured distance of the scan ( although more sophisticated fitting methods can be used to extract the central value ) . for a large number of distance measurements @xmath45 , the vibration effects can be greatly suppressed . of course , statistical uncertainties from fringe and frequency determination , dominant in our current system , can also be reduced with multiple scans . averaging multiple measurements in one scan , however , provides similar precision improvement to averaging distance measurements from independent scans , and is faster , more efficient , and less susceptible to systematic errors from drift . in this way , we can improve the distance accuracy dramatically if there are no significant drift errors during one scan , caused , for example , by temperature variation . this multiple - distance - measurement technique is called slip measurement window with fixed size , shown in figure [ multi_dist ] . however , there is a trade off in that the thermal drift error is increased with the increase of @xmath45 because of the larger magnification factor @xmath46 for a smaller measurement window size . the schematic of two multiple - distance - measurement techniques . the interference fringes from the femtowatt photoreceiver and the scanning frequency peaks from the fabry - perot interferometer(f - p ) for the optical fiber fsi system recorded simultaneously by daq card are shown in black and red , respectively . the free spectral range(fsr ) of two adjacent f - p peaks ( 1.5 ghz ) provides a calibration of the scanned frequency range.,width=415 ] in order to extract the amplitude and frequency of the vibration , another multiple - distance - measurement technique called slip measurement window with fixed start point is used , as shown in figure [ multi_dist ] . in eq . [ eq : rdist_vib ] , if @xmath35 is fixed , the measurement window size is enlarged one f - p peak for each shift , an oscillation of a set of measured @xmath44 values indicates the amplitude and frequency of vibration . this technique is not suitable for distance measurement because there always exists an initial bias term , from @xmath35 , which can not be determined accurately in our current system . the typical measurement residual versus the distance measurement number in one scan using the above technique is shown in figure [ rdist](a ) , where the scanning rate was 0.5 nm / s and the sampling rate was 125 ks / s . measured distances minus their average value for 10 sequential scans are plotted versus number of measurements ( @xmath45 ) per scan in figure [ rdist](b ) . the standard deviations ( rms ) of distance measurements for 10 sequential scans are plotted versus number of measurements ( @xmath45 ) per scan in figure [ rdist](c ) . it can be seen that the distance errors decrease with an increase of @xmath45 . the rms of measured distances for 10 sequential scans is 1.6 @xmath0 if there is only one distance measurement per scan ( @xmath47 ) . if @xmath48 and the average value of 1200 distance measurements in each scan is considered as the final measured distance of the scan , the rms of the final measured distances for 10 scans is 41 nm for the distance of 449828.965 @xmath0 , the relative distance measurement precision is 91 ppb . distance measurement residual spreads versus number of distance measurement @xmath45 ( a ) for one typical scan , ( b ) for 10 sequential scans , ( c ) is the standard deviation of distance measurements for 10 sequential scans versus @xmath45 . the frequency and amplitude of the controlled vibration source are 1 hz and 9.5 nanometers , ( d ) magnification factor versus number of distance measurements , ( e ) distance measurement residual versus number of distance measurements , ( f ) corrected measurement residual versus number of distance measurements.,title="fig:",width=245 ] distance measurement residual spreads versus number of distance measurement @xmath45 ( a ) for one typical scan , ( b ) for 10 sequential scans , ( c ) is the standard deviation of distance measurements for 10 sequential scans versus @xmath45 . the frequency and amplitude of the controlled vibration source are 1 hz and 9.5 nanometers , ( d ) magnification factor versus number of distance measurements , ( e ) distance measurement residual versus number of distance measurements , ( f ) corrected measurement residual versus number of distance measurements.,title="fig:",width=264 ] the standard deviation ( rms ) of measured distances for 10 sequential scans is approximately 1.5 @xmath0 if there is only one distance measurement per scan for closed box data . by using the multiple - distance - measurement technique , the distance measurement precisions for various closed box data with distances ranging from 10 cm to 70 cm collected are improved significantly ; precisions of approximately 50 nanometers are demonstrated under laboratory conditions , as shown in table 1 . all measured precisions listed in table [ tab : dist_single_laser ] are the rms s of measured distances for 10 sequential scans . two fsi demonstration systems , air fsi and optical fiber fsi , were constructed for extensive tests of multiple - distance - measurement technique , air fsi means fsi with the laser beam transported entirely in the ambient atmosphere , optical fiber fsi represents fsi with the laser beam delivered to the interferometer and received back by single - mode optical fibers . .[tab : dist_single_laser ] distance measurement precisions for various setups using the multiple - distance - measurement technique . [ cols="^,^,^,^,^ " , ] one possible silicon tracker alignment system is shown in figure [ sid_alignment ] for a generic tracker . the top left plot shows lines of sight for alignment in the r - z plane of the tracker barrel , the top right plot for alignment in x - y plane of the tracker barrel , the bottom plot for alignment in the tracker forward region . red lines / dots show the point - to - point distances need to be measured using fsis . there are 752 point - to - point distance measurements in total for the alignment system . more studies are needed to optimize the distance measurements grid . et.al._ , `` frequency scanning interferometry in atlas : remote , multiple , simultaneous and precise distance measurements in a hostile environment '' , meas . sci . technol.15 ( 11 ) : 2175 - 2187 ( 2004 ) hai - jun yang _ et.al._ , `` high - precision absolute distance measurement using dual - laser frequency scanned interferometry under realistic conditions '' , nucl . instrum . & meth . a575 ( 2007 ) 395 - 401 . [ physics/0609187 ]
|
in this paper , we report high - precision absolute distance and vibration measurements performed with frequency scanned interferometry .
absolute distance was determined by counting the interference fringes produced while scanning the laser frequency .
high - finesse fabry - perot interferometers were used to determine frequency changes during scanning .
a dual - laser scanning technique was used to cancel drift errors to improve the absolute distance measurement precision . a new dual - channel fsi demonstration system is also presented which is an interim stage toward practical application of multi - channel distance measurement . under realistic conditions ,
a precision of 0.3 microns was achieved for an absolute distance of 0.57 meters .
a possible optical alignment system for a silicon tracker is also presented .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
over the years , computation modeling and hydrodynamic calculations of supernova explosions due to core - collapse of massive stars have improved our theoretical understanding of how compact objects form . at present , two types of supernovae are actively discussed in the literature : ( i ) after iron is synthesized within a massive star , the core completes its final nuclear burning state . it then becomes degenerate and keeps growing in mass due to silicon shell burning outside the core . eventually , it crosses the chandrasekhar mass limit and ensues collapse . when the core reaches nuclear density , it bounces back and explodes the star as a supernova . ( bethe 1990 ; mezzacappa 2005 ; woosley & bloom 2006 ; kotake et al 2006 ; janka et al . ( ii ) on the other hand , if the core is not massive enough to ignite neon , nuclear burning within the core stops . the core becomes degenerate due to cooling by neutrino emission . because of carbon shell burning outside the core , the mass of the core approaches the chandrasekhar limit . when the core density and temperature rise above a certain threshold , electron capture by nuclei ( such as @xmath3 , @xmath4 , @xmath5 , and @xmath6 ) takes place , decreasing degenerate electron pressure support . as a result , the core contracts rapidly and eventually bounces back , driving a supernova explosion . this scenario is referred to as electron capture supernova ( ecs ) ( miyaji et al . 1980 ; nomoto 1987 ) . although the general idea of ecs has been established for several decades , the mechanism of how electron capture occurs in these relatively less massive cores is not yet settled . regardless of the supernova mechanism , it is expected that some asymmetries develop during the collapse , imparting a recoil kick to the nascent neutron star ( see harrison et al . 1993 ; frail et al . 1994 ; lyne & loimer 1994 ) . the observed proper motions of 233 galactic pulsars show that there is a class of neutron stars receiving high recoil velocities at birth ( hobbs et al . 2005 ) . on the other hand , pfahl et al . ( 2002 ) studied the orbital parameters of low eccentric be / x - ray binaries , and pointed out that these neutron stars could only have received low kick velocities ( < 50 km / s ) at the time of formation . there is thus evidence for the existence of two different neutron star formation mechanisms ( van den heuvel 2004 , 2007 ) , potentially distinguishable by the magnitude of the natal kick of the neutron star . currently , there are 8 dns observed in our galaxy . their orbital parameters are summarized in table 1 . the magnitude of the natal kick and the progenitor mass of the second - born neutron star ( ns2 ) in these systems , except psr j1906 + 0746 , have been studied previously by willems et . al . ( 2004 , 2006 ) , thorsett et . al . ( 2005 ) , stairs et al . ( 2006 ) , and wang et al . ( 2006 ) . in their study , wang et al . ( 2006 ) studied the formation of _ all _ dns except psr j1906 + 0746 , and derived constraints on the natal kick magnitude and progenitor mass of ns2 by studying the orbital dynamics of asymmetric supernova explosions , but not their kinematic history . the other authors focused on psr b1534 + 12 , psr b1913 + 16 , and psr j0737 - 3039 and used both orbital dynamics of asymmetric supernova explosions and the measured proper motion of these systems to constrain the natal kick magnitude and progenitor mass of ns2 . the authors traced the motion of the systems in the galaxy backwards in time and assumed each crossing of the binary through the galactic disk within the age of the dns to be a possible birth site of the dns . however , the backward calculation of the motion in the galaxy requires the knowledge of the radial velocity which can not be measured . willems et al . ( 2004 , 2006 ) , thorsett et al ( 2005 ) and stairs et al . ( 2006 ) therefore considered possible radial velocities drawn from either a uniform or gaussian radial velocity distribution . _ _ in this paper , we avoid the assumption of a present - day radial velocity distribution by carrying out monte carlo simulations of the motion of the observed dns in the galaxy that is forward in time instead of backward . for this purpose , we distribute populations of dns progenitors in the galaxy according to a double exponential distribution function . each system in the populations is assigned a velocity equal to the vector sum of the local galactic rotational velocity and an isotropic kick velocity with a magnitude generated from the kick velocity distribution function , derived from the supernova orbital dynamics constraints for each of the observed systems . compared to willems et al.(2004 ) , thorsett et al . ( 2005 ) , and wang et al . ( 2006 ) , we also relax the constraint that ns2 s progenitor needs to be more massive than 2.1 m@xmath7 , which is a convectional limit for a helium star that explodes in a core collapse supernova . instead , we restrict the progenitor to be more massive than ns2 only . furthermore we study _ all _ eight known dns in our galaxy , using the up - to - date observational parameters listed in table 1 . _ _ the methodology of the adopted analysis is outlined in more detail in 2 , while results for the individual systems and comparison with earlier studies are discussed in 3 . in 4 , we summarize our results and discuss their implications for the supernova forming the second - born neutron star in the observed dns . @l@ccccc@ccc@c@c@c@ psr b1534 + 12 & 15 37 09.96 & 11 55 55.55 & 1.02 & 1.34(1 ) & -25.05(2 ) & 250 & 1.3332(10 ) & 1.3452(10 ) & 3.28 & 0.274 & 25(155 ) @xmath8 3.8 + psr b1913 + 16 & 19 15 28.00 & 16 06 27.40 & 8.3(1.4 ) & -3.27(35 ) & -1.04(42 ) & 110 & 1.4408(3 ) & 1.3873(3 ) & 2.80 & 0.617 & 18(162 ) @xmath8 6 + psr j0737 - 3039 & 07 37 51.25 & -30 39 40.71 & 1.15 & -3.82(62 ) & 2.13(23 ) & 210 & 1.337(5 ) & 1.250(5 ) & 1.26 & 0.0878 & < 15 + psr j1518 + 4904 & 15 18 16.80 & 49 04 34.25 & 0.625 & -0.67(4 ) & -8.53(4 ) & 20000 & @xmath9 & @xmath10 & 24.7 & 0.249 & + psr j1756 - 2251 & 17 56 46.63 & -22 51 59.40 & 2.5 & -0.7(2 ) & & 443 & 1.312(17 ) & @xmath11 & 2.70 & 0.181 & + psr j1811 - 1736 & 18 11 55.03 & -17 36 37.70 & 6.0 & & & 1830 & @xmath12 & @xmath13 & 40.7 & 0.828 & + psr j1829 + 2456 & 18 29 34.60 & 24 56 19.00 & 1.2 & & & 12400 & @xmath14 & @xmath15 & 6.36 & 0.139 & + psr j1906 + 0746 & 19 06 48.67 & 07 46 28.60 & 5.4 & & & 0.112 & 1.365(18 ) & 1.248(18 ) & 1.75 & 0.0853 & + tauris @xmath16 van den heuvel ( 2004 ) reviewed the general scenario for forming dns . it starts with a binary consisting of two massive zams stars ( @xmath17 ) . the primary star , which is initially the more massive one , will leave the main sequence first and explode as the first supernova , in which it forms the first - born neutron star ( ns1 ) in this binary . there may be mass transfer from the primary to the secondary star before that supernova takes place . as the secondary star leaves the main sequence , wind mass transfer from the secondary to ns1 can occur , leading to the formation of a x - ray binary . while ns1 is accreting mass from its companion , it is spun up or `` recycled '' , and its spin axis is expected to align with the orbital angular momentum . later on , the x - ray binary may end up in a common envelope phase , during which ns1 is engulfed by the extended envelope of its companion . the orbit shrinks and circularizes rapidly due to the frictional forces acting on ns1 as it moves through the envelope of the companion . the lost orbital energy is deposited into the gas envelope , causing it to become gravitationally unbound and get ejected , leaving behind the companion s bare helium core . the helium star eventually explodes in a supernova and becomes ns2 , orbiting the recycled , milli - second pulsar ns1 in a compact binary orbit . mass transfer from the helium star to ns1 can also occur prior to the second supernova explosion . here , our goal is to use the current dns binary properties listed in table 1 to constrain as best possible the magnitude of the natal kick and the mass of the progenitor of ns2 in the 8 known dns in our galaxy at the time of the second supernova . we also predict the unknown radial velocity of each system , as well as the total transverse velocity and ns1 spin tilt angle for those systems where these properties have not been measured yet . we achieve these goals by means of monte carlo simulations , incorporating orbital dynamics during supernova explosions and kinematic of binaries in the galactic potential , as detailed in what follows . we start our analysis by deriving a probability distribution function ( pdf ) for the natal kick of ns2 , which depends on the semi - major axis and orbital eccentricity immediately after the second supernova ( @xmath18 and @xmath19 respectively ) . as the tidal interaction between two neutron stars in dns is extremely weak ( bildsten & cutler 1992 ) , the major mechanism that changes the orbital semi - major axis and eccentricity of dns is the emission of gravitational waves . to account for gravitational radiation driven orbital evolution , we use the current observational parameters as initial conditions and integrate equations ( 35 ) and ( 36 ) in junker & schfer ( 1992 ) backwards in time . because of the uncertainty in the true age of the observed dns ( see kiziltan et al . 2009 for details ) , we randomly draw the integration time @xmath20 from a uniform distribution of ages between 0 and @xmath21 , where @xmath21 is the characteristic age of the dns . if @xmath21 is greater than the age of our galaxy , which is about 10 gyr , we use 10 gyr as the upper limit for @xmath20 instead . for psr b1534 + 12 , psr b1913 + 16 , and psr j0737 - 3039 , different upper limits are adopted for @xmath20 , as discussed in more detail in the next section . once @xmath18 and @xmath22 are known from the gravitational radiation integration , we randomly draw the natal kick velocity ( @xmath23 ) from a uniform distribution of values between 0 and 2500 km / s ( i.e. , we assume no prior kick distribution derived from pulsar samples analysis ) , and the kick direction @xmath24 from an isotropic distribution . here @xmath25 is the polar angle between the natal kick velocity and the instantaneous orbital velocity of ns2 s progenitor at the time of the second supernova explosion , and @xmath26 is the corresponding azimuthal angle ( see figure 1 in kalogera 2000 for a graphical representation ) . using the conservation laws of orbital energy and angular momentum expressed by equations ( 19 ) and ( 20 ) in willems et al . ( 2005 ) , we obtain the orbital semi - major axis ( @xmath27 ) immediately before the second supernova and the mass of ns2 s helium star ( @xmath28 ) . the pre - supernova orbit is assumed to be tidally circularized at the time of the second supernova explosion . on the other hand , the spin tilt angle of ns1 , which is the angle between ns1 axis and the post supernova orbital angular momentum axis , can be calculated by following equation ( 3 ) in kalogera ( 2000 ) . also , using equations ( 12 ) in the same paper , we find the polar angle @xmath29 between the natal kick and the pre - supernova orbital angular momental axis . immediately before the second supernova explosion , there may be mass transfer from ns2 s progenitor to ns1 . if the radius @xmath30 of ns2 s progenitor is greater than its roche lobe radius @xmath31 , which is approximated by equation ( 2 ) in eggleton ( 1983 ) , roche lobe overflow ( rlo ) takes place . since ns2 s progenitor is a bare helium star before the second supernova , we use equation ( a10 ) in kalogera & webbink ( 1998 ) to calculate @xmath30 . note that the left hand side of that equation should be @xmath32 instead of @xmath33 . in order to have dynamically stable rlo , so that ns2 s progenitor does not coalesce with ns1 to form an isolated black hole , the mass ratio @xmath28 to @xmath34 needs to be less than 3.5 ( ivanova et al . even if ns2 s progenitor does not overflow its roche lobe , there may be mass transfer by stellar wind . in addition , @xmath28 is restricted to be less than 8 m@xmath7 , which is a conventional limit that a helium star will become a neutron star instead of a black hole ( see , e.g. , figure 1 in belczynski et al . 2002 ; table 16.4 in tauris & van den heuvel 2004 ) . the conservation laws of orbital energy and angular momentum allow for real solutions only for @xmath28 , @xmath23 , @xmath25 , @xmath26 , and @xmath27 values satisfying the orbital dynamics constraints given by equation ( 21)(27 ) in willems et al . ( 2005 ) . for psr b1534 + 12 , psr b1913 + 16 , and psr j0737 - 3039 , these constraints are supplemented with the measured ns1 spin tilt angle constraints . applying these constraints together with the constraints imposing stability of any mass transfer taking place right before the second supernova explosion yields a pdf of the natal kick velocity of ns2 . a corresponding pdf for the kick velocity @xmath35 imparted to the binary s center of mass is obtained using equations ( 28)(32 ) in willems et al . in all past studies , kinematic analyses have either been skipped ( wang et al . 2006 ) or simplified to one dimension ( piran & shaviv 2004 ) or have suffered by ad hoc assumptions about the un - measurable radial velocities . willems et al . ( 2006 ) used general dns population synthesis models to guide this assumption . in this sutdy we employ a new method that still uses forward in time kinematic simulations , but they are tied to each observed system instead of being general . we perform a monte carlo simulation of the motion of dns in the galactic potential . for this purpose , we adopt a reference frame with origin at the galactic center , with @xmath36-axis pointing to the northern galactic pole , and with the @xmath37-axis pointing in the direction from the sun to the galactic center . with respect to this reference frame , the sun is located at @xmath38 kpc ( ghez et al . 2008 , gillessen et al . 2009 , reid et al . 2009 , joshi 2007 ) and has a peculiar velocity of @xmath39km / s ( bienaym 1999 ) . with the given @xmath40 and peculiar velocity , the sun is moving at a velocity @xmath41 km / s . to simulate the motion of dns in the galaxy , we randomly distribute a population of newly formed dns in the galactic disk according to a double exponential distribution function @xmath42 here r and z are the cylindrical galactic coordinates , @xmath43 and @xmath44 are the galactic scale length and height respectively . the distribution is normalized to unity by setting @xmath45 . since the progenitors of dns are distributed in the same way as the massive stellar binaries in our galaxy , we choose @xmath43 = 2.8 kpc and @xmath44 = 0.07 kpc ( see joshi 2007 ) . for all of the observed dns except psr b1534 + 12 , we use a population of @xmath210@xmath46 simulated binaries . we use a population of @xmath210@xmath47 in the analysis of psr b1534 + 12 , due to the limiting chance of kicking it to the currently high galactic altitude and simultaneously satisfying the constraints derived from the orbital dynamics of supernova explosions . if the scale height @xmath44 were larger than 70 pc , it would reduce the difficulty in satisfying the galactic dynamics constraints of psr b1534 + 12 . a smaller population of simulated binaries would then be needed for the analysis of this system . the initial center - of - mass velocity of the dns is obtained by summing up the kick velocity @xmath48 imparted to the binary center of mass during the second supernova explosion and the local galactic rotational velocity @xmath49 . given the unknown orientation of the binary orbit in the galactic reference frame , the systemic kick velocity @xmath48 is assumed to be distributed isotropically in space . the magnitude of @xmath48 is drawn from the velocity distribution derived in the previous step specific to each of the observed systems . starting from the randomly generated initial position and velocity , the motion of the dns is calculated by numerically integrating the equations of motion from the dns birth time to the current epoch . the equations of motion are derived from the galactic potential of carlberg & innanen ( 1987 ) with updated model parameters of kuijken & gilmore ( 1989 ) . the dns birth time is chosen according to the dns age @xmath20 generated in the first step of the analysis . lastly , we use the measured current position of the observed dns to further constraint the ranges of possible @xmath23 and @xmath28 . we set the tolerance of this constraint to be within 100 pc from the observed position . we test the dependence of our results on this tolerance level by repeating the analysis with a 10 pc , or 50 pc tolerance , but our results do not show any significant changes . for psr j1518 + 4904 and psr j1756 - 2251 , the position constraint is supplemented with the measured current proper motion constraint . for psr b1534 + 12 , psr b1913 + 16 , and psr j0737 - 3039 , the position constraint is supplemented with both the measured proper motion and ns1 spin tilt angle constraints . the analysis outlined in the previous section allows us to derive pdfs for the kick velocity @xmath23 , ns2 s immediate pre - supernova progenitor mass @xmath28 , the spin tilt @xmath50 of ns1 , the systemic radial velocity @xmath51 , and the current transverse velocity @xmath52 . our primary interest is in @xmath23 and @xmath28 , because they can shed light on the type of supernova that formed ns2 , which will be discussed in more detail in the next section . the remaining quantities give us some predictions on currently unavailable parameters of the 8 known dns in our galaxy . instead of focusing on the most likely values as studies have done in the past , we derive confidence levels for the unknown parameters and they are summarized in table 2 . since the @xmath23 and @xmath28 constraints are correlated , we present our results in the form of confidence level plots of a 2 dimensional joint probability distribution of @xmath23 - @xmath28 , for each dns binary , in addition to the corresponding 1 dimensional pdf s of @xmath23 and @xmath53 . as pointed out by willems et al ( 2006 ) , physical parameter constraints are affected by the dimensionality of the pdf used , because of inherent correlations between parameters and projection effects . this variation is stronger when one solely examines the most likely values ( i.e. , values at the peak of the pdf ) and weakens as broader confidence levels are considered . in what follows , the confidence levels of @xmath23 and @xmath28 are derived from the 2d joint pdf . to calculate the 2d confidence levels , we first bin the data into a 2d grid and normalize the total probability within the 2d grid to unity . then , we add the probability of each bin from the highest to the lowest , until the sum best matches the desired confidence level . on the other hand , the ranges of @xmath51 , @xmath52 , and @xmath50 are derived from the corresponding 1d pdf . in this case , the confidence level is found by having the shortest range of bins that has a total probability that best matches the desired confidence level . lccccc psr b1534 + 12 & 170 - 260 ( 150 - 270 ) & 2.00 - 2.90 ( 1.34 - 3.40 ) & @xmath54 - 50 ( @xmath55 - 145 ) & 121 @xmath8 6 & 25.0 @xmath8 3.8 + psr b1913 + 16 & 200 - 410 ( 190 - 450 ) & 1.40 - 3.30 ( 1.39 - 5.00 ) & @xmath56 - 85 ( @xmath57 - 315 ) & 135 @xmath8 25 & 18 or 162 @xmath8 6 + psr j0737 - 3039 & 5 - 50 ( 5 - 120 ) & 1.25 - 1.55 ( 1.25 - 1.90 ) & 8 - 44 ( @xmath58 - 76 ) & @xmath59 & 0 - 2.0 ( 0 - 7.8 ) + psr j1518 + 4904 & 20 - 80 ( 5 - 110 ) & 1.80 - 3.30 ( 1.49 - 4.70 ) & @xmath60 - 8 ( @xmath61 - 48 ) & 25.4 @xmath8 3.6 & 2 - 18 ( 0 - 32 ) + psr j1756 - 2251 & 5 - 80 ( 5 - 185 ) & 1.25 - 1.90 ( 1.25 - 2.65 ) & @xmath62 - 32 ( @xmath55 - 98 ) & 6.5 - 28.5 ( 6.0 - 64.5 ) & 0 - 3.4 ( 0 - 16.4 ) + psr j1811 - 1736 & 0 - 170 ( 0 - 310 ) & 1.11 - 4.00 ( 1.11 - 8.00 ) & 0 - 112 ( @xmath63 - 210 ) & 30 - 114 ( 4 - 202 ) & 0 - 10.2 ( 0 - 57.6 ) + psr j1829 + 2456 & 5 - 85 ( 5 - 225 ) & 1.40 - 2.70 ( 1.36 - 6.10 ) & @xmath60 - 26 ( @xmath64 - 66 ) & 12 - 54 ( 4 - 102 ) & 0 - 8.5 ( 0 - 30.0 ) + psr j1906 + 0746 & 5 - 170 ( 5 - 510 ) & 1.25 - 2.90 ( 1.25 - 4.80 ) & 12 - 103 ( @xmath65 - 297 ) & 89 - 174 ( 21 - 376 ) & 0 - 6.7 ( 0 - 51.5 ) + in this binary , the masses of ns1 and ns2 are 1.33 @xmath66 and 1.35 @xmath66 respectively . the orbital period is @xmath67 days , with an eccentricity @xmath68 . the angle between ns1 s spin axis and the orbital angular momentum axis is @xmath69 or @xmath70 . the binary is currently located out of the galactic plane , at @xmath71 and @xmath72 , and 1.02 kpc away from us . the observed proper motion @xmath73 mas yr@xmath74 and @xmath75 mas yr@xmath74 . at the measured distance this implies @xmath76 km s@xmath74 and @xmath77 km s@xmath74 , in the reference frame of the sun . also , we use the spin down age , which is 210 myr given by arzoumanian et al . 1999 , as the upper limit of psr b1534 + 12 s age , instead of the characteristic age . furthermore , the current orientation of the orbital angular momentum axis is also constrained for this sytem . the orientation axis can be described by the inclination angle @xmath78 and the angle of the orbital ascending node on the plane of the sky ( @xmath79 ) . the sine of the inclination angle is @xmath80 = 0.975 ( stairs et al . the angle @xmath79 is @xmath81 or @xmath82 , reckoned north through east . the two solutions correspond to @xmath83 and @xmath84 respectively ( bogdanov et al . 2002 ) . since tidal effects between the two neutron stars are negligible , the orbital angular momentum axis keeps a fixed orientation in space . from the kinematic analysis of the galactic motion , the constraints on the present - day orbital inclination and proper motion can therefore be used to determine the systemic kick component @xmath85 parallel to the post - supernova orbital angular momentum axis . following kalogera ( 1996 ) and wex et al . ( 2000 ) , @xmath86 can also be expressed analytically as @xmath87 where @xmath88 @xmath89 @xmath90 here , @xmath91 ( @xmath92 ) is the @xmath37 , @xmath93 , or @xmath36 component of the natal kick velocity in the frame centered on ns2 s progenitor ( see figure 1 in kalogera for a graphical representation ) . note that the sign of @xmath86 needs to be consistent with the sign of @xmath94 . hence , through equation ( 2 ) , the @xmath86 derived from observations gives us a constraint on the y and z components of the natal kick , as well as @xmath50 . the results are shown in figures 1 and 2 . right before the second supernova , there must have been rlo from the progenitor of ns2 to ns1 , as noted previously by thorsett et al . as seen in the 2d @xmath23-@xmath28 joint probability distribution in figure 1 , the kick imparted to ns2 at birth is between 150 and 270 km / s , while the progenitor mass @xmath28 is between 1.34 and 3.40 m@xmath0 , both at 95% confidence . as seen in figure 2 , polar kicks ( i.e. , @xmath95 or @xmath96 ) are unlikely , and the possible kick directions are asymmetric about the pre - supernova orbital plane ( i.e. , @xmath29 = 90@xmath97 ) . this asymmetry comes from the constraint on the orientation of the post - supernova orbital angular momentum axis on the sky . moreover , we arrive at the same conclusion as thorsett et al . ( 2005 ) that @xmath98 is very unlikely . the current radial velocity of this binary is between -110 and 145 km / s , at 95% confidence , and it is likely to be moving away from us . within the observationally constrained range , the @xmath52 pdf is roughly flat . when comparing with recent analyses by willems et al . ( 2004 ) , thorsett et al . ( 2005 ) , and wang et . 2006 ) , our limits on @xmath23 and @xmath28 are consistent wtih thorsett et al . ( 2005 ) , and are more constrained than willlems et al . ( 2004 ) and wang et al . ( 2006 ) , because willems et al . ( 2004 ) did not have the spin tilt @xmath50 measurement and wang et al . ( 2006 ) did not have the proper motion and position constraints in their analysis . our limits on the angle @xmath29 , which is the polar angle between the natal kick direction and the pre - supernova orbital angular momentum , are not as constrained as the limits shown in thorsett et al . ( 2005 ) . through recent further investigation , it has been found that this difference is due to a code typo related to the galactic latitude of the source , which fortunately affected significantly only the @xmath29 results ( dewey & stairs 2010 , private communication ) . the masses of ns1 and ns2 in this binary are 1.44 @xmath66 and 1.38 @xmath66 respectively . the orbital period is @xmath99 days , with an eccentricity @xmath100 . the angle between the ns1 spin axis and the post supernova orbital angular momentum axis is @xmath101 or @xmath102 . at the current time , this binary is about 300 pc away the galactic plane , at @xmath103 and @xmath104 , and 8.3 kpc away from us . the observed proper motion in right ascension and declination is @xmath105 mas yr@xmath74 and @xmath106 mas yr@xmath74 . at the measured distance , this implies @xmath107 km s@xmath74 and @xmath108 km s@xmath74 , in the reference frame of the sun . in addition , instead of the characteristic age , we use the spin down age , which is 80 myr ( arzoumanian et al . 1999 ) , as the upper limit on psr b1913 + 16 s age . the results are displayed in figures 3 and 4 . as shown in the 2d @xmath23-@xmath28 joint probability distribution in figure 3 , the kick imparted to ns2 at birth is between 190 and 450 km s@xmath74 , while the progenitor mass @xmath28 is between 1.4 and 5.0 m@xmath0 , both at 95% confidence . as seen from figure 4 , both polar and planar kick directions are allowed . the current radial velocity is between @xmath57 and 315 km s@xmath74 at 95% confidence . within the observationally constrained range , the @xmath52 pdf is roughly flat , with a slight preference for transverse velocities of @xmath2120 km / s . furthermore , although observations show a degeneracy in @xmath50 at @xmath109 and @xmath110 , our analysis shows that @xmath50 is much more likely to be @xmath109 . in addition , although our confidence intervals of @xmath23 and @xmath28 are consistent with willems et al . ( 2004 ) and wang et . al . ( 2006 ) , our reported intervals are tighter , because willems et al . ( 2004 ) and wang et al . ( 2006 ) reported the allowed ranges of @xmath23 and @xmath28 instead of confidence intervals . the masses of the two neutron stars in this binary are 1.34 and 1.25 m@xmath0 , where the more massive one is ns1 . they are orbiting each other with a period of 0.102 days , and an eccentricity of 0.0878 . at the current time , the binary is close to the galactic plane , at @xmath111 and @xmath112 . through direct measurement of geometric parallax , deller et al . ( 2009 ) measured the distance to psr j0737 - 3039 to be 1.15 kpc , and the proper motion in r.a . and declination to be @xmath1133.82 and 2.13 mas yr@xmath74 respectively . at the measured distance , this implies @xmath114 km s@xmath74 and @xmath115 km s@xmath74 , in the reference frame of the sun . in addition , lorimer et al . ( 2007 ) constrained the possible age range to be either @xmath116 or @xmath117 myr . we adopt these age ranges in our analysis of psr j0737 - 3039 . as the radio pulse profile of psr j0737 - 3039a ( i.e. , ns1 ) has not changed significantly through six years of observation , the spin tilt angle of j0737 - 3039a with respect to the post supernova orbital angular momentum axis is believed to be small . however , there is no well constrained value for this angle . ferdman et al . ( 2008 ) and breton ( 2008 ) argue that there are two possible models to explain the observed pulse profile : the single - cone and two - cone emission model . in this context , the number of emission cones means the number of emitting cones that intersect with our line of sight . each model gives a different spin tilt angle estimate . ferdman et al . ( 2008 ) estimate this angle to be < 15@xmath97 for the single - cone model and < 6.1@xmath97 for the two - cone model , both at 68.3 % confidence . due to this uncertainty , we take the spin tilt angle @xmath50to be < 15@xmath97 in our work , which corresponds to 68.3 % confidence value in the single - cone model and 95.4 % confidence value in the two - cone model in ferdman et al ( 2008 ) . the results are displayed in figures 5 - 7 . as shown in the 2d @xmath23-@xmath28 joint probability distribution , the kick recieved by ns2 ( i.e. , psr j0737 - 3039b ) at birth is between 5 and 120 km s@xmath74 , while the progenitor mass @xmath28 is between 1.25 and 1.90 m@xmath0 , both at 95% confidence . the probability distribution of the kick direction is symmetric about the pre - supernova orbital plane ( i.e. , @xmath118 = 90@xmath97 ) , and planar kicks are more favorable than polar kicks . the pdf of the semi - major axis immediately before the second supernova ( @xmath27 ) peaks at two narrow ranges : @xmath119 and @xmath120 @xmath121 . the first peak corresponds to an age of @xmath116 myr , and the second peak to an age of @xmath117 myr . at the current time , psr j0737 - 3039 is more likely to be moving away from us , with a radial velocity between @xmath58 and 76 km / s with 95% confidence . within the observationally constrained range , the @xmath52 pdf peaks at @xmath220 km / s . furthermore , rlo from ns2 s progenitor to ns1 before the second supernova explosion can not be avoided . to test the dependence of the presented results on the adopted upper limit for @xmath50 , we repeated the analysis for the upper limits of @xmath122 and @xmath123 . the results do not show any significant differences compared to those obtained with the @xmath124 upper limit . this is not surprising , because the @xmath50 pdf as shown in figure 8 , which is obtained from the same constraints as before except raising the upper limit of @xmath50 to 60@xmath97 , peaks strongly at 0@xmath97 and decays exponentially with increasing values of @xmath50 . comparatively , our confidence intervals on @xmath23 and @xmath28 are consistent with stairs et al . ( 2006 ) and willems et al . ( 2006 ) , but are much more constrained than wang et al . ( 2006 ) , mainly because they did not have the proper motion constraint in their analyses . the masses of the two neutron stars in this binary are 0.72 and 2.00 m@xmath0 , where the less massive one is ns1 . however , each one has an uncertainty of @xmath20.5 m@xmath0 . in our analysis , we choose @xmath34 to be 1.23 and @xmath125 to be 1.49 , so that @xmath34 is greater than the smallest measured neutron star mass , which is about 1.2 @xmath126 . the characteristic age is 20 gyr , which is larger than a hubble time . thus , we set the upper limit on psr j1518 s age to be 10 gyr instead . the two neutron stars are orbiting each other with a period of 8.63 days , and an eccentricity of 0.249 . at the current time , this binary is out of the galatic plane , at @xmath127 and @xmath128 , and at a distance of 0.625 kpc from the sun . the proper motion in r.a . and dec . is @xmath129 and @xmath130 mas yr@xmath74 . at the measured distance , this implies @xmath131 and @xmath132 km s@xmath74 , in the reference frame of the sun . there is no spin tilt measurement available for this system . the results are shown in figures 9 and 10 . as seen in the 2d @xmath23-@xmath28 joint probability distribution , the kick imparted to ns2 at birth is between 5 and 110 km / s , while the progenitor mass @xmath28 is between 1.5 and 4.7 m@xmath0 , both at 95% confidence . both polar and planar kick directions are possible , but the planar kicks have a higher probability . the spin tilt angle @xmath50 is smaller than 32@xmath97 at 95% confidence . the radial velocity is found to be between -62 and 48 km / s at 95% confidence , and the system is currently likely moving towards us . within the observationally constrained ranges , the @xmath52 pdf is roughly flat . our limits of @xmath23 and @xmath28 are much more constrained than those of wang et al . ( 2006 ) , primarily because they did not have the proper motion and galactic position constraints included in their analysis . the masses of the two neutron stars in this binary are 1.31 and 1.26 m@xmath0 , where the more massive one is ns1 . the characteristic age is 443 myr . the two neutron stars are in an orbit with a period of 0.320 days , and an eccentricity of 0.181 . at the current time , this binary is close to the galactic plane , at @xmath133 and @xmath134 , and at a distance of 2.5 kpc away from us . only proper motion @xmath135 is measured , which is -0.7 mas / yr . at the measured distance , this implies @xmath136 km / s , in the reference frame of the sun . there is currently no spin tilt measurement available for this system . the results are shown in figures 11 and 12 . from the 2d @xmath23-@xmath28 joint probability distribution , the kick received by ns2 at birth is between 5 and 185 km s@xmath74 , while the progenitor mass @xmath28 is between 1.25 and 2.65 m@xmath0 , both with 95% confidence . both polar and planar kicks are allowed , but planar kicks are more favorable than polar ones . the spin tilt angle of ns1 is smaller than 16.4@xmath97 with 95% confidence . at the current time and at the measured distance , psr j1756 - 2251 has a total transverse velocity between 6.0 and 64.5 km s@xmath74 , and a radial velocity between @xmath55 and 98 km s@xmath74 , both with 95% confidence . furthermore , this binary is equally likely to be moving towards us as away from us . when comparing with wang et al . ( 2006 ) , we have tighter constrained limits on @xmath23 and @xmath28 , due to the fact that we have the additional constraint on the proper motion @xmath135 . the two neutron stars have masses of 1.62 and 1.11 m@xmath0 , where the more massive one is ns1 . however , each mass measurement has an uncertainty of @xmath20.5 m@xmath0 . psr j1811 has a characteristic age of 1830 myr . the neutron stars are in an orbit with a period of 18.8 days , and an eccentricity of 0.828 . at the current time , this binary is close to the galactic plane , at @xmath137 and @xmath138 , and at a distance of 6.0 kpc away from us . like psr j1756 - 2251 , there are currently no spin tilt nor proper motion measurements available . the results are displayed in figures 13 and 14 . the 2d @xmath23-@xmath28 joint probability distribution shows that the kick given to ns2 at birth is less than 310 km s@xmath74 , while the progenitor mass @xmath28 is between 1.11 and 8.0 m@xmath0 , both with 95% confidence . both polar and planar kicks are allowed , but the planar kicks are more favorable than the polar ones . the spin tilt angle @xmath50 is smaller than 57.6@xmath97 with 95% confidence . at the measured distance , the total transverse velocity of psr j1811 - 1736 at the current epoch is between 4 and 202 km s@xmath74 , and the radial velocity between -90 and 210 km s@xmath74 with 95% confidence , which means this binary is likely moving away from us . although our upper boundary of the 95% confidence intervals of @xmath28 is the same as the upper limit found by wang et al . ( 2006 ) , the value of our lower boundary is less than their lower limit . this is because wang et al . ( 2006 ) put a conservative lower limit of 2.1 m@xmath0 on the progenitor mass of a neutron star , which we did not . on the other hand , our limits on @xmath23 are more constrained than those of wang et al . ( 2006 ) , as we compute confidence levels instead of allowed ranges of solution as wang et al . ( 2006 ) . the two neutron stars of this binary have masses of 1.14 and 1.36 m@xmath0 , where the less massive one is ns1 . however , there is an uncertainty of @xmath20.5 m@xmath0 in the mass measurements . the characteristic age is 12.4 gyr . for the same reason as psr j1518 + 4904 , we set an the upper limit of 10 gyr on the age of this binary in our analysis . the neutron stars are orbiting each other with a period of 1.18 days , and an orbital eccentricity of 0.139 . at current time , this binary is out of the galactic plane , at @xmath139 and @xmath140 , and it is 1.2 kpc away from us . although there is no spin tilt nor proper motion measurements available at current time , lorimer et al . ( 2005 ) derived that the transverse velocity is smaller than 118 km s@xmath74 . the results are shown in figure 15 . the @xmath23-@xmath28 joint probability distribution shows that the kick given to ns2 at birth is between 5 and 225 km s@xmath74 , while the progenitor mass @xmath28 is between 1.4 and 6.1 m@xmath0 with 95% confidence . the pdfs of @xmath50 and @xmath29 are similar to those of psr j1756 - 2251 . both polar and planar kicks are allowed , but planar kicks are more probable . the spin tilt angle @xmath50 is less than 30.0@xmath97 with 95% confidence . at the known distance , the total transverse velocity of psr j1829 + 2456 at the current time is between 4 and 102 km s@xmath74 , and the radial velocity between @xmath64 and 66 km s@xmath74 with 95% confidence . in addition , the probability that this system is moving towards the sun is @xmath220% higher than that it is moving away from the sun . comparing our @xmath23 and @xmath28 results with wang et al . ( 2006 ) , we have more constrained limits , as they do not include any kinematic constraints . the masses of the neutron stars are 1.37 and 1.25 m@xmath0 , where the more massive one is ns1 . unlike the other 7 binaries , the observed pulsar is ns2 . this implies the binary is relatively young , with a charateristic age of 0.112 myr only . the neutron stars are in an orbit with a period of 0.166 days , and an orbital eccentrirctiy of 0.0853 . at current time , this binary is close to the galactic plane , at @xmath141 and @xmath142 , and it is 5.4 kpc away from us . again , there are no spin tilt nor proper motion measurements available yet . the results are displayed on figure 16 . the @xmath23-@xmath28 joint probability distribution shows that the kick given to ns2 at birth is between 5 and 510 km s@xmath74 , while the progenitor mass @xmath28 is between 1.25 and 4.80 @xmath126 with 95% confidence . the @xmath29 and @xmath50 pdf s are similar to those of psr j1756 - 2251 . both polar and planar kicks are possible , with the planar kicks having a higher probability . the polar angle @xmath50 is less than 51.5@xmath97 with 95% confidence . at the measured distance , the total transverse velocity of psr j1906 + 0746 at the current time is between 21 and 376 km s@xmath74 , and the radial velocity is between @xmath65 and 297 km s@xmath74 with 95% confidence . it is most likely that this binary is moving away from us . furthermore , right before its supernova , the progenitor of ns2 must have been overflowing its roche lobe , transferring mass to ns1 . our limits on @xmath23 and @xmath28 are tighter constrained than wang et al . ( 2006 ) , because they did not have a detailed analysis on this system . instead , they calculated the allowed @xmath23 with an assumption that @xmath28 was between 2.1 and 8.0 m@xmath0 . the anisotropic mass and neutrino ejection in a supernova explosion gives a recoil natal kick to the newly formed neutron star . iron core collapse supernova and ecs produce natal kicks of different strength because of the different physical conditions of the nascent neutron star immediately before the supernovae . there are several reviews ( bethe 1990 , mezzacappa 2005 , woosley & bloom 2006 , kotake et al 2006 , and janka et al . 2007 ) that summarize our current understanding of the iron core collapse supernova . after iron is synthesized in the core , the massive star reaches the final stage of hydrostatic nuclear burning , because the synthesis of any heavier elements requires energy instead of releasing energy . when the iron core grows by silicon shell burning around the core to a mass above the chandrasekhar mass limit , electron degeneracy pressure can not support the core any more and it starts to collapse . the iron core continues to collapse until it reaches nuclear density ( i.e. , @xmath143 g @xmath144 ) . since nuclear matter has a much lower compressibility , the collapse decelerates and the core bounces back because of the increased nuclear matter pressure . this drives a shock wave and eventually explodes the outer layers of the massive star away in a supernova explosion . on the other hand , the ecs scenario is described in the reviews of miyaji et al . ( 1980 ) and nomoto ( 1987 ) . after carbon burning within the core of a post main sequence massive star , a @xmath145 core is formed . neon can not be ignited if the mass of the @xmath145 core is less than the critical mass of 1.37 m@xmath0 for neon ignition . as the @xmath145 core is cooled by neutrino emissions , it becomes strongly degenerate . when the mass of the core grows by carbon shell burning and approaches the chandrasekhar mass limit , electron captures onto @xmath6 and @xmath5 take place . this leads to a decrease in the electron degeneracy pressure and the chandrasekhar mass limit , which induces a rapid contraction of the core . the rapid core contraction ignites the oxygen deflagration , which incinerates materials into nuclear statistical equilibrium ( nse ) . due to the rapid electron capture onto nse elements , the collapse of the core accelerates . similar to the iron core collapse picture , when the core reaches the nuclear matter density , it bounces back and drives a shock wave that eventually explodes the star in a supernova . gutirrez et al . ( 2005 ) show that the abundance of @xmath6 in @xmath145 core needs to be greater than 15% , in order to have nse developed through electron captures . however , in simulations of evolving massive agb stars by siess ( 2007 ) , the abundance of @xmath6 after carbon burning is smaller by a factor of @xmath210 than what is required to drive an explosion by electron captures . therefore , the mechanism of developing an ecs is not a settled issue yet . in addition , podsiadlowski et al . ( 2004 ) studied the binary evolution and the dynamics of core collapse , and suggested that ecs could only occur in interacting binaries , but not in single stars . in recent simulations of supernova explosions from the collapse of @xmath145 cores by kitaura et al . ( 2006 ) , they found these supernovae are powered by neutrino heating and neutrino - driven wind of the nascent neutron star . scheck et al . ( 2006 ) showed that the shock wave in these supernovae can propagate outwards on a relatively short timescale after the rebound of the core , which means the non - radial hydrodynamics instabilities do not have time to merge and grow to global asymmetry before the anisotropic pattern freezes out in the accelerating outward motion of the shock wave . as a result , _ _ the natal kick of an ecs due to anisotropic mass ejection is expected to be fairly small(see scheck et al . furthermore , as the @xmath145 core that eventually explodes in an ecs is not massive enough to ignite neon , _ _ the mass of the neutron star progenitor should be less than the progenitors that explode in an iron core collapse supernova . _ _ _ _ the present analysis is the first to include _ all _ currently known dns systems and account of all orbital and kinematic constraints . we also employ a novel method for dealing with the uncertainty due to the un - measurable radial velocities and we focus on the derived pdfs and associated confidence levels , instead of just the most likely values , which can be misleading at times . our results are summarized in table 2 ; the derived constraints are consistent with earlier studies ( when available ) but typically limit parameters to narrower ranges , and in this sense they represent the best available constraints on the formation of neutron stars in dns systems . _ _ in the context of our current understanding of massive star core collapse , we can use our results to draw a number of conclusions : + ( 1 ) psr j0737 - 3039 has a @xmath23 upper limit of 120 km / s and a @xmath28 upper limit of 1.9 @xmath126 at 95% confidence . therefore , the formation of ns2 ( i.e. , pulsar b ) likely occurred through an ecs event . this is consistent with the speculation of podsiadlowski et al . ( 2005 ) based on the space velocity and orbital eccentricity of psr j0737 - 3039 . for psr b1534 + 12 and psr b1913 + 16 , @xmath23 are 150 - 270 km / s and 190 - 450 km / s ( 95% ) respectively , which means ns2 in both system must have received a significant recoil natal kick at birth . psr b1534 + 12 has a @xmath28 upper limit of 3.4 @xmath126 ( 95% ) , while psr b1913 + 16 has a @xmath28 upper limit of 5.0 @xmath28 ( 95% ) . because of the relatively high @xmath23 ranges and @xmath28 upper limits , ns2 in both systems are probably formed through an iron core collapse supernova event . + ( 2 ) psr j1518 + 4904 has a @xmath146 upper limit of 110 km / s and a @xmath28 upper limit of 4.7 @xmath126 at 95% confidence . even though ns2 likely received a low recoil velocity at birth , we can not firmly conclude which type of supernova occurred during the formation of ns2 , due to the relatively high @xmath28 upper limit . + ( 3 ) psr j1756 - 2251 has a @xmath23 upper limit of 80 km / s and a @xmath28 upper limit of 1.90@xmath126 at 60% confidence , hence the formation of ns2 might possibly relate to an ecs event . however , the @xmath23 and @xmath28 upper limits are 185 km / s and 2.65 @xmath126 respectively at 95% confidence , which means the likelihood of ns2 formed through an ecs event is lower than that of psr j0737 - 3039 . + ( 4 ) psr j1811 - 1736 , psr j1829 + 2456 , and psr j1906 + 0746 have @xmath23 upper limits of several hundreds of km / s , and @xmath28 upper limits ranging from 4.80 to 8.0 @xmath126 ( i.e. , the conventional limit on neutron star progenitor mass ) at 95% confidence . since both low and high @xmath23 and @xmath28 are possible , we can not conclude which type of supernova event is more favorable for the formation history of ns2 in these systems . + ( 5 ) for all of the known dns except psr b1534 + 12 , all available constraints are consistent with imparting a polar or planar kick to ns2 at birth . for psr b1534 + 12 , the kick direction is constrained to be polar . + ( 6 ) for psr b1534 + 12 , psr j0737 - 3039 and psr j1906 + 0746 , the pre - supernova orbit is so tight that they can not avoid a rlo from ns2 progenitor to ns1 just before the second supernova explosion . + ( 7 ) furthermore , despite the low orbital eccentricity in psr j1829 + 2456 and psr j1906 + 746 , ns2 could have a high progenitor mass and have received a high recoil natal kick at birth , as shown in the 2d joint pdf of these systems ( see figures 15 , and 16 ) . + ( 8) we also tested the dependence of our psr j0737 - 3039 results on the current upper limit of the spin - orbital misalignment angle @xmath50 of ns1 ( i.e. , pulsar a ) . as shown in figure 8 , the @xmath50 pdf peaks strongly at @xmath147 , so choosing an upper limit of 15@xmath97 or 60@xmath97 does not affect our results noticably . arzoumanian , z. , cordes , j. m. , & wasserman , i. 1999 , , 520 , 696 belczynski , k. , kalogera , v. , bulik , t. 2002 , , 572 , 407 bethe , h. a. 1990 , rev . phys . , 62 , 801 bienaym , o. 1999 , , 341 , 86 bildsten , l. , cutler , c. 1992 , , 400 , 175 bogdanov , s. , pruszyska , m. , lewandowski , w. , & wolszczan , a. 2002 , , 581 , 495 breton , r. p. 2009 thesis at mcgill university burgay , m. , damico , n. , possenti , a. , manchester , r. n. , lyne , a. g. , et al . 2003 , , 426 , 531 champion , d. j. , lorimer , d. r. , mclaughlin , m.a . , cordes , j. m. , arzoumanian , z. , et al . 2004 , , 350 , l61 champion , d. j. , lorimer , d. r. , mclaughlin , m. a. , xilourix , k. m. , arzoumanian , z. , et al . 2005 , , 363 , 929 carlberg , r. g. , & innanen , k. a. 1987 , , 94 , 666 corongiu , a. , kramer , m. , stappers , b. w. , lyne , a. g. , jessner , a. , et al . 2007 , , 462 , 703 damour , t. , & taylor , j. h. 1991 , , 366 , 501 deller , a. t. , bailes , m. , & tingay , s. j. 2009 , science , 323 , 1327 eggleton , p. p. 1983 , , 268 , 368 faulkner , a. j. , kramer , m. , lyne , a. g. , manchester , r. n. , mclaughlin , m. a. , et al . 2005 , , 618 , l119 ferdman , r.d . 2008 , phd . thesis at the university of british columbia ferdman , r. d. , stairs , i. h. , kramer , m. , manchester , r. n. , lyne , a. g. , et al . in 40 years of pulsars : millisecond pulsar , magnetars and more . aip conference proceedings , 2008 , pp . 474 - 478 frail , d. a. , goss , w. m. , & whiteoak , j. b. z. 1994 , , 437 , 781 ghez , a. m. , salim , s. , weinberg , n. n. , lu , j. r. , do , t. , et al . , 2008 , , 689 , 1044 gillessen , s. , eisenhauer , f. , trippe , s. , alexander , t. , genzel , r. , et al . 2009 , , 692 , 1075 gutirrez , j. , canal , r. , garca - berro , e. 2005 , , 435 , 231 harrison , p. a. , lyne , a. g. , & anderson , b. 1993 , , 261 , 113 hobbs , g. , lorimer , d. r. , lyne , a. g. , kramer , m. 2005 , , 360 , 974 hulse , r. a. , & taylor , j. h. 1975 , , 195 , l51 ivanova , n. , belczynski , k. , kalogera , v. , rasio , f. a. , taam , r. e. 2003 , , 592 , 475 janka , h .- th . , langanke , k. , marek , a. , martnez - pindeo , g. , & mller , b. 2007 , , 442 , 38 janssen , g. h. , stappers , b. w. , kramer , m. , nice , d. j. , jessner , a. , et al . 2008 , , 490 , 753 joshi , y. c. 2007 , , 378 , 768 junker , w. , & schfer , g. 1992 , , 254 , 146 kalogera , v. 1996 , , 471 , 352 kalogera , v. 2000 , , 541 , 319 kalogera , v. , & webbink , r. f. 1998 , , 493 , 351 kasian , l. in 40 years of pulsars : millisecond pulsars , magnetars and more . aip conference proceedings , 2008 , pp . 485 - 487 kitaura , f. s. , jankka , h .- th . , & hillebrandt , w. 2006 , , 450 , 345 kiziltan b. , & thorsett s. e. 2009 , arxiv0909.1562 konacki , m. , wolszczan , a. , & stairs , i. h. 2003 , , 589 , 495 kotake , k. , sato , k. , & takahashi , k. 2006 , rep . , 69 , 971 kuijken , k. , & gilmore , g. 1989 , , 239 , 571 lorimer , d. r. , stairs , i. h. , freire , p. c. , cordes , j. m. , camilo , f. , et al . 2006 , , 640 , 428 lorimer , d. r. , freire , p. c. c. , stairs , i. h. , kramer , m. , mclaughlin , m. a. , et al . 2007 , , 379 , 1217 lyne , a. g. , & loimer , d. r. 1994 , , 369 , 127 lyne , a. g. , camilo , f. , manchester , r. n. , bell , j. f. , kaspi , v. m. , et al . 2000 , , 312 , 698 lyne , a. g. , burgay , m. , kramer , m. , possenti , a. , manchester , r. n. , et al . 2004 , science , 303 , 1153 mezzacappa , a. 2005 , annu . nucl . part . , 55 , 467 miyaji , s. , nomoto , k. , yokoi , k. , & sugimoto , d. 1980 , , 32 , 303 nomoto , k. 1987 , , 332 , 206 pfahl , e. , rappaport , s. , podsiadlowski , p. , spruit , h. 2002 , , 574 , 364 podsiadlowski , ph . , langer , n. , poelarends , a. j. t. , rappaport , s. , heger , a. , et al . 2004 , , 612 , 1044 podsiadlowski , ph . , dewi , j. d. m. , lesaffre , p. , miller , j. c. , newton , w. g. , et al . 2005 , , 361 , 1243 piran , t. , shaviv , n. j. 2004 , , 94 , 051102 reid , m. j. , menten , k. m. , zheng , x. w. , brunthaler , a. , moscadelli , l. , et al . 2009 , , 700 , 137 scheck , l. , plewa t. , janka , h .- th . , kifonidis , k. , & mller , e. 2004 , , 92 , 011103 scheck , l. , kifonidis , k. , janka , h .- th . , & mller , e. 2006 , , 457 , 963 siess , l. 2007 , , 476 , 893 stairs , i. h. 2004 , science , 304 , 547 stairs , i. h. in 40 years of pulsars : millisecond pulsars , magnetars and more . aip conference proceedings , 2008 , pp . 424 - 432 stairs , i. h. , thorsett , s. e. , & arzoumanian , z. 2004 , , 93 , 141101 stairs , i. h. , thorsett , s. e. , dewey , r. j. , kramer , m. , & mcphee , c. a. 2006 , , 373 , l50 stairs , i. h. , thorsett , s. e. , taylor , j. h. , & wolszczan , a. 2002 , , 581 , 501 tauris , t. m. , & van den heuvel e. p. j. 2004 , in compact stellar x - ray sources , ed . w. h. g. lewin & m. van der klis ( cambridge : cambridge univ . press ) , arxiv : astro - ph/0303456 taylor , j. h. , fowler , l. a. , & mcculloch , p. m. 1979 , , 277 , 437 taylor , j. h. , hulse , r. a. , fowler , l. a. , gullahorn , g. e. , & rankin , j. m. 1976 , , 206 , l53 taylor , j. h. , & weisberg , j. m. 1982 , , 253 , 908 taylor , j. h. , & weisberg , j. m. 1989 , , 345 , 434 thorsett , s. e. , & chakrabarty , d. 1999 , , 512 , 288 thorsett , s. e. , dewey , r. j. , & stairs , i. h. 2005 , , 619 , 1036 van den heuvel , e. p. j. , in the proceedings of the 5th integral workshop on the integral universe , 2004 , p. 185 van den heuvel , e. p. j. , in the multicolored landscape of compact objects and their explosive origins . aip conference proceedings , 2007 , pp . 589 - 606 wang , c. , lai , d. , & han , j. l. 2006 , , 639 , 1007 wex , n. , kalogera , v. , & kramer , m. 2000 , , 528 , 401 willems , b. , kalogera , v. , & henninger , m. 2004 , , 616 , 414 willems , b. , henninger , m. , levin , t. , ivanova , n. , kalogera , v. , et al . 2005 , , 625 , 324 willems , b. , kaplan , j. , fragos , t. , kalogera , v. , & belczynski , k. 2006 , , 74 , 043003 wolszczan , a. 1991 , , 350 , 688 woosley , s. e. , & bloom , j. s. 2006 , , 44 , 507
|
since the discovery of the first double neutron star ( dns ) system in 1975 by hulse and taylor , there are currently 8 confirmed dns in our galaxy . for every system , the masses of both neutron stars , the orbital semi - major axis and eccentricity are measured , and proper motion is known for half of the systems . using the orbital parameters and kinematic information , if available , as constraints for _ all system _ _ , we investigate the immediate progenitor mass of the second - born neutron star and the magnitude of the supernova kick it received at birth , with the primary goal to understand the core collapse mechanism leading to neutron star formation .
compared to earlier studies , we use a novel method to address the uncertainty related to the unknown radial velocity of the observed systems . for psr b1534 + 12 and psr b1913 + 16 ,
the kick magnitudes are 150 - 270 km / s and 190 - 450 km / s ( with 95% confidence ) respectively , and the progenitor masses of the 2nd born neutron stars are 1.3 - 3.4 m@xmath0 and 1.4 - 5.0 m@xmath0 ( 95% ) , respectively .
these suggest that the 2nd born neutron star was formed by an iron core collapse supernova in both systems . for psr j0737 - 3039 , on the other hand ,
the kick magnitude is only 5 - 120 km / s ( 95% ) , and the progenitor mass of the 2nd born neutron star is 1.3 - 1.9 m@xmath0 ( 95% ) . because of the relatively low progenitor mass and kick magnitude , the formation of the 2nd born neutron star in psr j0737 - 3039 is potentially connected to an electron capture supernova of a massive @xmath1 white dwarf . for
the remaining 5 galactic dns , the kick magnitude ranges from several tens to several hundreds of km / s , and the progenitor mass of the 2nd formed neutron star can be as low as @xmath21.5 m@xmath0 , or as high as @xmath28 m@xmath0 .
therefore in these systems , it is not clear which type of supernova is more likely to form the 2nd neutron star . _ _ _
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the de sitter ( ds ) and anti - de sitter ( ads ) space has a constant curvature with the maximal symmetry of a given dimension . the ( a)ds has thus attracted attention in understanding a quantum nature of spacetime and been applied to different physics . the present accelerating universe with a cosmological constant is an asymptotically pure ds space and the early inflationary universe also underwent a quasi - exponential expansion . the pure ds space has a cosmological horizon and emits the ds radiation with the gibbons - hawking temperature.@xcite the einstein equation for the ds space can be explained by the black hole thermodynamics.@xcite an intriguing feature of the ds radiation is the solitonic nature in the global coordinates in any odd - dimensional ds space.@xcite the discrimination of dimensions for ds radiation can be explained by the stokes phenomenon , in which a pair of instanton actions interferes destructively in odd dimensions and constructively in even dimensions.@xcite the stokes phenomenon can also be explained by the coherent destructive or constructive interference of the superadiabatic particle number.@xcite the one - loop effective action in an electromagnetic field and in a curved spacetime reveals the interplay between quantum electrodynamics ( qed ) and quantum nature of curved spacetime . a strong electromagnetic field changes the quantum vacuum structure,@xcite and an electric field with / without a parallel magnetic field pulls out virtual particles from the dirac sea and creates pairs of particles and antiparticles known as the schwinger effect.@xcite the schwinger effect in ( a)ds space entails the pair production by the electric field and the ds radiation@xcite or the suppression by the negative curvature of ads space.@xcite the one - loop effective action and the schwinger effect has been comprehensively studied in ref . . the @xmath0-wave of a scalar field in the nariari geometry of a rotating ds black hole is equivalent to a charge in a uniform electric field in @xmath1.@xcite interestingly , the near - horizon geometry @xmath2 of an extremal or near - extremal black hole deduces that the emission of charges from the black hole is related to the schwinger effect in @xmath3.@xcite the fermionic current due to the schwinger effect@xcite and the holographic schwinger effect@xcite have been studied in the ds space . the production of particles can be explained by the ( effective ) temperature . the hawking radiation has the bose - einstein or fermi - dirac distribution with the hawking temperature and the ds radiation has a distribution , whose leading boltzmann factor can be given by the gibbons - hawking temperature . the schwinger effect in a constant or pulsed electric field has an effective temperature given by the inverse of the period of the instanton action in the euclidean time.@xcite narnhofer , peter and thirring have shown that the effective temperature for an accelerating observer in ds space is the geometric mean of the unruh temperature and the gibbons - hawking temperature,@xcite which is extended to the ads space by deser and levin,@xcite as shown in fig . recently , cai and kim have shown that the schwinger effect in ( a)ds space also has a thermal interpretation in terms of an effective temperature for the unruh effect for accelerating charge by the electric field and the curvature effect,@xcite as summarized in fig . this implies that the schwinger effect from an extremal black hole , in which the hawking radiation is suppressed due to the zero temperature , may also have such a thermal interpretation and similarly for a near - extremal black hole.@xcite in this paper we elaborate the one - loop effective action of scalar qed in ref . and also advance a method to find the density of states for the one - loop action and the schwinger effect . in particular , the density of states is important to obtain the one - loop qed action density from the integrated one - loop action in the in - out formalism and the rate of pair production from the mean number by the bogoliubov transformation . the one - loop effective action is equivalent to all the one - loop diagrams with even number of photons and gravitons , as shown in fig . [ fig2 ] , whose summation is beyond the computational practicality . however , in the in - out formalism by schwinger and dewitt , the one - loop action is derived from the scattering amplitude between the in- and out - vacua@xcite @xmath4 in fact , the propagator or green function of the charge in the electric field and in the ( a)ds space gives the scattering amplitude . we study the one - loop effective action of a scalar with mass @xmath6 and charge @xmath7 in a uniform electric field in the planar coordinates of @xmath1 space @xmath8 and in @xmath3 space @xmath9 the one - loop action differs from quantum electrodynamic interaction by summing feynman diagrams . the gauge potentials are chosen to have the minkowski space limit ( @xmath10 or @xmath11 ) and the scalar curvature is @xmath12 . then the momentum mode @xmath13 in @xmath1 has the equation @xmath14 where the dimensionless parameters are @xmath15 on the other hand , the energy mode @xmath16 in the @xmath3 space is given by the equation @xmath17 where another set of dimensionless parameters @xmath18 in the @xmath1 space , the positive and negative solutions for the in - vacuum ( @xmath19 ) are given by the whittaker function as @xmath20 with @xmath21 and those for the out - vacuum ( @xmath22 ) are @xmath23 the solutions ( [ ds in - sol ] ) and ( [ ds out - sol ] ) are chosen to satisfy the quantization rule @xmath24 = i$ ] . similarly , the solutions for the in - vacuum ( @xmath25 ) for the tunneling boundary condition are @xmath26 with @xmath27 and those for the out - vacuum ( @xmath28 ) are @xmath29 the solutions ( [ ads in - sol ] ) and ( [ ads out - sol ] ) are chosen to satisfy the quantization rule @xmath30 = i$ ] for the tunneling boundary condition.@xcite using ( [ ds in - sol ] ) and ( [ ds out - sol ] ) , we recalculate the bogoliubov coefficients in the @xmath1 space@xcite @xmath31 and from eqs . ( [ ads in - sol ] ) and ( [ ads out - sol ] ) , we find the bogoliubov coefficients in @xmath3 space @xmath32 hence , the mean numbers of pairs by the schwinger effect are @xmath33 @xmath34 the binding nature of @xmath3 provides an effective mass @xmath35 and thus increases the critical strength @xmath36 while the separating nature of @xmath1 gives @xmath37 and thus lowers the critical strength . note the breitenlohler - freedman bound @xmath38 for the stability of ads against schwinger emission . recently , cai and kim have proposed a thermal interpretation of the schwinger effect in the @xmath5 space . note that the mean number for the ds space is determined by the actions@xcite @xmath39 here the effective temperature and the associated temperature are @xmath40 where the gibbons - hawking temperature and the unruh temperature for the accelerating charge by the electric field are , respectively , @xmath41 on the other hand , the mean number for the ads space is determined by the actions@xcite @xmath42 where @xmath43 in the minkowski limit @xmath44 , the mean number reduces to @xmath45 , as expected . the factor of two is intrinsic nature of the schwinger effect.@xcite the density of states is necessary to derive the qed action density and the pair - production rate in the in - out formalism . the energy - momentum integral of the off - shell mean number gives the density of states , @xmath46 , for a uniform electric field in the minkowski spacetime.@xcite in the case of a uniform magnetic field , the wave packet of landau states gives the density of states , @xmath47 . we now propose another method to find the density of states in the in - out formalism . for that purpose , we note that the field has the fourier integral @xmath48 using the asymptotic form @xmath49 and changing of variable @xmath50 , we rewrite eq . ( [ fourier ] ) as @xmath51 where @xmath52 is a function independent of @xmath53 , and @xmath54 the gaussian integral from the saddle point method gives @xmath55 in the tunneling picture , the decay of the vacuum due to the schwinger effect is determined by the magnitude square of the field @xmath56 and the momentum independent part @xmath57 determines the mean number of created pairs . therefore , the density of states is given by @xmath58 in the minkowski spacetime @xmath59 the density of states becomes @xmath60 . finally , we obtain the pair - production rate per unit two - volume @xmath61 the density of state has been proposed for spinor qed in the ds space.@xcite from the integrated one - loop action ( [ in - out scat ] ) follows the qed action density , which is given by the bogoliubov coefficient , which is in turn expressed by gamma functions with complex arguments @xmath62 to obtain the qed action density , one uses the gamma function in the proper - time integral @xmath63 , \label{gamma fn}\end{aligned}\ ] ] and then applies the schwinger subtraction scheme to regulate divergent terms via the renormalization of the vacuum energy and charge.@xcite remarkably , the consistence relation , @xmath64 , between the imaginary part of the effective action and the pair - production rate , is the consequence of the cauchy s residue theorem applied to the gamma function ( [ gamma fn ] ) . finally , we find the qed action density in the @xmath65 space @xmath66 , \nonumber\\ { \cal l}^{(1)}_{\rm ads } & = & \frac{k^2 \gamma_{\rm ads}}{2(2 \pi ) } \int_{0}^{\infty } \frac{ds}{s } e^ { -\gamma_{\rm ads}s } \cosh(\bar{\omega}_0 s ) \bigl(\frac{1}{\sin(s/2 ) } - \frac{2}{s } - \frac{s}{12 } \bigr ) . \label{(a)ds qed}\end{aligned}\ ] ] the scalar qed action and schwinger effect has been recalculated in the @xmath65 space . a new method has been advanced to find the density of states and to obtain the qed action density and the schwinger pair - production rate in the @xmath65 space , which is the main result . the author thanks christian schubert for useful discussion on the worldline instantons in de sitter space and won kim for drawing the figure . this work was supported by ibs ( institute for basic science ) under ibs - r012-d1 .
|
we review the one - loop effective action in scalar qed and the schwinger effect in a uniform electric field in a two - dimensional ( anti- ) de sitter space .
the schwinger effect has a thermal interpretation in terms of the effective temperature introduced by cai and kim .
we propose a method to find the density of states for the charged scalar and obtain the qed action density and the pair - production rate in the in - out formalism .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the tully - fisher relation ( tully & fisher , 1977 ) is a direct indication of a close relationship between the detected baryons and the total mass in spiral galaxies . the detected baryons consist of the stellar and gaseous content , i.e. the visible mass , and this sets the luminosity profile of the galaxy while the total gravitational mass , which includes the dark matter content ( and possibly a component of baryonic dark matter ) , sets its rotation velocity . numerous studies have been carried out to investigate this relation , crucial in determining extragalactic distances ( e. g. pierce & tully 1988 , tully & pierce 2000 ) , in the study of evolution of galaxies ( e.g. puech et al . 2008 ) and also in giving constraints on cosmological galaxy formation models ( e. g. portinari & sommer - larsen 2007 ) . the tully - fisher relation is undoubtedly a crucial test for galaxy evolution models and although it has been the focus of a number of studies , its origin is still being debated . a few authors argue a cosmological origin ( e. g. avila - reese , firmani & hernndez 1998 ) while others suggest that this relation is regulated by star forming processes ( e. g. silk 1997 ) . on the other hand , the tully - fisher relation is related to the stellar populations of galaxies as it is suggested by its steeper slope when the luminosity is measured in the near infrared ( nir ) bands , when compared to the slope measured in the optical ( e.g. tully & pierce , 2000 ) . the use of nir bands in the tully - fisher relation has shown to be extremely useful , because nir bands present lower internal extinction than optical bands ( verheijen 2001 ) and the mass - to - light ratio is less contaminated by younger stellar populations , giving a better reflection of the stellar mass of the galaxies . luminosities can be converted into stellar masses by scaling them by a given mass - to - light ratio . thus , instead of linking rotation velocities to luminosities , a few authors have chosen to show the correlation of rotation velocities to stellar masses . this relation is called stellar tully - fisher relation ( e.g. bell & de jong 2001 ) . in order to estimate the whole baryonic mass , the gaseous component should be added to the stellar mass . the relation between luminosities and baryonic masses is called the baryonic tully - fisher relation . this relation has been studied by several authors ( e. g. mcgaugh 2000 , verheijen 2001 , geha et al . 2006 , bell & de jong 2001 ) . the slope of the baryonic tf relation is an important test for galaxy formation models . a steeper slope indicates that the baryonic mass of massive galaxies tends to approximately match the total mass of the galaxy . to date , most of the works devoted to study the tully - fisher relation have used compilations of several galaxy surveys , observed in different ways ( hi line profiles and rotation curves ) adding factors of uncertainties due to sample non - homogeneity . beside the problems of using different samples , the use of certain observational techniques may add other uncertainties in the study of the tully - fisher relation . for example , an over or under prediction of the position angle in long - slit observation could produce an erroneous estimate of the maximum velocity of a galaxy , which will be reflected in the tully - fisher relation . in order to avoid the problems listed in the previous paragraph , we have made use of the homogeneous galaxy survey gassendi halpha survey of spirals ( ghasp ) to study the nir , stellar and baryonic tully - fisher relations and their implications for the total mass of galaxies . the ghasp survey represents a large effort to constitute a sample of field galaxies in an homogeneous way . first , a strict isolation criterion has been used to insure the isolation of the galaxies . rather close galaxies have been chosen in order to guarantee a high spatial resolution , compared to hi surveys . high and low inclination objects have been excluded in order to minimize uncertainties in the de - projected rotation curve . second , all ghasp galaxies have been observed using the same instrument ; a scanning fabry - perot attached to a focal reducer at the 193 cm at observatoire de haute provence ( ohp ) . in addition to obtaining data for the whole sample with the same instrument , which is a great advantage to insure homogeneity , the scanning fabry - perot is certainly the most adapted instrument to obtain rotation curves in the most proper way . because it gives a 2d velocity field , we can obtain the rotation curve ( and the maximum velocity ) without any previous assumptions about the position angle or the inclination like it is the case for long - slit observations . this technique avoids , in this way , a great factor of uncertainty that is common to not be taken into account using other techniques . third , the data reduction has been performed in an homogeneous way , in order to derive the rotation curves from the velocity fields in the cleanest and most rigorous possible way ( see epinat et al . 2008a , b for details ) . these three points allowed eliminating ( or at least greatly minimizing ) several problems that previous studies have encountered . using rotation curves to obtain the maximum rotation velocity is a more precise way compared to the hi line width profile technique , used by several others studies . the higher spatial resolution of optical velocity maps , compared to hi , avoids the problem of missing the maximum velocity because of lack of resolution ( beam smearing ) . together with the kinematic information , we have used h and k - band photometry from 2mass survey , mass - to - light ratios derived from stellar population models , hi fluxes and h@xmath2 masses from the literature to perform , for the first time , the nir , stellar and baryonic tully - fisher relations for the ghasp sample . in 2 we describe the data , including the method used to compute the stellar , gaseous and baryonic masses . in 3 , we present the results . in 4 we discuss and compare our results with previous works . finally , we summarize our main findings in 5 . ghasp is the largest sample of spiral and irregular galaxies observed to date using fabry - perot techniques . it consists of 3d h@xmath3 data cubes for 203 galaxies , covering a large range in morphological types and absolute magnitudes . all the ghasp galaxies have been recently reanalyzed in a homogeneous way in epinat et al . ( 2008a , b ) . these authors published velocity fields , monochromatic h@xmath3 images , dispersion velocity maps , rotation curves and maximum rotation velocities ( v@xmath4 ) for each galaxy . a sub - sample of 93 galaxies has been selected by removing from the sample : 1 ) galaxies with radial systemic velocities lower than 3000 km s@xmath5 ( to avoid the effect of the local group infall ) for which no other individual measurements of distances were available ( the references are indicated in epinat et al . 2008b ) and 2 ) galaxies with inclinations lower than 25 degrees for which the uncertainties on the rotational velocity is comparatively high . rotation curves shown in epinat et al . ( 2008a , b ) present a large variety of shapes ( from falling to rising ) and degrees of asymmetry . in order to study the influence of the shape of the rotation curves in the tully - fisher relation , we have made a classification of our sample in three subsamples , i. e. rising , flat and decreasing rotation curves , which will be described with the letters `` r '' , `` f '' and `` d '' , respectively . an `` + '' or ` ' sign has been added if the rotation curve is respectively more or less extended than the optical radius of the galaxy ( r@xmath6 ) . no symbol is added if the radius of the observed rotation curve is barely equal to r@xmath6 . a decreasing or flat rotation curve displays a clear v@xmath4 . this is not the case for a rotation curve which rises up to the very last observed radius , for which @xmath7 is possibly underestimated . this is even worse if the rising rotation curve does not reach the optical radius . in this case , v@xmath4 was computed at r@xmath6 by extrapolating an _ arctan _ function ( @xmath8 , where @xmath9 is the core radius ) to the rotation curves . if observed rotational velocities at r@xmath10r@xmath6 are higher than the modeled value , the observed values have been used . this could be the case if large scale bumps in the inner parts of rotation curves are present . table [ appendix1 ] ( column 8) presents the results of the rotation curve quality assessments . the main assumption necessary to derive a rotation curve from the observed velocity field is that rotation motions are dominant and non circular motions are not part of a large - scale pattern . thus , by construction , a rotation curve provides a measurement , for each radius , of the axi - symmetric component of the gravitational potential well of the galaxy . by consequence , if the motions in the galaxy disk are purely circular , the receding and the approaching sides of a rotation curve should match and the residual velocity field should not display any structure . once the parameters of the rotation curves are properly computed by minimizing the velocity dispersion in the residual velocity field ( epinat et al . 2008 ) , the remaining residuals are the signature of non - circular motions in or out the plane of the disk ( e. g. bars , oval distortions , spiral arms , local inflows and outflows , warps ) , including the intrinsic turbulences of the gas . to quantify the effects of these non - circular motions on the tully - fisher relations we have computed , for each galaxy , two indicators . the first one is based on the asymmetries between both sides of the rotation curves , it quantifies the mean difference of amplitude between the receding and approaching sides of the rotation curve . for each ring centered on the galaxy center , the weighted ( absolute ) velocity difference between both sides is computed . the weight is provided by the number of bins in each ring . each bin is an independent velocity measurement on the velocity field , it may be constituted by @xmath1150 pixels for low signal - to - noise region of the galaxy . depending of the spatial resolution , each ring contains from two to several hundreds bins . due to the fact that their radius are smaller , the central rings contains a number of pixels lower than the outer ones , their weights is thus smaller . the second indicator is based on the mean velocity dispersion extracted from the residual velocity field . this parameter is quantified by computing the average local velocity dispersion on the whole residual velocity field . to not overestimate the weight of non circular motion in slowly rotating systems with respect to high rotators , both indicators have been normalized by the maximum rotation velocity . we found that both indicators show the same trend on the tully - fisher relation , thus we will only illustrate the results using the indicator related to the asymmetries in the rotation curve . we computed the near - infrared magnitudes using 2mass data ( skrutskie et al . 2mass h and k - band data were available for 83 galaxies of the ghasp sub - sample defined in section 2.1 . absolute magnitudes were obtained using : @xmath12 distances ( _ d _ ) were taken from epinat et al . ( 2008b ) . they are computed from the systemic velocities ( from the ned database ) corrected from virgo infall and assuming h@xmath13=75 km s@xmath5 mpc@xmath5 , except when accurate distance measurements where available ( references are listed in epinat et al . the magnitudes m@xmath14 have been computed using the flux within the isophote of 20 mag arcsec@xmath5 ( where uncertainties were taken from 2mass ) . we corrected the magnitude for galactic extinction using the schlegel maps ( schlegel et al . k - corrections ( k@xmath14 ) , extinction due to the inclinations ( a@xmath14 ) and seeing were applied using the method given in masters et al . columns 1 , 2 and 3 in table [ appendix1 ] list the name , h and k - band absolute magnitudes for the sample . k - band luminosities were estimated using l@xmath15=10@xmath16 , where the k - band absolute solar magnitude of 3.41 was taken from allen ( 1973 ) . given the homogeneity of the _ sloan digital sky ( _ sdss _ ) , g and r - band optical magnitudes were extracted from this database . moreover , most of the mass - to - light ratio recipes use b - v colors as an input parameter . for this reason , we have converted g and r - band data into b and v - band magnitudes by using the recipes given in lupton ( 2000 ) . we have extracted the optical size for each galaxy from the _ sdss_. in this case , we used the parameter isoa ( in the r - band ) , which corresponds to the diameter of the isophote where the disk surface brightness profile drops to 25 mag arcsec@xmath17 . these values were converted in radii ( in kpc ) by using the distance to each galaxy . in order to compute the mass - to - light ratio for ghasp galaxies , g - r colors were corrected by galactic extinction using the values given in the _ sdss _ database and then converted into b - v colors . _ sdss _ colors were available for 45 galaxies from which we removed five objects for which their magnitudes and optical radius are obviously incorrect ( compared with the optical radius , and magnitudes , given in hyperleda ) . for other six galaxies , there were no radius measurement . all the analysis including the radius of the galaxies were thus performed with 34 objects . b - band luminosities were estimated by using l@xmath18=10@xmath19 , where the b - band absolute solar magnitude of 5.48 was taken from binney & merrifield ( 1998 ) . the main uncertainty in the study of the stellar and baryonic tully - fisher relations states in the stellar mass - to - light disk ratio @xmath20 . two main methods to estimate this parameter are used . spano et al . ( 2008 ) have modeled the stellar mass distribution of rotation curves , by scaling the r - band surface brightness profile to the rotation curve , obtaining an estimation of @xmath20 . bell & de jong ( 2001 ) have used stellar populations synthesis models to predict a relation between the colors of galaxies and their @xmath20 . although both approaches attempt to compute the same physical parameter , several authors have shown that surprisingly there seems to be no clear correlation between the @xmath20 obtained from these two methods ( e.g. barnes et al . other authors have invoked the modified newton dynamics ( mond ) to obtain the @xmath20 and study its implication on the baryonic tully - fisher relation ( mcgaugh 2005 ) . in this work , we have estimated @xmath20 and stellar masses using stellar population synthesis models recipes . one of them consists of simply fixing the value of @xmath20 . we have compared our results with other works available in the literature following the stellar populations synthesis models given by bell & de jong ( 2001 , b&j ) , bell et al . ( 2003 , be ) and portinari et al . ( 2004 , po ) ( equations [ eq1 ] and [ eq1b ] , [ eq2 ] and [ eq2b ] and [ eq3 ] respectively ) . @xmath21 @xmath22 @xmath23 @xmath24 @xmath25 - 0.115 } l_{k } \label{eq3}\ ] ] b&j and be suggested an uncertainty of 0.1 dex in the @xmath20 estimation . the adopted uncertainty in @xmath20 is larger than the uncertainties of the optical colors . we have adopted the same uncertainty for the po relation . b&j used a scaled salpeter initial mass function ( imf ) . po used a salpeter imf , with masses ranging between 0.1 and 100 m@xmath26 . these models are available for several colors ( to estimate @xmath20 ) and also for the luminosity in several bands ( to estimate the stellar mass ) . we have converted _ sdss _ g - r colors into b - v colors to obtain the @xmath20 parameter by using the recipes listed above . stellar masses were calculated using the k - band luminosities . this band is more likely to be reflective of the stellar mass of galaxies . in order to compare stellar masses derived from the k - band luminosities , we have also used the b - band luminosity , despite the fact that this photometric band could be contaminated with the emission of young stars . we removed from this analysis galaxies for which no _ sdss _ colors were available in the literature . therefore , in the stellar and baryonic analysis , we were left with 45 galaxies in total . k - band luminosities were estimated using l@xmath15=10@xmath16 ( see section 2.2 ) . stellar masses were also estimated using a fixed mass - to - light ratio following mcgaugh et al . ( 2000 , _ mg _ ) . these authors defined the mass - to - light ratio in the k - band as @xmath27 . in this case , the stellar mass is simply : @xmath28 it is interesting to note that gurovich et al . ( 2010 ) estimated the @xmath20 for a sample of local galaxies by modeling their stellar population histories . these authors did not find differences in the tully - fisher relation when the stellar masses were computed by using the modeled @xmath20 or when this parameter was fixed to @xmath20=0.8 ( mcgaugh et al . 2000 ) . the mass of a galaxy is constituted of its content in stars and stellar remnants , gas ( neutral , molecular and a negligible component of ionized gas ) , dust ( usually negligible ) and dark matter . the baryonic mass is the sum of the stellar and gas contents . the total mass in gas , m@xmath29 , is : @xmath30 where m@xmath31 is the neutral gas , m@xmath32 is the mass in helium and metals , m@xmath33 is the mass in molecular hydrogen and m@xmath34 is the ( negligible ) mass in ionized hydrogen . in order to obtain the baryonic mass for the ghasp sample we have calculated the observed hi mass for each galaxy using the corrected 21-cm line flux taken from hyperleda ( paturel et al . fluxes were converted into mass using the relation : @xmath35 where d is the distance to the galaxy in mpc , and f@xmath36 is the hi flux in jy km@xmath5 . to take into account the correction for helium and metals in the gas content ( e. g. mcgaugh et al . 2000 ) , m@xmath32 is related to the hi mass through : @xmath37 the h@xmath2 mass has been computed following the formula given by mcgaugh & de blok ( 1997 ) , using the morphological type of the galaxies ( young & knesek , 1989 ) . @xmath38 nevertheless , to avoid uncertainties linked to our bad knowledge in the h@xmath2 content , the baryonic mass studied in this paper does not include h@xmath2 , except when it is explicitly mentioned . the baryonic mass , m@xmath39 , is defined as : @xmath40 where m@xmath41 is the total stellar mass . uncertainties in the baryonic mass are the results of the quadratic sum of the uncertainties in stellar masses and uncertainties in the h i masses , which were taken from hyperleda . we have compared the baryonic mass to the total dynamical mass for each galaxy of our sample . although almost the whole baryonic mass is approximately within the optical radius , the total dark matter content is not , thus we compute the dynamical total mass only within the optical radius . to estimate the total dynamical mass , we assumed the mass has a spherical distribution , which is likely the case for the dark halo component , as supported by observations ( e.g. ibata et al . 2001 ) and n - body simulations ( e.g. kazantzidis et al . 2010 ) by using : @xmath42 where @xmath3 is a parameter depending on the mass profile distribution ( equal to one for an uniform distribution ) . to compute m(r ) we have used the r - band optical radius from _ sdss _ , as tabulated in the appendix . in order to obtain an estimation of the dark matter content at the optical radius , we have subtracted the baryonic mass ( as estimated in 2.5 ) from the dynamical total mass within the optical radius . galaxies having the same rotational velocity do not necessarily have the same luminosity ( or reciprocally ) , thus the observed tully - fisher relation presents a dispersion which may be produced by intrinsic properties of galaxies . beside this dispersion , uncertainties in magnitudes / masses and rotational velocities should be taken into account when the slope and zeropoint of this relation are computed ( see hogg et al . 2010 for details about fitting straight lines ) . several efforts have be performed to fit straight lines to fundamental relations , taken into account together the uncertainties in both axis and the intrinsic dispersion of the relation ( e. g. tremaine et al . 2002 , weiner et al . 2006 ) . in this paper , we have followed the prescription given by tremaine et al . ( 2002 ) , by adding ( in quadrature ) a dispersion factor to the uncertainties estimation of the nir magnitudes , stellar and baryonic masses . the value of the dispersion factor is chosen in order to reach a @xmath43 of unity per degree of freedom . to fit the tully - fisher relation , we used linear relation of the form : @xmath44 where , @xmath45 and @xmath46 ) , for the nir and stellar / baryonic tully - fisher relation , respectively , and @xmath47 . to obtain the slope and the zerpoint of this relation , we used the task fitexy ( press et al . in table [ appendix1 ] we list the nir magnitudes and the different rotational velocities that we obtained , from the observations and from the arctan model ( see 2.1 ) . columns 1 , 2 and 3 list the name , h and k - band magnitudes . column 4 shows the observed maximum rotational velocity for each galaxy . column 5 gives the modeled velocity at r@xmath6 . column 6 corresponds to the rotational velocity used in the tf relation . finally , in column 7 we classify the shape of rotation curves of the ghasp sample as shown in 2.1 . in table [ appendix2 ] we list the different determinations of mass - to - light ratios ( @xmath20 ) and masses for each galaxy . columns 1 , 2 and 3 indicate respectively the name , the radius of the galaxies ( taken from _ sdss _ ) and the b - v colors ( transformed from g - r colors ) . in columns 4 , 5 and 6 we list the mass - to - light ratios calculated from equations [ eq1 ] , [ eq2 ] and [ eq3 ] , respectively . values for the stellar masses are shown in columns 7 , 8 and 9 , following b&j , be and po , respectively . in column 10 , we list the m@xmath48+m@xmath32 , where m@xmath48 is calculated using the observed hi mass for the ghasp sample . column 11 corresponds to the baryonic masses excluding the h@xmath2 content ( m@xmath49+m@xmath48+m@xmath32 ) , m@xmath49 is here calculated following be given in column 8 . in fig . [ btf_ghasp_fig2a ] ( upper panels ) we plot the tully - fisher relations for the h and k - band ( left and right panels , respectively ) . in both panels , flat , decreasing and rising rotation curves are indicated by black dots , green triangles and red stars , respectively . galaxies having a rising rotation curve show a large dispersion on the tully - fisher relation , while most of the flat rotation curves lie on the relation . this may simply reflect the scatter in the determination of v@xmath4 for rising rotation curves , for which v@xmath4 may be uncertain . alternatively , this might indicated that rising rotation curves , that are usually dark matter dominated galaxies , show an intrinsic scatter in the tully - fisher relation . in the bottom panel of fig . [ btf_ghasp_fig2a ] we plot the k - band tully - fisher relation in which we divided the sample by their asymmetries in the rotation curve . galaxies displaying the largest non - circular motions ( red stars ) lie preferentially in the low velocity / luminosity region of the plot and present a larger scatter than galaxies less affected by non circular motion ( black dots ) . the conclusion is that non - circular motions contribute to the scatter in the nir tully - fisher relation , at least in the low luminosity ( and mass ) region of the plot . inspecting fig . [ btf_ghasp_fig2a ] , we observe a break in the tully - fisher relation at m@xmath1 - 20 . this effect has already been noted by mcgaugh et al . ( 2000 ) , gurovich et al . ( 2004 ) and amorn et al . ( 2009 ) . in order to quantify this break , we have applied a fit ( as discusses in 2.5 ) to all galaxies ( black dashed line ) and to the galaxies with m@xmath50 - 20 ( red dotted - dashed lines ) . for the first case , we have derived the followings equations : in the equations above we have included the one - sigma uncertainties in the slope and zero point . three galaxies ( @xmath114@xmath55 of the sample ) are 1@xmath56 off the tully - fisher relation . these galaxies could have slightly too high magnitudes for their rotational velocities or too low rotational velocities for their magnitudes ( upper left region in each panel of fig . [ btf_ghasp_fig2a ] ) . in spite of the rotation curves of these objects reach the optical radius r@xmath6 , two of them have rising rotation curves ( red stars in the left panels of fig . [ btf_ghasp_fig2a ] ) . in this sense , we can not exclude that both galaxies could have higher rotational velocities than the observed values , placing both galaxies on the tf relation . on the other hand , one galaxy has a flat rotation curve ( black dot in fig . [ btf_ghasp_fig2a ] ) , meaning that this object already reached its maximum rotational velocity . two possible scenarios could explain the high near - infrared magnitude of this galaxy . agn activity , that could enhance the near - infrared magnitude of seyfert 1 galaxies ( riffel et al . 2009 ) and/or the contribution of tp - agb stars into the k - band luminosity ( maraston 1998 ) . tp - agb stars are often not taken into account in the models and they are quite important specially for stellar populations with ages below 1 gyr . a detailed study of this galaxy is out of the scope of this paper . taking into account the dispersions , the slopes of the h and k - band tf relations are similar , being slightly steeper in the k - band . table [ table2 ] summarizes the fit parameters ( denoted by @xmath3 for slopes and @xmath57 for the zero points ) found for the tf relation in h and k - bands . in the same table we list values of @xmath3 and @xmath57 found in the literature . in order to compare different estimators of the stellar mass available in the literature , we have calculated the slope and zero points of the stellar tf relations of our ghasp sample using different prescriptions for mass determination given by b&j , be , po and mg ( see equations [ eq1 ] , [ eq2 ] , [ eq3 ] and [ eq3b ] ) . table [ table3 ] summarizes the fit parameters using the different stellar mass estimators . slopes from different estimators are consistent within 1@xmath56 . in the following , we used the be results to study the stellar and baryonic tf relations ( the be models are updated versions of the b&j models ) . in fig . [ btf_ghasp_fig2b ] ( left panel ) , we show the stellar tf relation for the ghasp sample . the central dashed line follows equation [ stellar ] ( and one-@xmath56 in the zeropoint ) and represents the best fit on the data . in this case , we use a dispersion factor of 0.31 dex in the stellar mass . we found a slope of [email protected] ( equation [ stellar ] ) when we calculated the stellar masses following be . b&j found a slope of [email protected] for the stellar tf relation , for a sample of galaxies in the ursa major cluster . as previously found in the h and k - band tf relation , low - mass galaxies in fig . [ btf_ghasp_fig2b ] ( with stellar masses lower than 10@xmath59 m@xmath26 ) lie below the relation defined by high - mass spirals . by using cosmological hydrodynamical simulations , de rossi et al . ( 2010 ) suggested that sn feedback is the main responsible for this behavior in low - mass systems . in fig . [ btf_ghasp_fig6 ] we plot stellar masses derived from the k and b - band luminosities , following the recipes of be . we found that the b - band luminosities slightly overestimate ( underestimate ) the stellar mass for the low - mass ( high - mass ) galaxies , when it is compared with the value derived from the k - band luminosity . such an overestimation ( and underestimation ) for the stellar masses of low - mass ( and high - mass ) galaxies could strongly affect the slope of the baryonic tully - fisher relation , adding a band - dependence effect " . the same trend was found when we use the recipes given in bj and po . in fig . [ btf_ghasp_fig2b ] ( right panel ) , we plot the baryonic tf relation for ghasp . the central dashed line represents the best fit for the data and one-@xmath56 in the zeropoint ( see equation [ bary ] ) . in this case , we use a dispersion factor of 0.21 dex in the baryonic mass . the shapes of the rotation curves are represented by different symbols ( circles , triangles and stars indicated flat , decreasing and rising rotation curves ) . it is interesting to note that low - mass galaxies now lie on the same relation defined by high - mass galaxies . this fact is attributed to the inclusion of the gaseous mass into the stellar budget . in this plot , the stellar mass was estimated from be . the slope of the baryonic tf that we derive for the ghasp sample is [email protected] ( which is in agreement with the slope obtained from an unweighted bisector fit , i.e. [email protected] ) . as done for the stellar mass tf relation , we obtained the slope and zero point of the baryonic tf relation when the stellar mass was calculated using b&j , be , po and mg . these values are listed in table [ tablehenri ] , where @xmath3 and @xmath57 correspond to the zero points and slopes , respectively . in table [ tablehenri ] we have included the resulting slopes and zero points obtained when h@xmath2 masses for the galaxies are included in the baryonic budget . we found that the slope of the baryonic tf relation , when h@xmath2 is included , does not change significantly . note , in particular , that the use of the bell & de jong ( 2001 ) or bell et al . ( 2003 ) mass - to - light recipes , on the ghasp sample , results in a very similar slope of the baryonic tf relation ( lines 1 and 9 of table [ tablehenri ] ) . in table [ tablehenri ] we also list the fit parameters for the baryonic tully - fisher relation when the stellar masses were derived from the b - band luminosities ( and the gaseous mass was corrected by the h@xmath2 gas mass ) . in this case , the slope is shallower ( @xmath61 ) than in the case when stellar masses are computed from the k - band luminosities . in fig . [ r25bary ] , we show that , even with some scatter in the relation , the baryonic mass @xmath62 grows almost linearly with the optical galactic radius r@xmath6 in log units ( @xmath63 ) . we used a weighted bisector least square fit to obtain the dependence between these parameters ( we used a bisector fit given that the _ sdss _ database does not quote errors in the radius ) . we found @xmath64 , where @xmath65 . these results suggest that the baryonic mass density , defined by @xmath66=@xmath67 , depends weakly on the sizes of the galaxies . more precisely , @xmath68 .
|
we studied , for the first time , the near infrared , stellar and baryonic tully - fisher relations for a sample of field galaxies taken from an homogeneous fabry - perot sample of galaxies ( the ghasp survey ) .
the main advantage of ghasp over other samples is that maximum rotational velocities were estimated from 2d velocity fields , avoiding assumptions about the inclination and position angle of the galaxies .
by combining these data with 2mass photometry , optical colors , hi masses and different mass - to - light ratio estimators , we found a slope of [email protected] and [email protected] for the stellar and baryonic tully - fisher relation , respectively .
we found that these values do not change significantly when different mass - to - light ratios recipes were used .
we also point out , for the first time , that rising rotation curves as well as asymmetric rotation curves show a larger dispersion in the tully - fisher relation than flat ones or than symmetric ones . using the baryonic mass and the optical radius of galaxies
, we found that the surface baryonic mass density is almost constant for all the galaxies of this sample . in this study
we also emphasize the presence of a break in the nir tully - fisher relation at m@xmath120 and we confirm that late - type galaxies present higher total - to - baryonic mass ratios than early - type spirals , suggesting that supernova feedback is actually an important issue in late - type spirals . due to the well defined sample selection criteria and the homogeneity of the data analysis , the tully - fisher relation for ghasp galaxies can be used as a reference for the study of this relation in other environments and at higher redshifts .
[ firstpage ] galaxies : evolution galaxies : kinematics and dynamics
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
dark energy ( de ) seems to play an important role of an agent that drives the present acceleration of the universe with the help of large negative pressure . an effective viscous pressure can also play its role to develop the dynamical history of an expanding universe @xcite-@xcite . it is found @xcite that viscosity effects are viable at low redshifts , which observe negative pressure for the cosmic expansion with suitable viscosity coefficients . in general , the universe inherits dissipative processes @xcite , but perfect fluid is an ideal fluid with zero viscosity . although , perfect fluid is mostly used to model the idealized distribution of matter in the universe . this fluid in equilibrium generates no entropy and no frictional type heat because its dynamics is reversible and without dissipation . the dissipative processes mostly include bulk and shear viscosities . the bulk viscosity is related with an isotropic universe whereas the shear viscosity works with anisotropy of the universe . the cmbr observations indicate an isotropic universe , leading to bulk viscosity where the shear viscosity is neglected @xcite . long before the direct observational evidence through the sn ia data , the indication of a viscosity dominated late epoch of accelerating expansion of the universe was already mentioned @xcite . the origin of the bulk viscosity in a physical system is due to its deviations from the local thermodynamic equilibrium . thus the existence of bulk viscosity may arise the concept of accelerating expansion of the universe due to the collection of those states which are not in thermal equilibrium for a small fraction of time @xcite . these states are the consequence of fluid expansion ( or contraction ) . the system does not have enough time to restore its equilibrium position , hence an effective pressure takes part in restoring the system to its thermal equilibrium . the measurement of this effective pressure is the bulk viscosity which vanishes when it restores its equilibrium @xcite-@xcite . so , it is natural to assume the existence of a bulk viscous coefficient in a more realistic description of the accelerated universe today . physically , the bulk viscosity is considered as an internal friction due to different cooling rates in an expanding gas . its dissipation reduces the effective pressure in an expanding fluid by converting kinetic energy of the particles into heat . thus , it is natural to think of the bulk viscous pressure as one of the possible mechanism that can accelerate the universe today . however , this idea needs a viable mechanism for the origin of the bulk viscosity , although there are many proposed best fit models . many models have been suggested to discuss the vague nature of de . during the last decade , the holographic dark energy ( hde ) , new agegraphic dark energy ( nade ) , their entropy corrected versions and correspondence with other de models have received a lot of attention . the hde model is based on the holographic principle which states that _ the number of degrees of freedom in a bounded system should be finite and has a relationship with the area of its boundary _ @xcite . moreover , in order to reconcile the validity of an effective local quantum field , cohen et al . @xcite provided a relationship between the ultraviolet ( uv ) and the infrared ( ir ) cutoffs on the basis of limit set by the formation of a black hole . this is given by @xcite @xmath1 where constant @xmath2 is used for convenience , @xmath3 is the reduced planck mass and @xmath4 is the ir cutoff . this model has been tested by using different ways of astronomical observations @xcite-@xcite . also , it has been discussed widely in various frameworks such as in the general relativity , modified theories of gravity and extra dimensional theories @xcite-@xcite . the nade model was developed in view of the heisenberg uncertainty principle with general relativity . this model exhibits that de originates from the spacetime and matter field fluctuations in the universe . in this model , the length measure is taken as the conformal time instead of age of the universe and its energy density is @xmath5 where @xmath6 is the conformal time . the causality problem occurs in the usual hde model , while it is avoided here . many people have explored the viability of this model through different observations @xcite-@xcite . another proposal to discuss the accelerating universe is the modified gravity theories @xcite . the @xmath0 gravity is the generalization of teleparallel gravity by replacing the torsion scalar @xmath7 with differentiable function @xmath0 , given by @xmath8 where @xmath9 is the coupling constant and @xmath10 . this leads to second order field equations formed by using weitzenb@xmath11ck connection which has no curvature but only torsion . the equation of state ( eos ) parameter , @xmath12 , is used to explore the cosmic expansion . bengochea and ferraro @xcite tested power - law @xmath0 model for accelerated expansion of the universe . they performed observational viability tests and concluded that this model exhibits radiation , matter and de dominated phases . incorporating exponential model along with power - law model , linder @xcite investigated the expansion of the universe in this theory . he observed that power - law model depends upon its parameter while exponential model acts like cosmological model at high redshift . bamba et al . @xcite discussed the eos parameter for exponential , logarithmic as well as combination of these @xmath0 models and they concluded that the crossing of phantom divide line is observed in combined model only . karami and abdolmaleki @xcite constructed this parameter for hde , nade and their entropy corrected models in the framework of @xmath0 gravity . they found that the universe lies in phantom or quintessence phase for the first two models whereas phantom crossing is achieved in entropy corrected models . sharif and rani @xcite described the graphical representation of k - essence in this modified gravity with the help of eos parameter . some other authors @xcite explored the expansion of the universe with different techniques in @xmath0 gravity . also , the effects of viscous fluid in modified gravity theories @xcite-@xcite are analyzed to display accelerating expansion . in this paper , we construct the viscous eos parameter for different viable de models in the framework of @xmath0 gravity with pressureless matter . for this purpose , we consider a time dependent viscous model with its constant viscous reduction to explore the de era in general fluid . the graphical behavior indicates the acceleration of the universe for suitable viscous coefficients . the scheme of paper is as follows : section * 2 * provides basic formalism and discussion about the field equations of @xmath0 gravity . in section * 3 * , the viscous eos parameter is constructed for different de models . also , we discuss the graphical behavior of this parameter for these models . the last section summarizes the results . the @xmath0 theory of gravity ( as the generalization of the teleparallel gravity ) is uniquely determined by the tetrad field @xmath13 @xcite . it is an orthonormal set of four - vector fields defined on lorentzian manifold . the metric and tetrad fields can be related as @xmath14 where @xmath15 is the minkowski metric for the tangent space . here we use greek alphabets @xmath16 to denote spacetime components while the latin alphabets @xmath17 are used to describe components of tangent space . the non - trivial tetrad field @xmath18 , yielding non - zero torsion , can be written as @xmath19 satisfying the following properties @xmath20 the variation of eq.([1 * ] ) with respect to the tetrad field leads to the following field equations @xcite @xmath21f_{t } + s_{i}~^{\mu\nu}\partial_{\mu}(t ) f_{tt}+\frac{1}{4}h^{\nu}_{i}f=\frac{1}{2}\kappa^{2}h^{\rho}_{i}t^{\nu}_{\rho},\ ] ] where @xmath22 . the torsion scalar is defined as @xmath23 where @xmath24 and torsion tensor @xmath25 are given as follows @xmath26 which are antisymmetric . the energy - momentum tensor for perfect fluid is @xmath27 where @xmath28 is the four - velocity in comoving coordinates , @xmath29 and @xmath30 denote the total energy density and pressure of fluid inside the universe . the flat homogenous and isotropic frw universe is described by @xmath31 where @xmath32 is the scale factor such that @xmath33 in the form of redshift @xmath34 . the corresponding tetrad components are @xcite-@xcite @xmath35 which obviously satisfies eq.([4 ] ) . using eqs.([6 ] ) and ( [ 10 ] ) , the torsion scalar turns out in the form of hubble parameter @xmath36 as @xmath37 . the corresponding modified friedmann equations become @xmath38 for the realistic model , we take viscosity term which introduces the effective pressure in the energy - momentum tensor @xcite , i.e. , @xmath39 defined by @xmath40 here @xmath41 is the time dependent bulk viscosity function . to avoid the violation of the second law of thermodynamics , @xmath42 . the field equations ( [ 13 ] ) and ( [ 14 ] ) may be rewritten as @xmath43 we assume here the pressureless ( dust ) matter , i.e. , @xmath44 and the expressions for torsion contributions @xmath45 and effective pressure become @xmath46 it is noted that if we insert @xmath47 in eq.([17 ] ) with non - viscous case , we arrive at the usual friedmann equations in general relativity . the corresponding viscous eos parameter becomes @xmath48 the phantom and quintessence regions are mostly described with the help of constant eos parameter such as , @xmath49 , which corresponds to the quintessence era whereas phantom era is referred to @xmath50 and the phantom divide line is given by @xmath51 . if we consider a torsion dominated universe , then eq.([13 ] ) reduces to @xmath52 inserting the above value in the energy conservation equation for torsion , it follows that @xmath53 @xmath54 the eos parameter @xmath55 describes a vacuum , phantom dominated or quintessence dominated universe for @xmath56 or @xmath57 respectively for viscous case . for the non - viscous case @xmath58 , these conditions reduce to @xmath59 or @xmath60 . viscous models have interesting insights about the evolution of the expanding universe . here we consider a simple time dependent bulk viscous model as follows @xcite @xmath61 where @xmath62 and @xmath63 are positive coefficients . the cosmological evolution can be explored for different values of these coefficients @xcite-@xcite . this bulk viscosity model is motivated due to the terms involved , i.e. , viscosity is related to the velocity and acceleration which give the phenomenon of scalar expansion in fluid dynamics . the viscous model having constant @xmath64 and velocity term @xmath65 are discussed in @xcite , thus a linear combination of these two with acceleration term @xmath66 may give more physical results . in general , the existence of viscosity coefficients in a fluid is due to the thermodynamic irreversibility of the motion . if the deviation from reversibility is small , the momentum transfer between various parts of the fluid can be taken to be linearly dependent on the velocity derivatives . this case corresponds to the constant viscous model . when viscosity is proportional to hubble parameter the momentum transfer involves second order quantities in the deviation from reversibility leading to more physical results . the proper choices of their coefficients may lead to the crossing of phantom divide line . to determine the evolution of effective eos parameter incorporating @xmath0 and viscous models , we assume the hubble parameter in the form @xcite @xmath67 here @xmath68 and @xmath69 are positive constants , the constant @xmath70 is either positive or negative and @xmath71 is guaranteed for the accelerated expansion of the universe due to the violation of strong energy condition @xmath72 . for @xmath73 , it leads to the scale factor @xmath74 which ends up the universe with future finite time big rip singularity . using eq.([a ] ) with @xmath73 , the torsion scalar becomes @xmath75 with @xmath76 . also , taking the value of @xmath36 , the energy conservation equation , @xmath77 for dust matter yields the solution @xmath78 where @xmath79 is an arbitrary constant . in the following , we discuss three de models by taking into account of the viscosity . first , we consider the following de model @xcite @xmath80 where @xmath81 and @xmath82 are constants . for @xmath83 , this model leads to the teleparallel gravity . it is interesting to note that the model ( [ 29 ] ) is the result of correspondence between energy densities of @xmath0 and hde model . in flat frw universe , the ir cutoff @xmath4 in eq.([1 ] ) becomes the future event horizon @xmath84 resulting the he d energy density . using eq.([a ] ) in the correspondence , @xmath81 takes the form @xmath85 and @xmath82 is an integration constant . replacing @xmath0 and viscous models in eq.([21 ] ) , the viscous eos parameter takes the form @xmath86}{(\kappa^2 \rho_{m0}(1+z)^{3+\frac{2}{h}}+3\alpha h^2)(1+z)^{\frac{1}{h}}}\\\label{31}&-&\frac{2\alpha h}{\kappa^2 \rho_{m0}(1+z)^{3+\frac{2}{h}}+3\alpha h^2}-1.\end{aligned}\ ] ] the graphical behavior of time dependent viscous eos parameter with respect to redshift is shown in * figure 1*. we draw this parameter by taking arbitrary values of the coefficients @xmath87 of viscous model , where @xmath81 depends upon the constant @xmath88 which is 0.818 for flat model @xcite . also , we fix the redshift range from 0 to 5 to discuss the behavior of the universe at low redshifts . the left graph in * figure 1 * shows the evolution of the universe initially from matter dominated era for higher values of @xmath34 and then converges to quintessence era at @xmath89 for @xmath90 and @xmath91 . the phantom divide line is being crossed by the @xmath92 as @xmath34 approaches to zero . by decreasing @xmath93 and @xmath63 from @xmath94 , the universe remains in phantom dominated era ( shown in the right graph ) . for the constant viscous case , we take @xmath95 in eq.([24 ] ) , thus the constant viscous eos parameter becomes @xmath96 * figure 2 * represents the same behavior as indicated by time dependent viscous eos parameter . however , the phantom crossing for the constant viscous coefficient occurs at @xmath97 , it shows phantom behavior for @xmath98 ( right graph ) . assuming the exponential @xmath0 model @xcite @xmath99 where @xmath100 is an arbitrary constant . inserting @xmath0 and viscous models in eq.([21 ] ) , the viscous eos parameter takes the form @xmath101 \left[2(1+z)^3+\frac{36h^4}{(1+z)^{\frac{4}{h}}}\exp(-\frac{6h^2b}{(1+z)^{\frac{2}{h}}})\right]^{-1}.\end{aligned}\ ] ] figure * 3 * represents the graphical behavior of time dependent viscous @xmath102 versus @xmath34 . in the left graph , the plot shows the evolution of the universe from matter to de phase for higher values of redshift , approximately for @xmath103 . at @xmath104 for particular values @xmath90 and @xmath105 , the eos parameter indicates the quintessence era and approaches to @xmath106 as @xmath107 . as we decrease the values of @xmath108 , the @xmath92 represents the phantom era of the universe . now for constant viscous eos parameter , we take @xmath109 in eq.([d ] ) yields @xmath110 \left[2(1+z)^3\right.\\\label{e}&+&\left.\frac{36h^4}{(1+z)^{\frac{4}{h } } } \exp(-\frac{6h^2b}{(1+z)^{\frac{2}{h}}})\right]^{-1}.\end{aligned}\ ] ] its plot versus @xmath34 is in figure * 4 * , showing same behavior as that of time dependent case . approximately , the universe meets the quintessence era at @xmath111 and converges towards @xmath112 as @xmath34 approaches to zero ( in left graph ) . in right graph , the evolution of eos parameter represents the phantom era of the universe for @xmath113 by decreasing the value of @xmath64 , i.e. , @xmath114 . finally , we take the model @xmath115 which includes linear and nonlinear terms of torsion scalar and @xmath116 are constants . similar to the first model ( [ 29 ] ) , this model comes through the correspondence of nade model with @xmath0 gravity . the energy density of the nade model inherits the conformal time @xmath117 . incorporating the correspondence , here @xmath118 is an integration constant and @xmath70 is @xmath119 where @xmath120 for flat universe . replacing eq.([38 ] ) in ( [ 21 ] ) , the viscous eos parameter becomes @xmath121}{(2\kappa^2 \rho_{m0}(1+z)^{5+\frac{2}{h}}+\gamma ( -6h^2)^{1+h})(1+z)^{\frac{1 - 2h}{h}}}\end{aligned}\ ] ] @xmath122 the graphical behavior of time dependent viscous @xmath92 is given in * figure 5*. initially , it shows the deceleration phase @xmath123 of the universe for higher values of @xmath34 . as we decrease the value of redshift up to @xmath124 , it meets the quintessence region for the particular values @xmath125 and @xmath126 , and crossing of the phantom divide line takes place for @xmath34 tends to zero . the right graph indicates that the universe remains in this era for @xmath127 . the constant viscous model for this case is @xmath128 * figure 6 * shows its plot versus redshift . it provides the crossing of phantom divide line for a high value @xmath129 , whereas @xmath130 corresponds to the phantom region for decreasing @xmath34 of the accelerating expansion of the universe . viscous models have been discussed in cosmological evolution of the universe as compared to the ideal perfect fluid . the term of shear viscosity vanished when a completely isotropic unverse is assumed and only the bulk viscosity contributes for the accelerating universe to get negative pressure . in this paper , we have considered viscosity by taking dust matter in the framework of @xmath0 gravity . we have taken three different viable de models and a time dependent viscous model to construct the viscous eos parameter for these models . the graphical representation is also developed by considering arbitrary values of the coefficients in viscous model for a specific expression of hubble parameter . the results and the comparison with non - viscous case are given as follows . all the three models in viscous fluid indicates the behavior of the universe from matter dominated phase to quintessence era and then converges to phantom era of the de dominated phase for decreasing @xmath34 . it shows the phantom universe by taking the particular values of viscous coefficients . the constant viscous cases also exhibit phantom behavior . the non - viscous case @xmath131 shows a universe which always stays in phantom for @xmath132 or quintessence for @xmath133 regions @xcite . however , the third model has resulted the phantom phase of the universe for the higher values of viscous coefficients as compared to the first and second @xmath0 models . in each case , the time dependent case shows the phantom crossing by taking small values of viscous coefficients while constant viscous case needs higher values for crossing . the combination of torsion and viscosity influences the accelerating expansion of the universe in such a way that it strictly depends upon the viscous coefficients of the model . we have to fix the ranges for these coefficients in order to get our desired results . we conclude that the viscosity model leads to different behavior of the accelerating universe in de era under the effects of viscous fluid . on the other hand , viscosity may result the crossing of the phantom divide line and phantom dominated universe @xcite as shown in * figures 1 * and * 6*. in the non - viscous case @xcite , the universe remains in the phantom and quintessence eras for the relevant scale factors . beyond the ideal situation , we remark that the de era of the universe in a real fluid may be observed and hence accelerating expansion of the universe is achieved .
|
we study the bulk viscosity taking dust matter in the generalized teleparallel gravity .
we consider different dark energy models in this scenario along with a time dependent viscous model to construct the viscous equation of state parameter for these dark energy models .
we discuss the graphical representation of this parameter to investigate the viscosity effects on the accelerating expansion of the universe .
it is mentioned here that the behavior of the universe depends upon the viscous coefficients showing the transition from decelerating to accelerating phase .
it leads to the crossing of phantom divide line and becomes phantom dominated for specific ranges of these coefficients . * keywords : * @xmath0 gravity ; viscosity ; effective equation of state . +
* pacs : * 04.50.kd ; 95.36.+x
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the development of large - scale structure had to take place at epochs coresponding to redshifts greater than 0.2 ; moreover structure is now known to extend beyond scales of @xmath2 mpc . both factors place the extent and evolution of large - scale structure beyond the reach of current optical and ir surveys and we must look outside these bands to address such key cosmological issues . it has now been established that _ extragalactic radio sources _ can trace structure both to very early epochs and on very large scales @xcite,@xcite,@xcite . this is a direct result of their uniform selection function : away from the galactic plane extragalactic radio sources are visible to very high redshifts ( @xmath3 ) with detection unaffected by obscuration . moreover the high radio luminosities and strong space - density evolution yield a peak selection range of very high redshifts ( @xmath4 ) . it was @xcite who showed how to determine the 3d 2-point spatial correlation function from 2d information in radio surveys themselves , a cosmological limber s equation , and an estimated redshift distribution for the radio sources to the survey limit . this and more recent analyses use the best estimates of @xmath1 available , namely those from the comprehensive analysis of @xcite , in which all the then - known redshift and source - count data were synthesized into a determination of the epoch - dependent luminosity functions . these in turn can be used to predict @xmath1 at any frequency and flux - density level . the statistical accuracy becomes rapidly poorer below flux densities equivalent to @xmath5 = 100 mjy , as the results are then an extrapolation . moreover the analysis considers steep - and flat - spectrum populations separately and independently , while not including at all the starburst galaxy population which dominates at levels below a few mjy . our recent analysis of radio source evolution in terms of a dual - population unified model @xcite,@xcite describes how the flat and steep - spectrum populations are physically related , and includes this latter starburst population . accordingly it promises to predict @xmath1 with somewhat greater reliability , particularly at the lower flux densities . our dual - population unified model is based on the two fanaroff - riley ( 1974 ) classes of radio galaxies as parent populations . anisotropic radiation mechanisms of relativistic beaming along the radio axes and dusty tori shrouding the nuclei result in core - dominated quasars and bllac objects from these , when radio axes coincide closely with our line - of - sight . we determined the cosmic evolution history of the two populations fri and frii from low - frequency ( 151 mhz ) survey statistics . we then determined beaming models for the related quasars and bllac objects by matching predicted and observed source counts at 5 ghz using monte carlo orientation of the parent population . in this process we adopted the pure luminosity evolution model for the starburst galaxy population derived by @xcite . in the process we found that a combination of strong cosmic evolution of the frii sources coupled with no evolution of the fri population is required to fit the low - frequency count data , while the best - fit beaming models for the two populations have parameters , jet - speeds in particular , which match those observed in vlbi observations of individual sources . the composite model which we found for evolution and beaming gave a natural explanation of the change in source count shape with frequency , while showing good agreement with several other independent data sets . our evolution and beaming model can be used predict at any frequency the intensity - dependent mix of the three underlying radio - source populations : the starburst galaxies , and the fri and frii radio galaxies together with their beamed ( on - axis ) counterparts . figure 1 shows this mix for 1.4 ghz . the model can also be used to predict the redshift distribution @xmath1 for any frequency and in a given intensity range . figure 2 shows the @xmath1 derived for 1.4 ghz in the flux - density range @xmath6 mjy . the dominant populations in this range ( see figure 1 ) are the fri sources ( beamed and unbeamed versions ) at moderate redshifts , with starburst galaxies at low redshift ; the high - power frii sources make only a minor contribution . the implication for the latest generation of radio surveys which extend to @xmath7 mjy is that these surveys _ completely _ sample this the most powerful radio - source population ; one entire population is uniformly sampled across its entire evolutionary history to the highest observable redshifts by these sensitive radio surveys . the difference between this estimation of @xmath1 and those used previously - in particular those from the models of @xcite - can be attributed to ( i ) the ` spike ' at @xmath8 due to the starburst galaxies , a population not included in previous analyses , and ( ii ) the lower median redshift due to the dominance of the unevolving fri population at this level . lccc + & first @xcite & nvss @xcite & sumss @xcite + + + frequency & 1400 mhz & 1400 mhz & 843 mhz + area ( sq deg ) & 10,000 & 33,700 & 8,000 + resolution ( @xmath9 ) & 5@xmath9 & 45@xmath9 & 43@xmath9 + detection limit & 1 mjy & 2.5 mjy & @xmath105 mjy + coverage & ngp & @xmath11 @xmath1240@xmath13 & @xmath14 @xmath1230@xmath13 + sources / sq deg & @xmath1090 & @xmath1060 & @xmath1040 + * figure 1.*the predicted integral population mix at 1.4 ghz from our dual - population unified model @xcite , @xcite . the contribution from the frii population ( frii radio galaxies and quasars ) is almost negligible at @xmath5=1 mjy . * figure 2.*the total predicted @xmath1 for sources in the flux density range 1 @xmath15 100 mjy ( solid line ) from evolution models of @xcite ( average of models 1 - 4,6 & 7 ) ( dashed line ) , and @xcite ( dotted line ) . our evolution model ( and in consequence the @xmath1 predictions ) will be further refined by ( i ) incorporating the results from a multi - object spectroscopy campaign ( wyffos + wht , 2df + aat ) which targets first survey sources , and ( ii ) redefining the local radio luminosity function using spectra obtained by the 2df galaxy redshift survey ( m colless , these proceedings ) .
|
extragalactic radio sources are a unique cosmological probe in that they trace large - scale structure on scales inaccessible to other wavelengths .
however as radio survey data is inherently 2@xmath0 , the redshift distribution , @xmath1 , is necessary to derive spatial information . to obtain this distribution either we measure thousands of radio source redshifts to directly determine @xmath1 _ or _ we derive @xmath1 from statistical analyses of radio source count and identification data . in this paper
we show how the dual - population unification scheme can be incorporated into a rigorous statistical analysis of radio source count data , with the result that our simple parametric evolution and beaming model revises previous estimations of @xmath1 , specifically at low flux densites .
this revision is particularly pertinent given that the new generation of radio surveys extend to milli - jansky flux density levels : sampling source densities high enough to reveal spatial structure . in turn
, these new radio surveys will provide potent tests which will refine our model .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
galaxy morphology is studied for the purpose of classification and analysis of the physical structures exhibited by galaxies in wide redshift ranges in order to get a better understanding of the structure and development of galaxies . while significant research has been done to study the morphology of galaxies with spiral arms @xcite , research efforts have been focused also on the analysis of elliptical and s0 galaxies using photometric measurement of the electromagnetic radiation , ellipticity , position angle , shape , and colour @xcite . these analyses were successful in acquiring information regarding the structure and development of some of these galaxies . however , these studies have done little analysis of the spirality of galaxies that were classified as elliptical . studying the morphology of large datasets of galaxies have attracted significant attention in the past decade @xcite , and was driven by the increasing availability of automatically acquired datasets such as the data releases of the sloan digital sky survey @xcite . however , attempts to automatically classify faint galaxy images along the hubble sequence have been limited by the accuracy and capability of computer learning classification systems , and did not provide results that met the needs of practical research @xcite . this contention led to the _ galaxy zoo _ @xcite project , which successfully used a web - based system to allow amateur astronomers to manually classify galaxies acquired by sdss @xcite , and was followed by other citizen science ventures based on the same platform such as _ galaxy zoo 2 _ @xcite , _ moon zoo _ @xcite , and _ galaxy zoo mergers _ @xcite . while it has been shown that amateurs can classify galaxies to their basic morphological types with accuracy comparable to professional astronomers @xcite , manual classification may still be limited to what the human eye can sense and the human brain can perceive . for instance , the human eye can sense only 15 to 25 different levels of gray , while machines can identify 256 gray levels in a simple image with eight bits of dynamic range . the inability of the human eye to differentiate between gray levels can make it difficult to sense spirality in cases where the arms are just slightly brighter than their background , but not bright enough to allow detection by casual inspection of the galaxy image . in fact , this limitation might affect professional astronomers as much as it affects citizen scientists . since the human eye can only sense the crude morphology of galaxies along the hubble sequence , and since the classification of galaxies is normally done manually , morphological classification schemes of galaxies are based on few basic morphological types . however , as these schemes are merely an abstraction of galaxy morphology , some galaxies can be difficult to associate with one specific shape , and many in - between cases can exist . here we use the ganalyzer method to transform the galaxy images into their radial intensity plots @xcite , and analyze the spirality of galaxies classified manually as elliptical and s0 by the _ galaxy zoo _ , rc3 , and na10 catalogues . the method that was used to measure the spirality of the galaxies in the dataset is the ganalyzer method @xcite . unlike other methods that aim at classifying a galaxy into one of several classes of broad morphological types @xcite , ganalyzer measures the slopes of the arms to determine the spirality of a galaxy . ganalyzer is a model - driven method that analyzes galaxy images by first separating the object pixels from the background pixels using the otsu graylevel threshold @xcite . the centre coordinates of the object are determined by the largest median value of the 5@xmath15 shifted window with a distance less than @xmath2 from the mass centre , where s is the surface area @xcite . this method allows the program to determine the maximum radial distance from the centre to the outermost point , as well as the major and minor axes by finding the longest distance between two points which pass through the centre for the major axis , and then assigning the perpendicular line as the minor axis @xcite . the ellipticity is defined as the ratio of the lengths of the minor axis to the major axis @xcite . comparison of the ellipticity of 1000 galaxies to the ellipticity computed by sdss ( using isoa and isob ) shows a high pearson correlation of @xmath30.93 between the two measurements . after the centre coordinates of the galaxy @xmath4 and the radius @xmath5 are determined , the galaxy is transformed into its radial intensity plot such that the intensity value of the pixel @xmath6 in the radial intensity plot is the intensity of the pixel at coordinates @xmath7 in the original galaxy image , such that @xmath8 is a polar angel of [ 0,360 ] , and @xmath5 is the radial distance that ranges from 0.4 to 0.75 of the galaxy radius , producing an image of dimensionality of 360@xmath135 @xcite . figure [ radial ] shows an example of two galaxies and their transformation such that the y axis is the pixel intensity and the x axis is the polar angle . as the figure shows , in the case of the elliptical galaxy the peaks are aligned on the same vertical line , while in the case of the spiral galaxy the peaks shift . the spirality is then measured by the slope of the groups peaks as described in @xcite , such that the peak in radial distance _ r _ is grouped with the peak in radial distance _ r_+1 if the difference between their polar angles is less than 5@xmath9 . this transformation makes it easier for machines to detect and measure the spirality , but can also detect spirality in galaxies that might look to the human observer as elliptical since the human eye can only recognize 15 - 25 gray levels , making it difficult to notice subtle spirality when looking at a raw galaxy image . for instance , tables 1 and 2 show several sdss galaxy images classified manually by _ galaxy zoo _ participants as elliptical , with their radial intensity plot transformation and their spirality as measured by ganalyzer . to test how the method analyzes tidally disrupted elliptical galaxies @xcite , we used several tidally disrupted galaxies from the na10 catalogue , displayed in table 3 . if the radial intensity plot does not feature peaks the galaxy is defined as pure elliptical . elliptical and lenticular galaxies in some cases can also have peaks in their radial intensity plot due to the position angle , but in these cases all peaks will be aligned on the same vertical line so that the slope will be very close to zero , and therefore the galaxy will be identified as elliptical . an exception can be in cases of s0 galaxies in which the position angle of the disk is different from the position angle of the galaxy , but the difference is not greater than 5@xmath9 . in that case ganalyzer might consider the disk and the galaxy as the same arm , but the difference in the position angles will lead to a certain slope in that arm . therefore , the arms of the galaxy will have a certain slope when measured using ganalyzer . the radial intensity plot can allow the detection of subtle curves in the arms that might not be easily detected by manual observation of the raw galaxy image , but becomes noticeable in its radial intensity plot . therefore , it is possible that many galaxies that were classified manually as elliptical might in fact feature a certain spirality @xcite . as the table shows , while the galaxies seem elliptical to the unaided human eye , the radial intensity plot transformations of the galaxies show that the peaks of maximal intensity shift , meaning that these galaxies feature certain curves in the arms . by defining spirality and ellipticity thresholds ganalyzer can also be used for classifying galaxies into their broad morphological types of elliptical , spiral and edge - on , and a thorough discussion and experimental results about galaxy classification with ganalyzer are described in @xcite . in previous experiments with ganalyzer @xcite thresholds were applied to the slopes in the radial intensity plots so that the decision whether a galaxy is spiral or not is in agreement with the perception of a person observing the raw galaxy image . however , as described above , in this study ganalyzer is not used as a classifier , but as a tool to measure and detect the existence of galaxy spirality . since the radial intensity plot provides a more sensitive view of galaxy spirality than the non - transformed raw image , no thresholds are used in this study in order to utilize the ability of the radial intensity plots to detect subtle slopes in the galaxy arms and to test whether galaxies that seem elliptical to the human eye are indeed ellipticals , or have a subtle spirality that is difficult to measure using the unaided eye . that is , the purpose of the method described in this section is not to mimic the human eye , but test whether the human eye observing the raw galaxy image is indeed the most accurate tool to determine whether a galaxy is spiral or elliptical . the data used in the experiment are galaxies acquired by sloan digital sky survey , and were classified manually by the participants of the galaxy zoo project @xcite . all galaxies in the dataset have redshift , and the classification results were based on the corrected _ super clean _ dataset described in @xcite . for the study , only galaxies that were classified by galaxy zoo participants as ellipticals were used , and the dataset consisted of 60,518 galaxies . the images were downloaded automatically by using the cas server . the galaxies were also divided into six bins based on their redshift , ranges from 0 to 0.3 , such that each bin had a redshift range of 0.05 . the number of galaxies in each redshift is specified in table 4 . the efficiency of ganalyzer can be affected by two or more galaxies that appear very close to each other in the image , either due to merging or superpositioning . since one galaxy can be segmented with part or all of the other galaxy , ganalyzer might detect the other galaxy as an arm . in most cases such arm " is not expected to be mistakenly identified as sharp spirality because the angle of the brightest point compared to the center is not expected to shift , but it can lead to the false detection of mild spirality that is not based on the morphology of the target galaxy . to avoid analyzing overlapping galaxies each image was scanned for psfs as done in @xcite , and when more than one psf is detected the image is ignored . out of the galaxies classified by galaxy zoo as elliptical @xmath312.05% were detected as galaxies with more than one nucleus and were therefore rejected from the analysis . other catalog that were used in this study were the rc3 catalog @xcite , of which 261 galaxies classified as ellipticals and 640 galaxies classified as s0 were used , and the na10 catalog @xcite , of which 2705 galaxies that were classified as ellipticals , and 1964 galaxies that were classified as s0 were used in the experiment . additionally , 7638 galaxies of the na10 catalog classified as spirals were also used in the analysis . figure [ spirality_distribution ] shows the distribution of the slopes of the arms of the galaxies classified manually as elliptical . as the figure shows , @xmath324% of the galaxies exhibit nonzero signal for spirality , and @xmath310% of the galaxies had a slope of the arms greater than 0.4 , indicating that many of the _ galaxy zoo _ galaxies that were classified manually as ellipticals actually have some spirality . expectedly , the fraction of galaxies that meet the spirality threshold decreases as the slope of the arms gets larger , and just less than 2% of the galaxies that were classified manually as ellipticals were detected to have a measured slope greater than 1 . as the graph shows , the measured slope of the arms of most galaxies is close to 0 . 60,000 galaxies classified as ellipticals ] figure [ redshift_0_015 ] shows the distribution of the slopes of the arms in a galaxy population of different redshifts . as can be learned from the figure , the slope of the arms increases with the redshift , showing that at higher redshifts human readers find it more difficult to detect spirality by eye and tend to classify more galaxies as ellipticals . correlating the apparent magnitude of these galaxies with the slope of the arms provided a weak pearson correlation value of -0.036 , showing that the analysis is merely weakly dependent on apparent magnitude in the redshift range used in this study . while the experiments above show that human readers can in some cases fail to notice mild spirality , we also tested the error rate of human readers when they determine that the galaxy that they observe is spiral . figure [ gz_spirals ] shows the distribution of the measured slope of the arms among galaxies that were classified by galaxy zoo participants as spiral . as the figure shows , when the human eye is able to detect spirality , spirality does exist , and galaxies classified by human observers as spirals are rarely galaxies that do not have any spirality in them . the reason could be the limited sensitivity of the human eye in detecting spirality , so that once spirality can be detected by the human eye it is above a certain spirality threshold that can be sensed by applying the analysis of the radial intensity plot of the galaxy as described in section [ methodology ] . the results obtained with the _ galaxy zoo _ data were compared also to the analysis using data taken by the rc3 catalogue @xcite and the na10 catalogue @xcite . unlike _ galaxy zoo _ that was classified by amateurs , rc3 and na10 were both classified by professional astronomers . figure [ rc3 ] shows the slopes of the arms of galaxies classified as ellipticals and s0 by rc3 . as the graph shows , galaxies classified as s0 have a higher arm slopes compared to galaxies classified as ellipticals . the figure also shows that like the _ galaxy zoo _ catalog , the arms of many of the galaxies classified by professional astronomers as ellipticals also has a certain slope . these results are in agreement with the observation of @xcite , according which professional astronomers do not outperform amateur astronomers in classification of galaxies into their broad morphological types . figure [ na10 ] shows the slopes of the arms of galaxies classified as ellipticals and s0 in the na10 catalog . these results are in agreement with the results of the rc3 catalog , showing higher arm slopes in galaxies classified manually as s0 . the distribution of the slopes of the arms was also analyzed for different redshifts , as displayed by figure [ na10_redshift ] , showing galaxies classified as ellipticals . as the figure shows , the fraction of galaxies with non - zero slope classified as ellipticals in the na10 catalog does not change significantly with the redshift in the tested redshift ranges . in this study we used computer - aided analysis based on the radial intensity plots of sdss galaxy images to examine spirality in galaxies that were classified manually as elliptical . while the unaided human eye provides a limited tool for analyzing elliptical galaxy images due to the limited sensitivity of the human vision to different gray levels , transforming the images to their radial intensity plots allows much easier detection of the spirality . the results suggest that more than a third of the galaxies that were classified manually by _ galaxy zoo _ participants as elliptical actually have a certain spirality . although in most cases the spirality was low , 10% of the galaxies classified as elliptical had a slope greater than 0.5 , suggesting that computer - aided analysis can in some cases be more sensitive to galaxy spirality compared to the human eye . these conclusions are also true for galaxies classified by professional astronomers , as was shown by using the rc3 and na10 catalogs . the results also exhibit redshift bias . this bias can be attributed to the quality of the images , as images of nearby galaxies provide higher image quality and therefore manual inspection of these images can be easier compared to images of galaxies with higher redshift , in which the ability of computer - aided analysis to detect subtle differences between gray levels can provide an advantage over the unaided human eye . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the us department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , university of cambridge , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max planck institute for astronomy ( mpia ) , the max planck institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory and the university of washington . abraham , r. g. , merrifield , m. r. , ellis , r. s. , tanvir , n. r. , brinchmann , j. , 1999 . the evolution of barred spiral galaxies in the hubble deep fields north and south . mnras 308 , 569576 . abraham , r. g. , van den bergh , s. , nair , p. , 2003 . a new approach to galaxy morphology . i. analysis of the sloan digital sky survey early data release . apj 588 , 218229 . ball , n. m. , brunner , r. j. , myers , a. d. , strand , n. e. , alberts , s. l. , tcheng , d. , 2008 . robust machine learning applied to astronomical data sets . probabilistic photometric redshifts for galaxies and quasars in the sdss and galex . apj , 683 , 1221 . ball , n. m. , loveday , j. , brunner , r. j. , 2008 . galaxy colour , morphology and environment in the sloan digital sky survey . mnras 383 , 907922 . banerji , m. , et al . galaxy zoo : reproducing galaxy morphologies via machine learning . mnras , 406 , 342353 . conselice , c.j . , 2003 . the relationship between stellar light distributions of galaxies and their formation histories . apjs , 147 , 128 . corwin , h. g. , jr . , buta , r. j. , de vaucouleurs , g. , 1994 . corrections and additions to the third reference catalogue of bright galaxies aj 108 , 21282144 . djorgovski , s. g. , davis , m. , 1987 . fundamental properties of elliptical galaxies . apj 313 , 5968 . dressler , a. , lynden - bell , d. , burstein , d. , davies , r. l. , faber , s. m. , terlevich , r. wegner , g. , 1987 . spectroscopy and photometry pf elliptical galaxies . i. a new distance estimator . apj 313 , 42 . huertas - company , m. , aguerri , j. a. l. , bernardi , m. , mei , s. , sanchez almeida , j. , 2011 . sdss automated morphology classification . a&a 525 , 157 joy , k. et al . , 2011 . moon zoo : citizen science in lunar exploration . astronomy & geophysics 52 , 1012 . kormendy , j. , fisher , d. b. , cornell , m. e. , bender , r. , 2009.structure and formation of elliptical and spheroidal galaxies . apjs 182 , 216309 . kormendy , j. , bender , r. , 2012 . a revised parallel - sequence morphological classification of galaxies : structure and formation of s0 and spheroidal galaxies . apjs 198 , 2 . lintott , c. j. , et al . galaxy zoo : morphologies derived from visual inspection of galaxies from the sloan digital sky survey . mnras 389 , 11791189 . lintott , c. j. , et al . 2011 . galaxy zoo 1 : data release of morphological classifications for nearly 900,000 galaxies . mnras 410 , 166178 . loveday , j. , 1996 . the apm bright galaxy catalogue . mnras 278 , 10251048 . masters , k. l. , et al . 2011 . galaxy zoo : bars in disk galaxies . mnras 411 , 20262034 . nair , p. , 2009 , phd dissertation , university of toronto nair , p. , abraham , r. , 2010 . on the fraction of barred spiral galaxies . apj , 714 , l260 otsu , n. , 1979 . a threshold selection method from gray level histograms . ieee trans . sys . man . 9 , 6266 . shamir , l. , nemiroff , r.j . a lossy fits image compression algorithm that protects user - defined levels of photometric integrity . aj 129 , 539546 . shamir , l. , nemiroff , r.j . , 2005b . a fuzzy logic based algorithm for finding astronomical objects in wide - angle frames . pasa 22 , 111117 . shamir , l. , 2009 . automatic morphological classification of galaxy images . mnras 399 , 13671372 . shamir l. , 2011a . ganalyzer : a tool for automatic galaxy image analysis . apj 736 , 141 . shamir l. , 2011b . astrophysics source code library , ascl:1105.011 shamir l. , 2012 , handedness asymmetry of spiral galaxies with [email protected] shows cosmic parity violation and a dipole axis , physics letters b , 715 , 2529 . scorza , c . , bender , r. , 1990 , a&a 235 , 49 . thorsten , l. , 2008 . is the gini coefficient a stable measure of galaxy structure ? apjs 179 , 319 tyson , j.a . , 2002 . large synoptic survey telescope : overview . proceedings of the spie , 4836 , 1020 . wallin , j. , holincheck , a. , borne , k. , lintott , c. , smith , a. , bamford , s. , fortson , l. , 2010 . tasking citizen scientists from galaxy zoo to model galaxy collisions . asp conference series , 423 , 217222 . van den bergh , s. , 2009 . what are s0 galaxies ? apj , 694 , l120l122 . van den bergh , s. , 2009 . lenticular galaxies and their environment . apj 702 , 1502 van dokkum , p. g. 2005 . the recent and continuing assembly of field elliptical galaxies by red mergers . aj 130 , 2647 - 2665 . york , d.g . et al . , 2000 . the sloan digital sky survey quasar catalog . ii . first data release . aj , 120 , 25792593 .
|
we use an automated galaxy morphology analysis method to quantitatively measure the spirality of galaxies classified manually as elliptical . the data set used for the analysis
consists of 60,518 galaxy images with redshift obtained by the sloan digital sky survey ( sdss ) and classified manually by _ galaxy zoo _ , as well as the rc3 and na10 catalogues .
we measure the spirality of the galaxies by using the ganalyzer method , which transforms the galaxy image to its radial intensity plot to detect galaxy spirality that is in many cases difficult to notice by manual observation of the raw galaxy image .
experimental results using manually classified elliptical and s0 galaxies with redshift @xmath00.3 suggest that galaxies classified manually as elliptical and s0 exhibit a nonzero signal for the spirality .
these results suggest that the human eye observing the raw galaxy image might not always be the most effective way of detecting spirality and curves in the arms of galaxies .
* keywords * : galaxies : elliptical and lenticular techniques : image processing .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
within the famous total variation ( tv ) regularization framework , image reconstruction in low - dose ct is formulated as minimizing the cost function expressed as @xmath0 subject to @xmath1 , and that in few - view ct is formulated as @xmath2 , where @xmath3 is the tv norm which aims at regularizing the ill - conditioned reconstruction problem . a variety of iterative algorithms have been proposed for solving the above problems in ct reconstruction fields based on modifying classical iterative algorithms such as the gradient method , art ( algebraic reconstruction technique ) , sirt ( simultaneous iterative reconstruction technique ) , and sart ( simultaneous algebraic reconstruction technique ) , often combined with the ordered - subsets ( os ) technique [ 1]-[10 ] ( and many others ) . however , it is fair to say that there exist very few iterative algorithms in engineering literatures , which not only converge fast but also can exactly solve the above tv - regularized problems . it is known that , from mathematical point of view , this difficulty partly comes from the non - differentiability of tv penalty . however , in applied mathematics fields , the research progress on this topic is very fast so that many nice algorithms to handle the tv regularization have been developed since 2010 . therefore , we believe that it would be time for ct engineers like us to develop and use more rigorous iterative algorithms for the tv - regularized image reconstructions . in this paper , we propose a very fast iterative algorithm which can be applied to the above two typical tv - regularized image reconstructions in a unified way . the algorithm derivation is outlined as follows . first , the original minimization problem is reformulated into the standard saddle point ( primal - dual ) problem by using the lagrangian duality , to which we apply the alternating projection proximal ( app ) algorithm which belongs to a class of first - order primal - dual methods including chambolle - pock algorithm , generalized iterative soft - thresholding ( gist ) algorithm , and alternating extragradient ( ae ) algorithm [ 11]-[14 ] . however , the resulting algorithm converges very slowly , mainly because its overall structure is same as the simultaneous iterative reconstruction methods such as the classical sirt and sart algorithms . to overcome this drawback , we precondition the iteration formula using the ramp filter of filtered backprojection ( fbp ) reconstruction algorithm in such a way that the solution to the preconditioned iteration perfectly coincides with the solution to the original problem . the final algorithm can be interpreted as the first - order primal - dual method accelerated by the fbp - type preconditioning using the ramp filter . unlike the famous known fbp - based acceleration technique called the iterative fbp algorithm [ 15],[16 ] , for both the above two different formulations ( low - dose ct and few - view ct ) , the proposed algorithm converges to the solution exactly minimizing the cost function . we evaluate this algorithm for both the low - dose ct case and the few - view ct case with 32 projection data . in the low - dose ct experiment , the proposed algorithm converged with approximately 10 iterations to the almost same image as the gist algorithm with 1,000 iterations . in the few - view ct experiment , the proposed algorithm required only 3 iterations to reach to the almost same image as the chambolle - pock algorithm with 1,000 iterations . these accelerations were achieved by the introduction of fbp - type preconditioning in the primal - dual space ( in the saddle point formulation ) , which has not been investigated yet and is an original contribution of this paper . in this section , we formulate image reconstruction in the low - dose ct and review typical existing iterative algorithms . we denote an image by a @xmath4-dimensional vector @xmath5 and denote a corresponding projection data ( sinogram ) by an @xmath6-dimensional vector @xmath7 . we denote the system matrix which relates @xmath8 to @xmath9 by @xmath10 , where we assume that @xmath11 and @xmath9 is contaminated with statistical noise . throughout this paper , we assume that the projection data is acquired by the standard parallel - beam geometry so that @xmath9 consists of uniform samples of continuous projection data @xmath12 . normally , image reconstruction in this setup can be formulated as the problem of minimizing the penalized weighted least - squares ( p - wls ) cost function expressed as @xmath13 where @xmath14 denotes @xmath15 diagonal matrix in which each diagonal element @xmath16 is the inverse of noise variance @xmath17 , and @xmath18 is the penalty function to smooth the image [ 17],[18 ] . although there exist several choices in @xmath18 , throughout this paper , we assume that @xmath18 is the tv penalty function defined by @xmath19 where @xmath20 and @xmath21 are inner product representations of finite difference operations around the @xmath22-th pixel along the horizontal and vertical directions , respectively . see fig . 1 for the detailed definitions of @xmath23 and @xmath24 . since 2008 when ge healthcare developed the asir ( advanced statistical iterative reconstruction ) software , iterative low - dose ct reconstruction has been progressed according to the following three stages . [ image space denoising ( 1-st generation ) ] this class of reconstruction methods first perform an fbp reconstruction followed by a smoothing by using an iterative edge - preserving denoising algorithm such as the tv - denoising and the mrf - based denoising . [ iterative fbp algorithm ( 2-nd generation ) ] typically , this class of reconstruction methods are based on the following iteration formula @xmath25,\end{aligned}\ ] ] where @xmath26 $ ] denotes the projection operator onto the positive orthant @xmath1 and @xmath27 is the 1-d ramp filter of fbp reconstruction algorithm having the frequency response @xmath28 . thanks to the introduction of @xmath27 during the iteration , eq . ( 3 ) converges very fast . however , it is well - known that is can not exactly minimize the p - wls cost function of eq . another limitation is that the penalty function @xmath29 needs to be differentiable so that the tv penalty , which is non - differentiable , can not be used . [ true iterative reconstruction ( ir ) algorithm ( 3-rd generation ) ] this class of reconstruction methods try to exactly minimize the p - wls cost function of eq . ( 1 ) . for example , if we use the gradient method , its iteration formula can be expressed as @xmath30.\end{aligned}\ ] ] the drawback of this method is that its convergence is rather slow compared with the iterative fbp algorithm when no acceleration techniques are incorporated . another limitation is that the penalty function @xmath29 needs to be differentiable so that the tv penalty can not be used again . [ preconditioned gradient method ] an alternative approach which is known in image reconstruction community is to use the iteration formula as @xmath31,\end{aligned}\ ] ] where @xmath32 denotes the 2-d ramp filter in image space having the frequency response @xmath33 [ 19 ] . unlike eq . ( 3 ) , this method converges to an exact minimizer of eq . ( 1 ) , but it is rarely used mainly because the 2-d filter is computationally intensive and its design is troublesome . we note that the proposed algorithm derived in the following sections also uses the fbp - type acceleration as in eq . ( 3 ) , but it exactly converges to a minimizer of eq . ( 1 ) and also allows to use the tv penalty . therefore , we believe that the proposed algorithm overcomes the most major drawbacks of existing low - dose ct iterative reconstruction algorithms . in fig . 2 , we show a comparison between the true ir reconstruction and the iterative fbp reconstruction for a numerical chest phantom without a regularization term in the low - dose ct setup . as expected from the theory , it can be observed that the noise property of iterative fbp reconstruction is worse compared with the true ir reconstruction . this result motivates us to develop an alternative fbp - type acceleration technique which is able to exactly minimize the wls and p - wls cost functions . and the vertical difference @xmath34 used in the tv penalty function.,height=264 ] from this section , we derive the proposed algorithm , which is based on reformulating the minimization of eq . ( 1 ) into a saddle point ( primal - dual ) problem , precondition it , and applying the alternating projection proximal ( app ) algorithm which belongs to a class of first - order primal - dual methods [ 11 ] . first , eq . ( 1 ) can be reformulated into the following equality constrained minimization by introducing an additional variable @xmath35 . @xmath36 where we note that @xmath35 can be interpreted as forward - projected projection data computed from @xmath8 . next , we perform a preconditioning on the constraint @xmath37 . when we solve eq . ( 6 ) using an iterative algorithm , its convergence speed strongly depends on the condition of linear constraint @xmath37 . so , we perform the preconditioning by multiplying @xmath38 by some non - singular @xmath15 matrix @xmath39 . then , eq . ( 6 ) can be converted into @xmath40 the detailed form of @xmath39 is not shown here , but we discuss later which choice of @xmath39 is best to accelerate the convergence . intuitively , if we choose @xmath39 in such a way that @xmath41 approximates the ramp filter in the fbp reconstruction , the final structure of iteration formula resembles the iterative fbp algorithm leading to a fast convergence . furthermore , we remark that the rewriting from eq . ( 6 ) to eq . ( 7 ) does not alter the problem solution @xmath42 . next , we reformulate eq . ( 7 ) into a form of saddle point problem [ 11]-[14 ] . the procedure is as follows . the lagrangian function @xmath43 corresponding to eq . ( 7 ) with respect to the constraint @xmath44 is defined by @xmath45 where @xmath46 is the lagrange multiplier vector , which is also called the dual variable . therefore , eq . ( 7 ) can be converted into the saddle point problem@xmath47 furthermore , the variable @xmath35 can be eliminated in eq . ( 9 ) by solving the minimization with respect to @xmath35 . by performing this computation , we obtain @xmath48 where @xmath49 denotes the inner product . in some literatures , ( 10 ) is called the standard form of saddle point problem [ 12 ] . it is also called the primal - dual ( pd ) problem whereas the original problem of eq . ( 1 ) is called the primal ( p ) problem [ 20]-[22 ] . by using the so - called strong duality theorem in optimization literatures , we can prove the following theorem [ 20]-[22 ] . [ theorem ] assume that the penalty function @xmath18 is a possibly non - differentiable convex function . we denote the solution to the primal problem ( eq . ( 1 ) ) by @xmath50 . we denote the solution to the primal - dual problem ( eq . ( 10 ) ) by @xmath51 . then , the following two properties hold . \(1 ) @xmath52 \(2 ) @xmath53 from this theorem , if we succeed in exactly solving the saddle point problem of eq . ( 10 ) , its solution vector @xmath54 is also a solution of the original problem of eq . ( 1 ) , _ i.e. _ @xmath50 . in summary of the above discussion , we need to solve the following saddle point problem . @xmath55 where the non - negativity constraint @xmath1 was put into the cost function @xmath56 as the indicator function @xmath57 defined by @xmath58 the proposed algorithm is based on solving eq . ( 11 ) instead of eq . there exist a variety of iterative methods to solve eq . ( 11 ) in optimization literatures such as the classical multiplier method , augmented lagrangian method , alternating direction method of multiplier ( admm ) method , etc [ 20]-[23 ] . among them , we use a class of iterative methods called the first - order primal - dual methods , because they require relatively simple computations per iteration [ 11]-[14 ] . the basic overall structure of this iterative method is to repeat an update of primal variable @xmath8 along the descent direction of lagrangian @xmath59 , _ i.e. _ the descent direction of @xmath60 , and an update of dual variable @xmath46 along the ascent direction of lagrangian @xmath61 , _ i.e. _ the ascent direction of @xmath62 , alternately with some additional extrapolation step . the role of extrapolation step is to guarantee that each update from @xmath63 to @xmath64 becomes a non - expansive mapping leading to a convergence to the saddle point . in each update along the descent or ascent direction , either the gradient step or the proximal step can be used dependent on differentiabilities of @xmath56 and @xmath65 . typically , there exist four variations in the first - order primal - dual methods , which are the chambolle - pock algorithm , generalized iterative soft - thresholding ( gist ) algorithm , alternating projection proximal ( app ) algorithm , and alternating extragradient ( ae ) algorithm [ 11]-[14 ] . the differences in used operations among the four algorithms are briefly summarized in table 1 . .differences among the four first - order primal - dual algorithms . [ cols="<,<,<,<,<",options="header " , ] in our problem , as is clear from eq . ( 11 ) , @xmath56 is non - differentiable because it corresponds to the tv penalty ( combined with the non - negativity constraint ) whereas @xmath65 is differentiable ( quadratic form ) . thus , we decided to employ the app algorithm [ 11 ] . for general ( non - differentiable ) @xmath56 and ( differentiable ) @xmath65 , the iteration formula of app algorithm in its original form is summarized as follows . @xmath66 } \nonumber \\ & & \ \ \ \ \ \ \ \ \overline{\vec{\mu}}^{(k+1)}=\vec{\mu}^{(k)}+\sigma(d^{1/2}a\vec{x}^{(k)}-\nabla h(\vec{\mu}^{(k ) } ) ) \\ & & { \rm [ primal\ update\ ( proximal ) ] } \nonumber \\ & & \ \ \ \ \ \ \ \ \vec{x}^{(k+1)}={\rm prox}_{\tau g}(\vec{x}^{(k)}-\tau a^td^{t/2}\overline{\vec{\mu}}^{(k+1)})\\ & & { \rm [ dual\ update\ ( gradient ) ] } \nonumber \\ & & \ \ \ \ \ \ \ \ \vec{\mu}^{(k+1)}=\vec{\mu}^{(k)}+\sigma(d^{1/2}a\vec{x}^{(k+1)}-\nabla h(\vec{\mu}^{(k)})),\end{aligned}\ ] ] where @xmath67 is the iteration number , and @xmath68 , @xmath69 are stepsize parameters corresponding to the primal update and the dual update , respectively . to guarantee the convergence , we assume that @xmath70 and @xmath71 are selected such that @xmath72 where @xmath73 is lipschitz constant of the gradient @xmath74 [ 11 ] . furthermore , the operator @xmath75 appearing in eq . ( 14 ) denotes the proximity ( prox ) operator defined by @xmath76 where @xmath70 is called the stepsize parameter [ 24 ] . we will describe later on how to compute the prox operator when @xmath18 is the tv penalty . we note that eq . ( 13 ) represents the extrapolation step to mathematically guarantee the convergence , eq . ( 14 ) is the primal update using the prox operator , and eq . ( 15 ) is the dual update using the gradient step . the iteration formula in our special case of eq . ( 11 ) is obtained by applying the general algorithm of eqs . ( 13)-(15 ) followed by the variable change @xmath77 and rewriting the extrapolation step into a much simpler form . the resulting iteration formula is expressed as @xmath66 } \nonumber \\ & & \ \ \ \ \ \ \ \ \overline{\vec{\mu}}^{(k+1)}=\vec{\mu}^{(k)}+\sigma d(a\vec{x}^{(k)}-\vec{b}-w^{-1}\vec{\mu}^{(k)})=2\vec{\mu}^{(k)}-\vec{\mu}^{(k-1)}-\sigma dw^{-1}(\vec{\mu}^{(k)}-\vec{\mu}^{(k-1 ) } ) \\ & & { \rm [ primal\ update ] } \nonumber \\ & & \ \ \ \ \ \ \ \ \vec{x}^{(k+1)}={\rm prox}_{\tau g}(\vec{x}^{(k)}-\tau a^t\overline{\vec{\mu}}^{(k+1)})\\ & & { \rm [ dual\ update ] } \nonumber \\ & & \ \ \ \ \ \ \ \ \vec{\mu}^{(k+1)}=\vec{\mu}^{(k)}+\sigma d(a\vec{x}^{(k+1)}-\vec{b}-w^{-1}\vec{\mu}^{(k)}),\end{aligned}\ ] ] where @xmath78 and we simplified the extrapolation step using the fact that the @xmath79 term in the extrapolation step is same as @xmath80 appearing in the previous dual update . the algorithm derivation was finished . with respect to computational requirements , the dominant computations in implementing eqs . ( 18)-(20 ) are a forward projection @xmath81 , a backprojection @xmath82 , and a prox operator associated with the tv penalty in each iteration . therefore , we expect that the computational costs of the proposed algorithm are similar to those of the other iterative reconstruction algorithms which use the tv penalty . in implementing eqs . ( 18)-(20 ) , there still exist two unclear points . the first point is how to compute @xmath75 in the primal update , which amounts to solving the minimization of eq . when @xmath18 is the tv penalty of eq . ( 2 ) , eq . ( 17 ) is same as the so - called tv denoising problem ( rof problem ) investigated in [ 25],[26 ] , to which a variety of efficient algorithms are available . in our implementations , we use chambolle s projection algorithm for this purpose modified in such a way that the non - negativity constraint @xmath1 is incorporated [ 26 ] . the second unclear point concerns how to design the preconditioning matrix @xmath83 to achieve a fast convergence . intuitively , from eqs . ( 18)-(20 ) , it can be inspired that a nice choice of @xmath83 is the ramp filter in the fbp reconstruction , because , in this case , the first iteration initialized with @xmath84 coincides with the fbp reconstruction followed by the tv denoising . however , this is not the best choice when the projection data @xmath9 contains statistical noise . we discuss this issue below . first , we note that @xmath83 needs to be a positive definite matrix , because @xmath83 is required to be expressed as @xmath78 . by forgetting the extrapolation step ( for some moment ) and combining eqs . ( 19 ) and ( 20 ) , the approximate iteration formula with respect to the dual variable @xmath46 is expressed as @xmath85 furthermore , neglecting the prox operator , we have @xmath86 equation ( 22 ) implies that the meaning of @xmath46 is the noise compoment in the projection data and the app method is implicitly solving the equation @xmath87 at each iteration @xmath67 by using the preconditioned iteration of eq . ( 22 ) . from this observation , the best choice of @xmath83 is clearly given by @xmath88 however , eq . ( 23 ) is not practical because its computation requires a large matrix inverse . so , we use a shift - invariant approximation @xmath89 where @xmath90 is the average value of diagonal elements of @xmath91 . then , noting that @xmath92 is the blurring operator @xmath93 in projection data space ( @xmath94 is the number of view angles over @xmath95 in the parallel - beam projection data ) , @xmath83 becomes the frequency domain filter expressed as @xmath96 where @xmath97 and @xmath98 denote the 1-d fourier transform with respect to the radial variable in the projection data space and its inverse , respectively . we note that eq . ( 24 ) is the smoothed ramp filter . we show the frequency response @xmath99 with varying parameter @xmath70 in fig . 3 . finally , we summarize that the smoothed ramp filter is the best preconditioning matrix @xmath83 in the case of low - dose ct reconstruction . we experimentally confirmed that there exists an essential difference between the ordinary ( non - smoothed ) ramp filter and the smoothed ramp filter in the convergence speed of cost function value @xmath100 . used in the proposed algorithm for the low - dose ct.,height=188 ] we summarize the proposed algorithm for the low - dose ct in algorithm 1 . as a special case of @xmath101 , _ i.e. _ absence of the tv penalty term , we obtain a fast fbp - preconditioned algorithm to exactly minimize the wls cost function , which is also summarized in algorithm 1. p40em algorithm 1 : low - dose ct tv - regularized wls reconstruction + prepare the preconditioning matrix @xmath83 . give initial primal and dual vectors by @xmath102 and @xmath103 . set stepsize parameters @xmath104 such that @xmath105 and @xmath106 . execute the following steps for @xmath107 . [ step 1 ] extrapolation step + @xmath108 + [ step 2 ] primal update + @xmath109 + @xmath110 + [ step 3 ] dual update + @xmath111 + p40em algorithm 1 : low - dose ct wls reconstruction + prepare the preconditioning matrix @xmath83 . give initial primal and dual vectors by @xmath102 and @xmath103 . set stepsize parameters @xmath104 such that @xmath105 and @xmath106 . execute the following steps for @xmath107 . [ step 1 ] extrapolation step + @xmath108 + [ step 2 ] primal update + @xmath112 $ ] + [ step 3 ] dual update + @xmath111 + the algorithm derivation used in section 2 based on the first - order primal - dual methods combined with the preconditioning is a rather general framework so that it can be applied to develop a fast convergent reconstruction algorithm in the few - view ct formulation . we denote an image by a @xmath4-dimensional vector @xmath5 and denote a corresponding projection data ( sinogram ) by an @xmath6-dimensional vector @xmath7 . we denote the system matrix which relates @xmath8 to @xmath9 by @xmath10 , where we assume that @xmath113 and @xmath9 does not contain statistical noise . normally , image reconstruction in this setup can be formulated as the problem of minimizing a penalty function @xmath18 under the linear constraint @xmath114 as @xmath115 where we assume that @xmath29 is the tv penalty defined by eq . ( 2 ) throughout this section . very often , in engineering literatures , the problem of eq . ( 25 ) is converted into the unconstrained minimization @xmath116 followed by solving it using a technique of unconstrained optimizations . however , the solution obtained in this way can be quite different from that of the original constrained problem of eq . the proposed algorithm below allows us to exactly solve eq . ( 25 ) . we derive the proposed algorithm to solve eq . ( 25 ) using the same mathematical framework as in section 2 below . first , we introduce the @xmath15 non - singular preconditioning matrix @xmath39 into eq . ( 25 ) as @xmath117 we note that the solution to eq . ( 25 ) is same as the solution to eq . . the lagrangian function @xmath118 corresponding to eq . ( 26 ) with respect to the constraint @xmath119 is defined by @xmath120 where @xmath46 is the lagrange multiplier vector ( the dual variable ) . therefore , eq . ( 26 ) can be converted into the saddle point problem @xmath121 where @xmath57 is the indicator function defined by eq . ( 12 ) . since the structure of eq . ( 28 ) is same as that of eq . ( 11 ) ( there exists only a difference in the form of @xmath65 ) , we can use the same approach as in the low - dose ct case to develop an iterative algorithm . applying the app algorithm given by eqs . ( 13)-(15 ) to eq . ( 28 ) yields the following iterative algorithm . @xmath66 } \nonumber \\ & & \ \ \ \ \ \ \ \ \overline{\vec{\mu}}^{(k+1)}=\vec{\mu}^{(k)}+\sigma d(a\vec{x}^{(k)}-\vec{b})=2\vec{\mu}^{(k)}-\vec{\mu}^{(k-1 ) } \\ & & { \rm [ primal\ update ] } \nonumber \\ & & \ \ \ \ \ \ \ \ \vec{x}^{(k+1)}={\rm prox}_{\tau g}(\vec{x}^{(k)}-\tau a^t\overline{\vec{\mu}}^{(k+1)})\\ & & { \rm [ dual\ update ] } \nonumber \\ & & \ \ \ \ \ \ \ \ \vec{\mu}^{(k+1)}=\vec{\mu}^{(k)}+\sigma d(a\vec{x}^{(k+1)}-\vec{b}).\end{aligned}\ ] ] the algorithm derivation was finished . we note that the only difference between the low - dose ct case and the few - view ct case lies in the absence of @xmath122 term in the latter from the comparison between eqs . ( 18)-(20 ) and eqs . ( 29)-(31 ) . next , we discuss about the preconditioning matrix @xmath83 to achieve a fast convergence . by following the same discussion as in the low - dose ct case , thanks to the absence of @xmath122 term , the best preconditioning matrix @xmath83 in eqs . ( 29)-(31 ) can be shown to be the ramp filter in the fbp reconstruction , which is expressed as @xmath123 where we note again that @xmath94 is the number of view angles over @xmath95 in the parallel - beam projection data , and @xmath97 and @xmath98 denote the 1-d fourier transform with respect to the radial variable in the projection data space and its inverse , respectively . with this choice of @xmath83 , the first iteration initialized with @xmath84 coincides with the fbp reconstruction followed by the tv smoothing ( denoising ) . we summarize the proposed algorithm for the few - view ct in algorithm 2 . p40em algorithm 2 : few - view ct tv - regularized reconstruction + prepare the preconditioning matrix @xmath83 . give initial primal and dual vectors by @xmath102 and @xmath103 . set stepsize parameters @xmath104 such that @xmath124 . execute the following steps for @xmath107 . [ step 1 ] extrapolation step + @xmath125 + [ step 2 ] primal update + @xmath109 + @xmath126 + [ step 3 ] dual update + @xmath127 + we have performed two simulation studies to demonstrate performances of algorithm 1 in the low - dose ct reconstruction and algorithm 2 in the few - view ct reconstruction . the details of simulation studies are summarized below . [ low - dose ct simulation ] we used a numerical phantom called the spot phantom consisting of @xmath128 ( pixels ) and a single slice of chest - scan ct image consisting of 320@xmath129320 ( pixels ) . the simulated projection data was computed with 1,200 ( angles ) by the parallel - beam geometry followed by adding noise which follows transmission poisson statistics , from which image reconstructions were performed . we compared the proposed algorithm with the following three competitive algorithms . \(a ) gradient descent algorithm : the standard gradient descent algorithm to minimize the tv - regularized wls cost function of eq . ( 1 ) was implemented . to deal with the non - differentiability of tv penalty , the tv norm was approximated by the standard smoothing technique as @xmath130 where @xmath131 is a small positive number to control the degree of smoothing . \(b ) iterative fbp algorithm [ 15],[16 ] : the iterative fbp algorithm was implemented mainly to demonstrate that it can not accurately minimize the tv - regularized wls cost function of eq . ( 1 ) and there exists an image degradation although it converges very fast . \(c ) generalized iterative soft - thresholding ( gist ) algorithm [ 13 ] : we also implemented one of newest iterative algorithms developed by applied mathematicians , which is able to exactly minimize the tv - regularized wls cost function of eq . since this algorithm is of simultaneous update type such as the sirt and sart algorithms , we expect that its convergence is rather slow . [ few - view ct simulation ] we used the spot phantom same as in the low - dose ct simulation . the simulated projection data was computed with 32 ( angles ) by the parallel - beam geometry , from which image reconstructions were performed . we compared the proposed algorithm with the following two competitive algorithms . \(a ) chambolle - pock algorithm [ 4],[12 ] : we implemented one of newest iterative algorithms developed by applied mathematicians , which is able to exactly solve the minimization of tv penalty under the constraints @xmath132 . since this algorithm is of simultaneous update type as in the gist algorithm , we expect that its convergence is rather slow . \(b ) art algorithm [ 27 ] : we also implemented the standard art algorithm without the tv penalty , which is known to converge fast thanks to its row - action ( sequential update ) structure . in figs . 4 - 5 , we show reconstructed images in the low - dose ct case together with corresponding root mean squares ( rmse ) reconstruction errors defined by @xmath133 where @xmath134 denotes the reconstructed pixel value and @xmath135 denotes the true pixel value . also , in fig . 6 , we show the corresponding convergence properties of each reconstruction algorithm . these figures clearly demonstrate that algorithm 1 converges to the exact minimizer of the tv - regularized wls cost function very fast . in particular , it can be observed that the iterative fbp algorithm converges to an image with worse rmse value although its convergence is very fast , whereas algorithm 1 converges to the exact minimizer with a much faster speed compared with the other slow algorithms such as the gradient descent and gist algorithms thanks to the fbp - type preconditioning . in fig . 7 , we show reconstructed images in the few - view ct case together with corresponding rmse reconstruction errors . also , in fig . 8 , we show the corresponding convergence properties of each reconstruction algorithm . these figures clearly demonstrate that algorithm 2 converges to an image with much smaller rmse value compared with the art algorithm , and its convergence is much faster than the chambolle - pock algorithm again thanks to the fbp - type preconditioning . was plotted.,height=566 ] in this paper , we proposed new image reconstruction algorithms for two typical tv - regularized reconstruction problems in tomography . the first algorithm ( algorithm 1 ) minimizes the tv - regularized wls cost function for the low - dose ct , and the second algorithm ( algorithm 2 ) minimizes the tv penalty under the data fidelity linear constraint for the few - view ct . the both algorithms were derived from the saddle point reformulation of each optimization problem followed by the preconditioning using the ramp filter of fbp reconstruction algorithm and applying the alternating projection proximal ( app ) iterative algorithm . we demonstrated that both algorithm 1 and algorithm 2 converge very fast while its convergence to the exact solution of each optimization problem is guaranteed . on the low - dose ct reconstruction , very recently , we began to use a shift - variant preconditioner instead of the smoothed ramp filter , which approximates @xmath136 better , and confirmed an essential improvement . on the few - view ct reconstruction , we began to extend the formulation to @xmath137 by introducing the error tolerance @xmath138 . these newest results will be published elsewhere . this work was partially supported by jsps kakenhi grant number 15k06103 . 1 e.y.sidky and x.pan `` image reconstruction in circular cone - beam computed tomography by constrained , total - variation minimization , '' _ phys.med.biol._ , vol.53 , pp.4777 - 4807 , 2008 . h.yu and g.wang `` a soft - thresholding filtering approach for reconstruction from a limited number of projections , '' _ phy.med.biol._ , vol.55 , pp.3905 - 3916 , 2010 . s.ramani and j.a.fessler `` a splitting - based iterative algorithm for accelerated statistical x - ray ct reconstruction , '' _ ieee trans.med.imaging_ , vol.31 , pp.677 - 688 , 2012 . e.y.sidky , j.h.jorgensen , and x.pan `` convex optimization problem prototyping for image reconstruction in computed tomography with the chambolle - pock algorithm , '' _ phys.med.biol._ , vol.57 , pp.3065 - 3091 , 2012 . l.ritschl , f.bergner , c.fleischmann , m.kachelriess `` improved total variation - based ct image reconstruction applied to clinical data , '' _ phys.med.biol._ , vol.56 , pp.1545 - 1561 , 2011 . j.song , q.h.liu , g.a.johnson , and c.t.bade `` sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro - ct , '' _ med.phys._ , vol.34 , pp.4476 - 4483 , 2007 . z.chen , x.jin , l.li , and g.wang `` a limited - angle ct reconstruction method based on anisotropic tv minimization , '' _ phys.med.biol._ , vol.58 , pp.2119 - 2141 , 2013 . z.tian , x.jia , k.yuan , t.pan , and s.b.jiang `` low dose ct reconstruction via edge - preserving total variation regularization , '' _ phys.med.biol._ , vol.56 , pp.5949 - 5967 , 2011 . m.defrise , c.vanhove , and x.liu `` an algorithm for total variation regularization in high - dimensional linear problems , '' _ inverse problems _ , vol.27 , paper no . 065002 , 2011 . v.p.gopia , p.palanisamya , k.a.wahidb , p.babync , and d.cooperd `` micro - ct image reconstruction based on alternating direction augmented lagrangian method and total variation , '' _ comput.med.imaging graph . _ , vol.37 , pp.419 - 429 , 2013 . p.tseng `` alternating projection - proximal methods for convex programming and variational inequalities , '' _ siam j.opt._ , vol.7 , pp.951 - 965 , 1997 . a.chambolle and t.pock `` a first - order primal - dual algorithm for convex problems with applications to imaging , '' _ j.math.imaging vis . _ , vol.40 , pp.120 - 145 , 2011 . i.loris and c.verhoeven `` on a generalization of the iterative soft - thresholding algorithm for the case of non - separable penalty , '' _ inverse problems _ , vol.27 , paper no.125007 , 2011 . s.bonettini and v.ruggiero `` an alternating extragradient method for total variation - based image restoration from poisson data , '' _ inverse problems _ , vol.27 , paper no.095001 , 2011 . k.zeng , z.chen , l.zhang , and g.wang `` an error - reduction - based algorithm for cone - beam computed tomography , '' _ med.phys._ , vol.31 , pp.3206 - 3212 , 2004 . j.sunnegardh , p-e.danielsson `` regularized iterative weighted filtered backprojection for helical cone - beam ct , '' _ med phys _ , vol.35 , pp.4173 - 4185 , 2008 . j.a.fessler `` penalized weighted least - squares image reconstruction for positron emission tomography , '' _ ieee trans.med.imaging_ , vol.13 , pp.290 - 300 , 1994 . e.u.mumcuoglu , r.leahy , s.r.cherry , and z.zhou `` fast gradient - based methods for bayesian reconstruction of transmission and emission pet images , '' _ ieee trans.med.imaging_ , vol.13 , pp.687 - 701 , 1994 . g.chinn and s-c.huang `` a general class of preconditioners for statistical iterative reconstruction of emission computed tomography , '' _ ieee trans.med.imaging_ , vol.16 , pp.1 - 10 , 1997 . d.p.bertsekas `` nonlinear programming , '' athena scientific , 1999 . d.p.bertsekas `` convex optimization algorithms , '' athena scientific , 2015 . r.glowinski and p.letallec `` augmented lagrangian and operator - splitting methods in nonlinear mechanics , '' siam , 1987 . s.boyd , n.parikh , and e.chu `` distributed optimization and statistical learning via the alternating direction method of multipliers , '' foundations and trends(r ) in machine learning , now publishers inc . , 2011 . n.parikh and s.boyd `` proximal algorithms , '' foundations and trends(r ) in optimization , now publishers inc . , 2013 . l.i.rudin , s.osher , and e.fatemi `` nonlinear total variation noise removal algorithm , '' _ physica d _ , vol.60 , pp.259 - 268 , 1992 . a.chambolle `` an algorithm for total variation minimization and applications , '' _ j.math.imaging vis . _ , vol.20 , pp.89 - 97 , 2004 . g.t.herman and l.b.meyer `` algebraic reconstruction techniques can be made computationally efficient , '' _ ieee trans.med.imaging_ , vol.12 , pp.600 - 609 , 1993 .
|
this paper concerns iterative reconstruction for low - dose and few - view ct by minimizing a data - fidelity term regularized with the total variation ( tv ) penalty .
we propose a very fast iterative algorithm to solve this problem .
the algorithm derivation is outlined as follows .
first , the original minimization problem is reformulated into the saddle point ( primal - dual ) problem by using the lagrangian duality , to which we apply the first - order primal - dual iterative methods .
second , we precondition the iteration formula using the ramp filter of filtered backprojection ( fbp ) reconstruction algorithm in such a way that the problem solution is not altered .
the resulting algorithm resembles the structure of so - called iterative fbp algorithm , and it converges to the exact minimizer of cost function very fast .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
odd - frequency ( of ) superconductivity @xcite , which is characterized by pair potential or pair amplitude with odd functions in time and frequency , has been recognized as a candidate of intriguing quantum states of matter . while its realization has been theoretically proposed in a variety of systems @xcite there has been a long - standing problem : a thermodynamic instability arises if we apply a conventional approach to of superconductivity which has succeeded in describing ordinary even - frequency ( ef ) superconductors @xcite . at the same time , the sign of electromagnetic response function is reversed from the usual diamagnetic one , indicating a paramagnetic meissner response and negative superfluid density . therefore , its realization in bulk of condensed matter has been questioned . on the other hand , without such difficulties the of superconductivity can exist as a surface state . while the ordinary ef superconductivity is dominant in bulk , the of pairing state is present as an induced state @xcite . it has been reported that the paramagnetic meissner response @xcite is observed in this induced of pairing state @xcite . recent theoretical studies show that the of pairing can also be stabilized in a bulk , if we reconsider some conditions which are usually assumed in the theory of conventional superconductors . namely , the sign arising from the of pair potential @xmath0 , which causes the thermodynamic instability , can be canceled by introducing an additional sign . one of the solutions is to re - examine the conjugate relations of the frequency - dependent pair potential @xcite . using a path - integral formalism , it has been shown in refs . and that the problem can be resolved by using an unusual conjugate relation for pair potential , and consequently a description based on hermitian mean - field hamiltonian is impossible . subdominant even - frequency pairings induced in inhomogeneous systems such as surface or defect have been classified for this kind of of pairing based on symmetry arguments @xcite . with this situation , however , recently josephson junctions have also been studied by fominov _ et al . _ and peculiar properties are revealed @xcite . whereas a real current is obtained for the junction with ef superconductor , the current becomes imaginary if we make a junction with paramagnetic of superconductivity realized at e.g. the edges of @xmath1- or @xmath2-wave superconductors . furthermore , an electromagnetic response function shows that the superfluid weight becomes complex number when these diamagnetic and paramagnetic odd - frequency pairings coexist @xcite . these unphysical behaviors at least indicate that the conventional approach fails in describing the coexistence of the above of superconductor . on the other hand , it has also been established that there is another type of stable of superconductivity , which can be described in the mean - field theory with hermitian hamiltonian . here , the additional minus sign to resolve the thermodynamic problem comes from a spatially oscillating phase of the pair amplitude @xcite , which is called staggered pairing @xcite . the existence of staggered of pairing has been clearly demonstrated in the two - channel kondo lattice ( tckl ) @xcite . in this paper , for a deeper understanding of this type of of superconductors , we take tckl as a concrete example and explore novel properties of the junction systems to clarify the difference from already known superconductors . . classification of uniform and staggered pairs . the sign represents even or odd character of the exchange symmetry . the symmetry in bracket is a secondarily induced pair from a purely of or ef pair potential . [ cols="^,^,^,^,^,^",options="header " , ] for transformed hamiltonian given in eq . , the of pair has a uniform character . in terms of the classification in tab . [ tab : list ] , the transformed state belongs to uniform oseo . since the spin symmetry is broken in this picture , the uniform pair with eteo is mixed at the same time . thus the secondarily induced pairs are transformed from esoo to eteo by the local gauge transformation . in the following we explain how the of pairs in tckl give the ordinary diamagnetic meissner effect , although the odd - frequency superconductors have long been considered to give a paramagnetic meissner kernel . while the numerical calculation @xcite shows the diamagnetic response , here we discuss it by focusing on the structure of the meissner kernel and do not enter the details . following the derivation in ref . , only the anomalous part contributes to the meissner kernel @xmath3 which can be written in the form @xmath4 where @xmath5 is an electric charge , and we define the velocity @xmath6 along the @xmath7-direction . we have also introduced the ` daggered ' anomalous green function by @xmath8 . from hermiticity of the hamiltonian , we have the relation @xmath9 this relation can be explicitly shown by using the spectral representation . we assume the inversion symmetry in the original lattice : @xmath10 . let us consider the conventional spin - singlet @xmath11-wave ( ef ) superconductor as a reference . the anomalous green s function has the structure in the form @xmath12 here we do not have to know the detailed functional form of @xmath13 . the orbital degree of freedom is not included here . the meissner kernel is then given by @xmath14 for the velocity we have the relation @xmath15 , which gives the minus sign . in addition , another sign comes from the spin factor @xmath16 , and hence in total the electromagnetic response is diamagnetic : @xmath17 . on the other hand , if we had @xmath11-wave spin - triplet of superconductivity , there would be no sign from spin - factor . hence in this case the sign of the meissner kernel is reversed to give a paramagnetic response ( or sometimes called negative meissner effect ) . for @xmath1-wave superconductors , the minus sign comes from spatial part , i.e. @xmath18 , instead of spin part . now we consider the kernel in tckl . substituting eq . into eq . , we obtain @xmath19 although the factors from spin and orbital parts give the minus sign as @xmath20 , the sign operates twice and does not affect the total meissner kernel . for the velocity , we have @xmath21 originating from @xmath22 , which gives no minus sign in contrast to the above @xmath11-wave spin - singlet superconductor . we further transform the expression in terms of ef and of pair amplitudes @xmath23 originating from eqs . and . using the relation in eq . , the final expression is written as @xmath24 . \label{eq : tckl_kernel}\end{aligned}\ ] ] namely the of pair ( oseo ) gives a diamagnetic contribution and the ef pair ( esoo ) shows a paramagnetic response , which is contrary to the standard wisdom . although it is not trivial to determine which parts give the dominant contribution , the numerical calculation shows that the of part is more dominant to give the total diamagnetic response @xcite . this fact implies the importance of the of pair in tckl . the characteristic diamagnetic response by of pairs in tckl is closely related to @xmath21 with finite center - of - mass momentum @xmath25 . otherwise we would have another minus sign from @xmath26 and then the of pair gives paramagnetic contribution . this point has also been numerically demonstrated in ref . . at the end of this section , let us also consider the meissner kernel in the modified tckl given by eq . . in this case the uniform pair amplitudes have the form @xmath27 . \label{eq : pair_sym_mod}\end{aligned}\ ] ] the first and second terms in the right - hand side respectively correspond to oseo and eteo pairs . the meissner kernel has the form @xmath28 instead of eq . . the important point here is that the velocity is dependent on spin : @xmath29 and @xmath30 . substituting eq . into the kernel , we obtain the essentially same result as eq . which shows diamagnetic response . while we have no staggered phase here , the additional minus sign comes from the spin - dependent velocity . thus we have explicitly demonstrated that the staggered nature is not the only way to stabilize of superconductivity . in this section we consider a tunneling conductance in normal metal ( n)/superconductor ( s ) junction , where s is a superconducting tckl in one dimension . tunneling conductance can be calculated based on the blonder - thinkham - klapwijk theory @xcite , and a similar method is developed also in the tight - binding system @xcite . we choose the bulk wave functions of tckl satisfying a proper boundary condition and calculate both the andreev and normal reflections in this n / s junction . for simplicity we take the half - filled case ( @xmath31 ) in the following , and qualitatively same results can be obtained for @xmath32 . the mean - field hamiltonian introduced in the previous section can be decomposed into two sets of subsystems : @xmath33 and @xmath34 with @xmath35 . we focus on the former set , where the eigenenergies @xmath36 are given by @xmath37 and the corresponding eigenoperators by @xmath38 ( @xmath39 ) where @xmath40 with @xmath41 and @xmath42 . the dispersion relation is illustrated in fig . [ fig : dispersion ] . when we take another subsystems , the behaviors discussed in this section remain unchanged . we note that the gapless part @xmath43 contributes to the diamagnetic meissner kernel @xcite . this is possible because the fermi surface is composed of both electron and hole to form a bogoliubov particle , as is distinct from an ordinary metal . now we consider the n / s junction . the normal metal with @xmath44 and staggered of pairing state with @xmath45 in eq . are placed at the left- ( @xmath46 ) and right - hand ( @xmath47 ) sides , respectively . when the ( @xmath48 ) conduction electron with the energy @xmath49 is injected from the left , the wave function in n is written in the vector form as @xmath50 the results for @xmath51 are obtained from the ones for @xmath49 by using the particle - hole symmetry . the coefficients @xmath52 and @xmath53 correspond to andreev and normal reflection weights , respectively . the wave vector @xmath54 is determined by the condition @xmath55 . here we have only the two components because the localized pseudofermions are decoupled in n. a part of injected electron transmits into s , whose wave function is written as @xmath56 the wave vectors satisfy the relations @xmath57 and @xmath58 . here only the wave functions with positive group velocity appear . we note that @xmath59 becomes imaginary for @xmath60 , where it exists as a quickly damping evanescent wave . the n part at left and the s part at right are connected at the origin by the following tunnel hamiltonian : @xmath61 here we consider the barrier potential @xmath62 at the edge of the normal metal . the present setup of the system is schematically illustrated in fig . [ fig:1d_junction](a ) . is considered at the edge of the normal metal in ( a ) . in ( b ) the josephson current is calculated at the two sites located at the center of the chain . , width=302 ] we assume the sites for @xmath46 and for @xmath47 are described by the wave functions @xmath63 and @xmath64 , respectively . at the sites @xmath65 , @xmath66 and @xmath67 we have the relations @xmath68 the diagonal matrices are made from eqs . and as @xmath69 , @xmath70 and @xmath71 . these matrices are @xmath72 matrices , and they operate for the upper two components of @xmath73 , since pseudofermions have no inter - site hopping . we also define the matrix @xmath74 which originates from the local part of eq . . the function @xmath75 can not be described in general by either @xmath76 or @xmath73 due to the presence of the potential @xmath62 . to determine the coefficients we need to have another relations . this situation is similar to a usual quantum mechanics which requires smooth wave functions at the boundary . we consider the extrapolated wave functions @xmath77 and @xmath78 which satisfy the relations @xcite @xmath79 by solving these equations we can explicitly derive the coefficients @xmath52 , @xmath53 , @xmath80 , @xmath2 and the wave function @xmath75 at the interface . the normal reflectance @xmath81 , andreev reflectance @xmath82 , and transmittances @xmath83 of quasiparticles are defined by @xmath84 which satisfy the sum rule of probability flow : @xmath85 note that the evanescent wave does not contribute to this sum rule . from these quantities we define the conductance by @xmath86 where the factor @xmath87 originates from spin and orbital degrees of freedom and @xmath88 is the planck constant . the condition @xmath89 means the existence of an excess current due to andreev reflection , or cooper pair tunneling into the s part . we note that in actual systems the energy is given by @xmath90 with electric charge @xmath5 and bias voltage @xmath91 . ) and transmittances ( @xmath83 ) as a function of energy . the transfer integrals at the junction are chosen as ( a ) @xmath92 and ( b ) @xmath93 . the barrier potential is not included in this figure . , width=302 ] we first discuss the n / s junction for the @xmath94 case . figure [ fig : probability ] shows the reflectances and transmittances defined by eqs . ( [ eq : prob1][eq : prob4 ] ) . we take the hybridization strength @xmath95 , and the gap is then @xmath96 . in the high transmissivity limit with @xmath92 for @xmath97 shown in fig . [ fig : probability](a ) , a half of the injected electron transmits into the tckl superconducting state ( @xmath98 ) . the other half is reflected into the normal metal both as electron ( @xmath99 ) and hole ( @xmath100 ) . this behavior is in contrast to the ordinary @xmath11-wave superconductor , where the perfect andreev reflection ( @xmath101 ) can be observed . for small @xmath102 case , the energy dependence is modified while the behavior at low energy remains nearly unchanged . the presence of normal reflection in tckl is related to the form of the mean - field hamiltonian given by eq . . namely , the gapped structure in spectrum has the characters of both hybridization ( normal ) gap and superconducting ( anomalous ) gap . consequently both the normal and anomalous self energies are present as in eqs . and , which cause normal and andreev reflections simultaneously . another characteristic behavior different from ordinary superconductors is that the transmittance into the superconducting tckl is finite even at zero energy . this is due to the presence of the fermi surface as shown in fig . [ fig : dispersion ] . hence bound states e.g. in s / n / s junction or at vortex core are unlikely formed even in the clean limit at low temperatures . ) and ( b ) attractive ( @xmath103 ) barrier potentials . we take @xmath92 . , width=321 ] now we consider the situation with finite barrier potential at the edge of the normal metal . the conductances are shown in fig . [ fig : conductivity2 ] for @xmath92 , where we normalize them by the normal conductivity @xmath104 . it is characteristic that the peaked structure is observed for @xmath105 while there is no such behavior for @xmath103 . the effect of the sign of the barrier potential is remarkable near the gap edge ( @xmath106 ) , but it is irrelevant in the low - energy limit . the solutions at low energies can be obtained in a simple form . in the limit with @xmath107 , we can use the relations @xmath108 , @xmath109 and @xmath110 . we then explicitly derive the reflectances for @xmath111 as @xmath112 , \\ b(e ) & \sim \frac 1 4 \left [ 1 - \left ( \frac v t \right ) \frac{e}{\gamma_0 } \right].\end{aligned}\ ] ] the magnitude of the reflectance of andreev reflection is enhanced with increasing energy @xmath113 for @xmath105 and is diminished for @xmath103 , while the normal reflection shows the inverse behavior . thus the results are sensitive to the setup at the boundaries . by contrast , the conventional spin - singlet @xmath11-wave superconductor does not show such a sign - sensitive behavior for barrier potential , and there is no difference between repulsive and attractive potentials . on the other hand , for sufficiently large magnitude of potentials both with @xmath105 and @xmath103 , the line shape of the resulting conductance becomes similar to that of the local density of states ( ldos ) at the edge as will be shown in fig . [ fig : dos](a ) . this nonzero value of @xmath114 at zero energy clearly characterizes the present superconducting state as distinct from ordinary superconductors . the present conductance in tckl is also different from that in spin - singlet @xmath2-wave or spin - triplet @xmath1-wave superconductor junctions @xcite . in these junctions , surface andreev bound state ( sabs ) produces a zero bias conductance peak @xcite and the magnitude of odd - frequency pairing amplitude is significant at the surface @xcite . on the other hand , in the present tunneling spectroscopy of tckl , the presence of the odd - frequency pairing does not produce a clear zero bias conductance peak . the staggered of pairing state of tckl is coupled to the other types of superconductors in josephson junctions . we consider the simple spin - singlet @xmath11-wave superconductivity ( esee ) , whose hamiltonian is given by @xmath115 in one dimension . the conduction electron part is written as @xmath116 . as we mentioned in the introduction , there is a paramagnetic of superconductivity which is induced only at the edge from ef superconductivity in bulk . here as one of such examples we take the spin - triplet @xmath117-wave superconductor ( etoe ) . the hamiltonian is explicitly written as @xmath118 when we make an edge with this hamiltonian , the local ( @xmath11-wave ) electron pair is generated at the edge . since the spin structure of this pair is a triplet or even with respect to the spin exchange , the time dependence of the pair must be odd ( otee ) . this induced of pair at the surface which is closely connected to the andreev bound state @xcite . the josephson junction can be constructed by connecting tckl with one of the above superconductors . the hamiltonian of this system is given by @xmath119 each of which describes the semi - infinite left - hand system ( @xmath46 ) , semi - infinite right - hand system ( @xmath120 ) and the middle junction part , respectively . we take the spin - singlet @xmath11- or spin - triplet @xmath117-wave superconductors as @xmath121 and the tckl as @xmath122 . the hamiltonian for the junction part is given by @xmath123 the present setup of the system in one dimension is schematically illustrated in fig . [ fig:1d_junction](b ) . the josephson current @xmath124 is calculated at the center of this junction : @xmath125 here the josephson current is well defined because the gauge - symmetry breaking terms are not included at the junction region , and the equation of continuity locally holds only by quasiparticle flow . the josephson current can be calculated by using the semi - infinite green function @xcite . as an alternative method one can approximate this by the green function at the edge of the finite chain . we take the number of sites as @xmath126 in the following . the semi - infinite left- and right - hand surface green functions @xmath127 and @xmath128 are explicitly derived from the hamiltonians and , which can be written in a nambu matrix form with respect to spin / orbital index . the local green functions at the site @xmath66 and @xmath129 without the connection by @xmath130 are given by @xmath131^{-1 } , \label{eq : gl0 } \\ \hat g^{\rm r}_1 & = [ z\hat 1 - \hat { \gamma } ' \hat g^{\rm r}_{\infty } ( \hat { \gamma}')^{\dagger}]^{-1 } , \label{eq : gr1}\end{aligned}\ ] ] respectively . the indices 0 and 1 mean the site index at the junction part . the matrices @xmath132 , @xmath133 and @xmath134 are made from eq . in a manner similar to n / s junction . using these quantities , the green functions at the junction are given by @xmath135^{-1 } , \label{eq : g10 } \\ \hat g_{01 } & = \hat g^{\rm l}_{0 } \hat { \gamma}_{\rm i } [ ( \hat g^{\rm r}_1)^{-1 } - \hat { \gamma}_{\rm i}^{\dagger}\hat g^{\rm l}_{0 } \hat { \gamma}_{\rm i}]^{-1 } .\label{eq : g01}\end{aligned}\ ] ] the josephson current defined in eq . is then calculated at finite temperatures from @xmath136 . -wave superconductor in ( c ) and ( d ) . the parameters are chosen as @xmath137 for ( a , b ) and @xmath138 for ( c , d ) . the infinitesimal imaginary part @xmath139 is taken as @xmath140 . , width=283 ] first we show the ldos and local pairing amplitude at the edge of the semi - infinite chain calculated from @xmath141 and @xmath142 . figures [ fig : dos](a ) and ( b ) show the ldos proportional to @xmath143 and pair amplitude @xmath144 , respectively , for tckl . the values are normalized by @xmath145 which is the density of states for a normal metal . in contrast to the conventional spin - singlet @xmath11-wave superconductor , the ldos is nonzero at the fermi level . this is because the half of the bogoliubov particles in the of pairing state have an energy gap and the others still have the fermi surface as shown in fig . [ fig : dispersion ] . the frequency dependence of the real part of pair amplitude ( or anomalous green function ) shown in fig . [ fig : dos](b ) is odd with respect to real frequency . these behaviors are similar to the ones in bulk @xcite . although here we can not see ef components , it appears as the inter - site green functions . on the other hand , the ldos at the edge of the spin - triplet @xmath117-wave superconductor has the sharp peak as shown in fig . [ fig : dos](c ) , which is known as a consequence of the andreev bound state @xcite . this non - trivial localized edge state is formed when the sign of the gap function felt by quasiparticle is reversed at the reflection process . figure [ fig : dos](d ) displays the local pair amplitude which is odd in frequency ( otee ) , although in bulk only the spin - triplet @xmath117-wave ef pair ( etoe ) is formed @xcite . . the parameters are same as the ones in fig . [ fig : dos ] . , width=283 ] let us discuss how the edge state is connected to the bulk state . figure [ fig : spat_dos ] shows the spatial dependence of the ldos at the fermi level . here the ldos at the fermi level has the spatial dependence in the form @xmath146 where @xmath147 and @xmath148 are smooth functions in space . this oscillating behavior in a staggered manner originates from half - filled situation with @xmath149 , and the period of oscillation changes for @xmath32 reflecting the change of fermi wave vectors . as shown in fig . [ fig : spat_dos](a ) , the ldos at the edge of tckl is continuously connected to the bulk . a slow relaxation is characteristic for the metallic state , and is consistent with the presence of fermi surface in superconducting state of tckl . ( in numerical simulation , the healing length , which may be physically regarded as mean free path , is given by @xmath150 with small but finite @xmath139 . ) hence the character of this zero - energy state can be regarded as similar to the one in bulk tckl . for the spin - triplet @xmath117-wave superconductor shown in fig . [ fig : spat_dos](b ) , on the other hand , the zero - energy state located at the edge vanishes quickly as we go into the bulk state . this edge state has a different character from the bulk state in this case . with these preliminaries , now we consider the josephson junction . in the following we consider the zero barrier potential case ( @xmath94 ) unless explicitly stated otherwise . the phase of the pair amplitude in the left - hand system is taken as @xmath151 , while it is set as zero in the right - hand system as illustrated in fig . [ fig:1d_junction ] . we begin with the spin - singlet @xmath11-wave superconductor / tckl junction . however , the josephson current completely vanishes in the present simple setup . as explained later , the absence of josephson current is related to the fact that symmetries of the induced pairs located at the edges do not match between the left- and right - hand sides . in order to have finite current , the simplest modification without changing bulk properties , is to change the tunnel matrix at the interface as @xmath152 with @xmath153 . we call this the setup ( i ) . note that with this tunnel matrix both the spin and orbital symmetries are broken but their product is not broken ( see also tab . [ tab : induced_pair ] ) . on the other hand , the more realistic setup giving finite currents is to modify the bulk nature of tckl with keeping the tunnel matrix @xmath154 . we consider the orbital field both for conduction electrons and localized pseudospin , whose hamiltonain is given by @xmath155 ( called the setup ( ii ) in the following ) this term breaks the orbital symmetry and experimentally corresponds to the uniaxial pressure effect . when we make the junction in real materials , some stress should be applied to the edge of tckl . hence the effect of eq . will reasonably appear . for simplicity we take @xmath156 in the following , but this assumption does not affect qualitative aspect of the results . dependences of the josephson current @xmath124 for ( a ) spin - singlet @xmath11-wave and ( b ) spin - triplet @xmath117-wave superconductors connected to tckl with the setup ( i ) [ @xmath157 , @xmath158 . the results for the setup ( ii ) [ @xmath159 , @xmath160 are shown in ( c ) and ( d ) . the parameters are chosen as @xmath138 , @xmath161 , @xmath162 and @xmath163 . the results for finite barrier potentials ( @xmath164 ) are also shown . , width=321 ] figure [ fig:1d_current ] shows the phase @xmath165 dependence of josephson currents , which is normalized by @xmath166 . let us first discuss the case with the setup ( i ) . as shown in fig . [ fig:1d_current](a ) , the josephson current has the functional form of @xmath167 for spin - singlet @xmath11-wave superconductor / tckl junction . this indicates that the first - order coupling vanishes in this case . the josephson current for the spin - triplet @xmath117-wave superconductor / tckl junction have the form @xmath168 as seen in fig . [ fig:1d_current](b ) . on the other hand , the results are changed for the setup ( ii ) as shown in fig . [ fig:1d_current](c , d ) . the first - order coupling survives for @xmath11-wave superconductor / tckl junction , while it vanishes in the junction using @xmath117-wave superconductor . these behaviors can be qualitatively understood by considering the two - site model ( zero - dimensional system ) that simulates the edges of right- and left- superconductors . here we focus on the case with the setup ( ii ) , and the more detailed analysis including the setup ( i ) is given in appendix . the local spin - singlet @xmath11-wave pairing field term is given by eq . , and the pairing field for tckl by eq . . we directly connect these two sites by the matrix defined by the third line of eq . . the josephson current is explicitly evaluated as @xmath169 within the lowest - order approximation . the left- and right - anomalous green functions are given by @xmath170 where @xmath171 is the orbital - dependent hybridization function . for @xmath172 the anomalous green function of tckl is a purely odd function with respect to frequency , but the even - frequency component mixes in the presence of orbital fields . from these expressions it is clear that the josephson current becomes zero if we take zero orbital field ( @xmath172 ) . with finite orbital field , on the other hand , @xmath173 and @xmath174 have the same parity in frequency space , and the first - order contribution to the josephson current becomes finite . namely , the induced ef pair in tckl contributes to the josephson coupling . for spin - triplet @xmath117-wave superconductor / tckl junction , the josephson coupling is expressed by odd - frequency spin - triplet @xmath11-wave ( otee ) and odd - frequency spin - singlet @xmath11-wave ( oseo ) pairing . then , the first - order contribution with respect to @xmath175 vanishes . thus we obtain consistent results with numerical calculations for a chain discussed above . next we discuss the above josephson junction from symmetry point of view . originally , the oseo+esoo pairs are present in tckl without any field as discussed in sec . iii . on the other hand , for spin - singlet @xmath11-wave superconductor the esee pair and the induced osoe pair are present at the edge . in a similar manner the etoe and induced otee pairs exist for spin - triplet @xmath117-wave superconductor . thus , no symmetries match between tckl and the other superconductors , and the first - order coupling in josephson junction becomes zero . in fact , this vanishing current persists to higher orders . to explain this behavior , we must specify the component of orbital - triplets in @xmath11- and @xmath117-wave superconductors . in the present setup , since we do not include the orbital degrees of freedom explicitly , the triplet component is identified as @xmath176 or @xmath177 and no @xmath178 component . thus the mismatch between orbital - singlet in tckl and orbital - triplet ( @xmath179 ) in @xmath11- and @xmath117-wave superconductors gives exactly zero current in the present system . with the tunnel matrix in the setup ( i ) , the spin - orbital symmetry breaking is present and the induced pair is otee+etoe according to tab . [ tab : induced_pair ] . hence , the first - order josephson coupling survives for @xmath117-wave superconductor / tckl junction , but it vanishes for @xmath11-wave superconductor / tckl case . similarly , with uniaxial pressure in the setup ( ii ) , the esee+osoe pairs are newly generated at the edge of tckl , where orbital - triplet component include @xmath176 . hence the first - order contribution to josephson current becomes nonzero for tckl / spin - singlet @xmath11-wave superconductor junction . since we rely only on the symmetry of cooper pairs , the above discussion should be applicable also to systems in higher dimensions . finally we make a comment on the effect of a barrier potential @xmath62 at the junction part ( @xmath180 ) . the phase @xmath165 dependence of the currents with repulsive and attractive potentials are shown in fig . [ fig:1d_current ] . the functional forms are not influenced qualitatively by the sign of @xmath62 , since the barrier potential does not create any new species of cooper pairs . in addition , we do not observe the difference between @xmath105 and @xmath103 . this behavior is consistent with results in the n / s junction : the sign of the barrier potentials does not affect the behaviors in the low - energy limit as shown in fig . [ fig : conductivity2 ] . we have investigated the staggered of pairing realized in tckl from a symmetry point of view . although the pair potential is purely odd function with respect to time ( frequency ) , both of and ef components of pair amplitude are present due to the absence of translational invariance even in the bulk . the existing pairs in bulk are identified as primary oseo and secondary esoo . we have also shown that a local gauge transformation changes the staggered state into uniform one with spin - symmetry broken state . the mechanism for the diamagnetic meissner effect has been explained by focusing on the symmetry of pair amplitude and structure of the meissner kernel . in addition to time / spin / space / orbital structures of cooper pairs , the finite center - of - mass momentum , which affects the sign of the velocity , plays an important role for diamagnetic response . the n / s junction has been constructed and it is shown that the normal reflection is always present in addition to andreev reflection . this behavior is in contrast with ordinary bcs superconductors , where only the andreev reflection is observed for high transmissivity limit . the difference lies in the presence of normal self energy in the superconducting state of tckl . due to a finite density of states , the transmittance into tckl is also nonzero even at low energies . hence the bound state at e.g. superconducting vortex core is unlikely to be formed . when we consider the barrier potential at the interface , the conductance shows the difference between attractive and repulsive potentials , although no such difference is observed in conventional superconductors . we have also investigated the josephson junction using green function formalism . we connect tckl both with spin - singlet @xmath11-wave and spin - triplet @xmath117-wave pairing states . here a uniaxial pressure effect is considered for tckl , which is naturally expected at the edge of real materials . for tckl / spin - singlet @xmath11-wave superconductor junction , the relative phase @xmath165 dependence of josephson current becomes @xmath181 . it becomes @xmath182 for tckl/@xmath117-wave superconductor junction , where no first - order coupling appears . these josephson currents can be qualitatively described by a zero - dimensional system . the symmetry of the pairs induced at the edge is a key ingredient to understand the current phase relations of josephson junctions . this work was financially supported by a grant - in - aid for jsps fellows ( grant no . 13j07701 ) , a grant - in aid for scientific research on innovative areas `` topological material science '' ( grant no . 15h05853 ) , and a grant - in - aid for scientific research b ( grant no . 15h03686 ) . we consider the simple two - site model given by @xmath183 where @xmath184 here @xmath121 and @xmath122 simulate the edge of the @xmath11-wave superconductor and tckl , respectively . the tunnel matrix is given by eq . , and the current is simply defined by @xmath185 we can solve this model analytically , which helps us understand the basic properties of josephson junction . we define the green function @xmath186 the first component of this vector is relevant to current . its fourier transformation @xmath187 satisfies the equation @xmath188 where @xmath189 is the constant vector and @xmath190 the other contributions to current can also be calculated in a similar manner . from the above equations , we can obtain eq . . for the special case with @xmath191 and @xmath192 , namely without any symmetry breaking fields , the green function matrix @xmath193 becomes diagonal . correspondingly , the anomalous parts in the second term of the eq . , which is relevant to higher - order josephson couplings , are effectively dropped from the equation and the josephson coupling terms vanish in general . this behavior is consistent with the results discussed in sec . v. v.l . berezinskii , jetp lett . * 20 * , 287 ( 1974 ) . kirkpatrick and d. belitz : phys . lett . * 66 * , 1533 ( 1991 ) . d. belitz and t.r . kirkpatrick : phys . b * 46 * , 8393 ( 1992 ) . a. balatsky and e. abrahams , phys . rev . b * 45 * , 13125 ( 1992 ) . v.j . emery and s. kivelson , phys . b * 46 * , 10812 ( 1992 ) . m. vojta and e. dagotto : phys . b * 59 * 713 ( 1999 ) . y. fuseya , h. kohno , and k. miyake : j. phys . jpn . * 72 * 2914 ( 2003 ) . t. hotta : j. phys . . jpn . * 78 * 123710 ( 2009 ) . k. shigeta , s. onari , k. yada , and y. tanaka : phys . b * 79 * 174507 ( 2009 ) ; k. shigeta , y. tanaka , k. kuroki , s. onari , and h. aizawa : phys . rev . b * 83 * 140509 ( 2011 ) . j. otsuki : phys . lett . * 115 * , 036404 ( 2015 ) . p. coleman , e. miranda , and a. tsvelik , phys . lett . * 70 * , 2960 ( 1993 ) . r. heid , z. phys . b * 99 * , 15 ( 1995 ) . bergeret , a.f . volkov , and k.b . efetov , rev . phys . * 77 * , 1321 ( 2005 ) . y. tanaka and a.a . golubov , phys . . lett . * 98 * , 037003 ( 2007 ) . s. higashitani , s. matsuo , y. nagato , and k. nagai , phys . b * 85 * , 024524 ( 2012 ) . y. tanaka , m. sato , and n. nagaosa , j. phys . . jpn . * 81 * , 011013 ( 2012 ) . a. di bernardo , z. salman , x. l. wang , m. amado , m. egilmez , m. g. flokstra , a. suter , s. l. lee , j. h. zhao , t. prokscha , e. morenzoni , m. g. blamire , j. linder and j. w. a. robinson . x , * 5 * , 041021 , ( 2015 ) . d. belitz and t.r . kirkpatrick , phys . b * 60 * , 3485 ( 1999 ) . d. solenov , i. martin , and d. mozyrsky , phys . rev . b * 79 * , 142502 ( 2009 ) . h. kusunose , y. fuseya , and k. miyake , * 80 * , 054702 ( 2011 ) . y. asano , y.v . fominov , and y. tanaka , phys . b * 90 * , 094512 ( 2014 ) . fominov , y. tanaka , y. asano , and m. eschrig , phys . b * 91 * , 144514 ( 2015 ) . v. martisovits and d.l . cox , phys . b * 57 * , 7466 ( 1998 ) . r. heid , ya.b . bazaliy , v. martisovits , and d.l . cox , phys . * 74 * , 2571 ( 1995 ) . s. hoshino and y. kuramoto , phys . lett . * 112 * , 167204 ( 2014 ) . s. hoshino , phys . b * 90 * , 115154 ( 2014 ) . a.m. black - schaffer and a.v . balatsky , phys . b * 88 * , 104514 ( 2013 ) . y. asano and a. sasaki , phys . b * 92 * , 224508 ( 2015 ) . c. triola and a.v . balatsky , arxiv:1512.07031 ( 2015 ) . d. podolsky , e. demler , k. damle , and b. i. halperin , phys . b * 67 * , 094514 ( 2003 ) . chen , o. vafek , a. yazdani , and s .- c . zhang , phys . lett . * 93 * , 187002 ( 2004 ) . f.j . ohkawa , phys . b * 73 * , 092506 ( 2006 ) . k. seo , h .- chen , and j. hu , phys . b * 78 * , 094510 ( 2008 ) . d.f . agterberg and h. tsunetsugu , nat . phys . * 4 * , 639 ( 2008 ) . e. berg , e. fradkin , and s.a . kivelson , phys . lett . * 105 * , 146403 ( 2010 ) . t. yoshida , m. sigrist , and y. yanase , phys . rev . b * 86 * , 134514 ( 2012 ) . cho , r. soto - garrido , and e. fradkin , phys . lett . * 113 * , 256405 ( 2014 ) . yang , phys . lett . * 63 * , 2144 ( 1989 ) . r.r.p . singh and r.t . scalettar , phys . * 66 * , 3203 ( 1991 ) . j. de boer , v.e . korepin , and a. schadschneider , phys . lett . * 74 * , 789 ( 1995 ) . japaridze , a.p . kampf , m. sekania , p. kakashvili , and ph . brune , phys . b * 65 * , 014518 ( 2001 ) . h. zhai , phys . b * 71 * , 012512 ( 2005 ) . m. jarrell , h. pang , d.l . cox , and k.h . luk , phys . lett . * 77 * , 1612 ( 1996 ) . cox and a. zawadowski : adv . phys . * 47 * , 599 ( 1998 ) . p. fulde , and r.a . ferrell , phys . rev . * 135 * , a550 ( 1964 ) . a.i . larkin , and y.n . ovchinnikov , sov . jetp * 20 * , 762 ( 1965 ) . blonder , m. tinkham , and t.m . klapwijk , phys . b * 25 * , 4515 ( 1982 ) . zhu and h. kroemer , phys . b * 27 * , 3519 ( 1983 ) . a.v . burmistrova , i.a . devyatov , a.a . golubov , k. yada , and y. tanaka , j. phys . . jpn . * 82 * , 034716 ( 2013 ) . s. kashiwaya and y. tanaka , rep . * 64 * 1641 ( 2000 ) . c. r. hu , phys . * 72 * , 1526 ( 1994 ) . y. tanaka and s. kashiwaya , phys . lett . * 74 * 3451 ( 1995).y . tanaka , a. a. golubov , s. kashiwaya , and m. ueda , phys.rev . lett . * 99 * , 037005 ( 2007 ) .
|
junction systems of odd - frequency ( of ) superconductors are investigated based on a mean - field hamiltonian formalism . one - dimensional two - channel kondo lattice ( tckl ) is taken as a concrete example of of superconductors .
properties of normal and andreev reflections are examined in a normal metal / superconductor junction . unlike conventional superconductors ,
normal reflection is always present due to the normal self energy that necessarily appears in the present of pairing state .
the conductance reflects the difference between repulsive and attractive potentials located at the interface , which is in contrast with the preexisting superconducting junctions .
josephson junction is also constructed by connecting tckl with the other types of superconductors .
the results can be understood from symmetry of the induced cooper pairs at the edge in the presence of spin / orbital symmetry breaking .
it has also been demonstrated that the symmetry argument for cooper pairs is useful in explaining meissner response in bulk .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in 1957 g.a . askaryan pointed out that ionisation and cavitation along a track of an ionising particle through a liquid leads to hydrodynamic radiation @xcite . in the 1960s , 1970s and 1980s , theoretical and experimental studies have been performed on the hydrodynamic radiation of beams and particles traversing dense media @xcite . the interest in characterising the properties of the acoustic radiation was , among other reasons , lead by the idea that the effect can be utilised to detect ultra - high energy ( @xmath1 ) cosmic , i.e. astrophysical neutrinos , in dense media like water , ice and salt . in the 1970s this idea was discussed within the dumand optical neutrino detector project @xcite and has been studied in connection with cherenkov neutrino detector projects since . the detection of such neutrinos is considerably more challenging than the search for high - energy neutrinos ( @xmath2 ) as currently pursued by under - ice and under - water cherenkov neutrino telescopes @xcite . due to the low expected fluxes , detector sizes exceeding 100km@xmath3 are needed @xcite . however , the properties of the acoustic method allow for sparsely instrumented arrays with @xmath4100 sensors / km@xmath3 . to study the feasibility of a detection method based on acoustic signals it is necessary to understand the properties of the sound generation by comparing measurements and simulations based on theoretical models . according to the so - called thermo - acoustic model @xcite , the energy deposition of particles traversing liquids leads to a local heating of the medium which can be regarded as instantaneous with respect to the hydrodynamic time scales . due to the temperature change the medium expands or contracts according to its bulk volume expansion coefficient @xmath5 . the accelerated motion of the heated medium generates an ultrasonic pulse whose temporal signature is bipolar and which propagates in the volume . coherent superposition of the elementary sound waves , produced over the cylindrical volume of the energy deposition , leads to a propagation within a flat disk - like volume in the direction perpendicular to the axis of the particle shower . in this study , the hydrodynamic signal generation by two types of beams , interacting with a water target , was investigated : pulsed protons and a pulsed laser , mimicking the formation of a hadronic cascade from a neutrino interaction under laboratory conditions . with respect to the aforementioned experimental studies of the thermo - acoustic model , the work presented here can make use of previously unavailable advanced tools such as geant4 @xcite for the simulation of proton - induced hadronic showers in water . good agreement was found in the comparison of the measured signal properties with the simulation results , providing confidence to apply similar simulation methods in the context of acoustic detection of ultra - high energy neutrinos . a puzzling feature observed in previous studies a non - vanishing signal amplitude at a temperature of 4@xmath6c , where for water at its highest density no thermo - acoustic signal should be present was investigated in detail . such a residual signal was also observed for the proton beam experiment described in this article , but not for the laser beam , indicating that the formation of this signal is related to the charge or the mass of the protons . in the following , the thermo - acoustic model @xcite is derived from basic assumptions , using a hydrodynamic approach . basis is the momentum conservation , i.e. , the euler equation @xmath7 for mass density @xmath8 , velocity vector field of the medium @xmath9 and momentum - density tensor @xmath10 including the pressure @xmath11 @xcite . equation ( [ eq_euler ] ) can be derived from momentum conservation . in the derivation , energy dissipation resulting from processes such as internal friction or heat transfer are neglected . motions described by the euler equation hence are adiabatic . taking the three partial derivatives of eq . ( [ eq_euler ] ) with respect to @xmath12 and using for the density the continuity equation @xmath13 a non - linear wave equation can be derived : @xmath14 to solve this equation , the problem is approached in two separated spatial regions : firstly , a region @xmath15 ( _ ` beam ' _ ) , where the energy is deposited in the beam interactions with the fluid and thus the wave excited in a non - equilibrium process ; and secondly , a hydrodynamic ( _ ` acoustic ' _ ) region @xmath16 , where the acoustic wave propagates through the medium and where linear hydrodynamics in local equilibrium can be assumed . this splitting can be reflected by the momentum density tensor , rewriting it as @xmath17 in local equilibrium the changes in mass density are given by @xmath18 with the bulk volume expansion coefficient @xmath19 , the energy deposition @xmath20 , the adiabatic speed of sound @xmath21 in the medium and the specific heat @xmath22 . in the acoustic regime , where @xmath23 , the momentum density tensor can be expressed as @xmath24 ( using eqs.([eq_densitytensor ] ) and ( [ eq_density_change ] ) ) , where we assume an adiabatic density change with pressure . the non - linear kinetic term @xmath25 entering @xmath26 according to eq . ( [ eq_densitytensor ] ) can be neglected for small deviations @xmath27 from the static density @xmath28 and small pressure differences @xmath29 from the static pressure @xmath30 @xcite . in the region b , where non - equilibrium deposition occur , one may make the _ ansatz _ @xmath31 with the direction @xmath32 of the beam which breaks the isotropy of the energy - momentum tensor and describes with the parameter @xmath33 in an effective way the momentum transfer on the fluid . although in non - equilibirum we apply eq . ( [ eq_density_change ] ) with the energy deposition density @xmath34 of the beam . then , with the additional energy - momentum tensor due to the beam @xmath35 the wave equation ( [ eq_wave_base ] ) reads @xmath36 the general solution for the wave equation can be written using a green function approach as @xmath37 \nonumber \end{aligned}\ ] ] with the components of the unit vector @xmath38 and the retarded time @xmath39 . for the last conversion , partial integration and the total derivative @xmath40 have been used repeatedly . note that @xmath41 for @xmath42 , so that the integration is carried out over the volume of the energy deposition region @xmath15 . assuming an energy deposition without momentum transfer to the medium , the kinetic term in the _ ansatz _ ( [ eq : ansatz ] ) can be neglected ( @xmath43 ) yielding @xmath44 for a thermo - acoustic wave generated solely by heating of the medium . the signal amplitude @xmath45 can be shown to be proportional to the dimensionless quantity @xmath46 when solving eq . ( [ eq_pressure ] ) for the case of an instantaneous energy deposition . equation ( [ eq_pressure ] ) is equivalent to the results obtained from the approaches presented in @xcite . the derivation pursued above , however , uses a different approach starting with the euler equation and an anisotropic energy - momentum tensor , yielding a more general expression in eq . ( [ eq : pressure - approx ] ) . only if assuming an isotropic energy deposition one arrives at the expression for the pressure deviation @xmath45 given in eq . ( [ eq_pressure ] ) . note that the validation of the last assumption @xmath43 being a good approximation would require a detailed knowledge of the momentum transfer from the beam to the medium . taking it into account would result in an additional dipol term @xmath47 in eq . ( [ eq_pressure ] ) which may become the dominant contribution to wave generation if @xmath48 close to 4.0@xmath49c . however , for @xmath43 the pressure field resulting from a beam interaction in a medium is determined by the spatial and temporal distribution of the energy deposition density @xmath50 alone . the amplitude of the resulting acoustic wave is governed by the thermodynamic properties @xmath21 , @xmath51 and @xmath5 , the latter three depending primarily on the temperature of the medium . a controlled variation of these parameters in the conducted laboratory experiments and a study of the resulting pressure signals therefore allows for a precise test of the thermo - acoustic model . simulations based on the thermo - acoustic model , as performed to interpret the results of the experiments described in the next section , will be discussed in sec . [ sec_simulation ] . note that the energy deposition density @xmath50 and its temporal evolution for the proton and laser beam interactions discussed in this paper are quite different from those expected for the interaction of ultra - high energy neutrinos . however , if a simulation starting from basic principles allows for a good reproduction of the experimental results , it is reasonable to assume that these simulation methods are transferable to neutrino interactions , as they are governed by the same underlying physical processes . the experiments presented in this paper were performed with a pulsed infrared nd : yag laser facility ( @xmath52 ) located at the erlangen centre for astroparticle physics ( ecap ) of the university of erlangen , and the pulsed @xmath53 proton beam of the `` gustaf werner cyclotron '' at the `` theodor svedberg laboratory '' in uppsala , sweden . the beam properties allow for a compact experimental setup . in both cases , the beams were dumped into a dedicated @xmath54 water tank , where the acoustic field was measured with several position - adjustable acoustic sensors ( see fig . [ fig_test_setup ] ) . the sensors ( also called hydrophones ) could be positioned within the tank with absolute uncertainties below 1 cm . the temperature of the water could be varied between @xmath55 and @xmath56 with a precision of @xmath57 . the temperature was brought to a particular value by first cooling the water with ice ; subsequently the whole water volume was heated to the desired temperature in a controlled , gradual procedure . once the water temperature had been established , at least 10min remained for measurements until the water volume heated up by @xmath57 through heat transfer from the environment . this time span was sufficient for all measurements conducted at water temperatures below the ambient temperature . the explored range of spill energies for the proton beam was from @xmath58 to @xmath59 , the beam diameter was approximately @xmath60 and the spill time @xmath61 . for @xmath62 protons , the energy deposition in the water along the beam axis ( @xmath63-axis , beam entry into the water at @xmath64 ) is relatively uniform up to @xmath65 ending in the prominent bragg - peak at @xmath66 ( see fig . [ fig_energy_deposition_z ] ) . to adjust the spill energy , the number of protons per bunch was varied . the total charge of a bunch was calibrated with two independent methods ( faraday cups and scintillation counters ) , leading to an uncertainty on the order of 15% , with some higher values for low spill energies . to obtain the beam intensity and profile for the proton interactions in the water tank , the distance of about 1.2 m that the beam was travelling from the exit of the beam pipe through air and its entering into the water tank were included in the geant4 simulation . for the laser experiment , the pulse energy was adjusted between @xmath67 and @xmath68 and calibrated using a commercial power meter . the beam had a diameter of approximately @xmath69 mm and the pulse length was fixed at @xmath70 . for the infrared light used , the laser energy density deposited along the beam axis has an exponential decrease with an absorption length of @xmath71 ( see fig . [ fig_energy_deposition_z ] ) . for both beam types the lateral energy deposition profile was gaussian ( the aforementioned beam diameters are the @xmath72 s of the profiles ) . the two experiments allow to explore different spatial and temporal distributions of the energy deposition as well as two different mechanisms of energy transfer into the medium . for both beams , energy is deposited via excitation , in addition the medium is ionised in the case of the proton beam . , width=294 ] for the signal recording , sensors based on the piezo - electric effect @xcite were used . a full characterisation of these sensors had been performed prior to the experiments . they are linear in amplitude response , the frequency response is flat starting from a few khz up to the main resonance at @xmath73 with a sensitivity of @xmath74 ( @xmath40.02v / pa ) . the main resonance is more sensitive by @xmath45@xmath75 and sensitivity drops rapidly at higher frequencies ; at 90khz , the sensitivity has dropped by 20db . the absolute uncertainty in the determination of the sensitivity is at a level of 2db in the frequency range of interest . above 90khz the uncertainty exceeds 5db . to calculate the response of the sensors to an external pressure pulse a parametrised fit of an equivalent circuit model as described in @xcite was used . the sensitivity dependence on temperature was measured and the relative decrease was found to be less than @xmath76 ( or about 0.13db ) per @xmath77 . for every set of fixed experimental parameters ( temperature , energy , sensor position , etc . ) the signals of 1000 beam pulses were recorded with a digital oscilloscope at a sampling rate in excess of @xmath78 . this rate is sufficient for the signals with spectral components up to 100khz , where the sensitivity of the sensors is negligibly small . these individual pulses were averaged to reduce background and environmental noise in the analysis , thereby obtaining a very high statistical precision . figure [ fig_signals ] shows typical signals measured in the proton and the laser experiment , respectively , using the same sensors and experimental setup . the general shapes of the two signals differ : a typical signal for the proton beam shows a bipolar signature , the one for the laser deviates from such a generic form . the laser signal has high frequency components up to several mhz due to the high energy deposition density at the point of beam entry and the almost instantaneous energy deposition compared to the @xmath79s pulse of the proton beam ; therefore the resonance of the sensor is excited causing a ringing in the measured signal . the spatial distribution of the energy density @xmath50 deposited by the laser leads to the two separate signals : the first originates in the beam area at the same @xmath63-region as the sensor placement ( _ ` direct signal ' _ ) , the second from the beam entry , a point of discontinuity where most of the energy is deposed ( _ ` beam entry signal ' _ ) . the signal of the proton beam is deteriorated with respect to an ideal bipolar signal . three main contributions to this distortion can be discerned : the recorded signal starts before the expected onset of the acoustic signal ( 55.2@xmath79s for the given position , see fig . [ fig_signals ] , given by the sonic path length ) ; reflections of the acoustic wave on the beam entry window overlay the original wave starting in the first rare - faction peak ; and finally there are frequency components of the signal exciting a resonant response of the sensor , slightly changing the signal shape and causing ringing . the first point was studied and found to be consistent with an electric charge effect in the sensors caused by the proton beam . its starting time was always coincident with the beam pulse entry into the water , even for sensor distances of up to 1 m , hinting at an electromagnetic origin of the distorting signal . the shape of this non - acoustic signal is consistent with the integrated time - profile of the beam pulse with a subsequent exponential decay . this deformation of the main signal is considered a systematic uncertainty on the signal properties and treated as such in the analysis . for the most part , its shape was fitted and subtracted from the signal . in order to minimise the impact of the signal deformation caused by the described effects on the analysis of the recorded signals , robust characteristics were used : the peak - to - peak amplitude and the signal length from maximum to minimum of the signal . for the laser experiment these features were extracted for the direct signals only . = 10 cm and @xmath63=20 cm and are shown within the same time interval . the dash - dotted line in the upper graph indicates the charge effect described in the text . [ fig_signals ] , width=264 ] for an in - depth validation of the thermo - acoustic model , comparisons of the signal properties with simulation results based on the model are essential . to this end , a simulation of the expected signals was developed . it is based on the thermo - acoustic model using a numeric solution of eq . ( [ eq_pressure ] ) . the input parameters to the simulation were either measured at the experiments , i.e. medium temperature and beam profiles , or simulated , i.e. the energy deposition of the protons ( using geant4 ) . the thermodynamic parameters bulk volume expansion coefficient , heat capacity and speed of sound were derived from the measured water temperature using standard parametrisations . tap water quality was assumed . a series of simulations was conducted , where the input parameters were varied individually in the range given by the experiment , including uncertainties . especially the spatial and temporal beam profile have a substantial impact on amplitude , duration and shape of the signal . simulated signals and the respective sensor response corresponding to the measured signals of fig . [ fig_signals ] are shown in fig.[fig_signals_sim ] . to minimise systematic effects from the setup caused e.g. by reflections on the surfaces , the sensor response was convoluted onto the simulated signals , rather than deconvoluted from the measured ones . thus in the analysis voltage rather than pressure signals are compared . . the corresponding dashed - lined signals mark the simulated signals after convolution with the sensor response . for better comparability , the signal maxima were normalised to 1 . [ fig_signals_sim ] , width=264 ] the shapes of the simulated pulses are altered by the sensor response , especially the high - frequency components above the resonant frequency of the sensor . in the case of the laser pulse , mostly the resonance of the sensor is excited , leading to a strong ringing . for the proton beam , the primarily bipolar shape is again prominent , whereas the laser pulse is segmented into the two parts described above . the direct pulse of the laser experiment exhibits a bipolar shape as well , albeit less symmetric than for the proton beam . figure [ fig_systematics ] exemplifies the dependency of the signal amplitude on the input parameters of the simulation : water temperature , pulse energy , beam profile in @xmath80 and @xmath81 , pulse length and the position of the sensor . as nominal positions of the sensor @xmath82 m , @xmath83 m , and @xmath84 m were used . all parameters were varied by @xmath85 around the value of the best agreement with measurement , i.e. the values used for the simulations of the signals shown in fig . [ fig_signals_sim ] . some of the characteristics of thermo - acoustic sound generation are observable . as discussed in sec . [ sec_model ] , the dependence on temperature enters through the factor @xmath46 ( where the dependence on the speed of sound is negligible ) and is roughly linear in the range investigated for this study . the dependence on energy is strictly linear . the dependency on the beam pulse parameters is diverse . it is governed by the integral in eq.([eq_pressure ] ) and therefore depends on both the spatial and the temporal beam profiles and the interaction of the particles with the water . for a given point in space and time , the elementary waves produced in the volume of energy deposition may interfere constructively or destructively depending on the beam properties . accordingly , the length of the laser pulse has no influence on the amplitude , as with 9ns it is much shorter than the transit time of the acoustic signal through the energy deposition area . for the several ten @xmath79s long proton spill , the spill time is comparable to the transit time . thus the acoustic signal shows a strong dependence on the spill time . the dependence on the radial coordinate @xmath86 w.r.t . the beam axis @xmath86 follows roughly the expected 1/@xmath87 and @xmath88 fall - off of a cylindric source in the near and the far field , respectively . the @xmath81-position was varied between @xmath892 and 2 cm , as the signals were recorded within the @xmath90-plane . the resulting change in amplitude is below 1% . the @xmath63-dependency for the laser experiment follows the exponential fall - off expected from the light absorption . the one for the proton experiment shows the only non - strictly monotonic behaviour due to the form of the energy deposition with the prominent bragg peak . using these dependencies the systematic uncertainties of the model were obtained using the experimental uncertainties of the various parameters . the main uncertainty for the proton beam is given by the temporal profile of the pulse , which was simplified to a gaussian profile for this chapter ( however not for the rest of this work ) . for the laser beam experiment this parameter influences the signal amplitude only on a one percent level . the second main influence on the amplitude is the sensor position along the beam axis ( @xmath63-direction ) . table [ tab_systematics ] gives the parameters and their uncertainties ( @xmath91 ) used for the simulation of the signals . the resulting systematic uncertainties in the amplitude ( @xmath92 ) are given as well . the combined uncertainties are @xmath93 for the proton signal and @xmath94 for the laser signal , respectively . .beam parameters used for the simulated signals in fig . [ fig_signals_sim ] with their associated experimental uncertainties ( @xmath91 ) and resulting uncertainty in signal amplitude ( @xmath92 ) . the pulse length of the laser is set to a value much higher than in the experiment , to save calculation time . the resulting uncertainty in the signal amplitude is below @xmath95 [ cols="<,^,^,^,^ " , ] in the following , the results obtained in the comparison of experiment and simulation are presented . figure [ fig_proton_sim_meas ] shows a comparison between simulated and measured signals for the proton beam experiment at different sensor positions . for better visibility only the main part of the signals ( first bipolar part ) is plotted . the input parameters of the simulation were varied within the experimental uncertainties until the best agreement with the measured signal in amplitude and duration was obtained for the reference point at @xmath80=0.40 m and @xmath63=0.11 m . for this optimisation , a simple procedure of adjusting the parameters manually and scanning the resulting agreement visually was found to be sufficient . the other signals were simulated with the same parameter set , only the sensor positions were changed . the signal shapes differ for different @xmath63-positions due to the geometry of the energy deposition profile described in sec . [ sec_setup ] with cylindric form in the @xmath96-plane and almost flat energy density in @xmath63-direction up to the bragg peak at @xmath63=0.22 m . due to this geometry an almost cylindrical wave is excited in the medium , with coherent emission perpendicular to the beam axis . in the region @xmath97 m the signals are of bipolar shape . along the beam axis ( @xmath98 m , @xmath99 m ) the main part of the observed signal originates from the bragg peak as a nearly spheric source and no clear bipolar shape evolves . the agreement between simulation and measurement is good for all positions . not only amplitude and duration match ( see also the following sections ) but also the signal shape is reproduced to a very high degree . the small discrepancies , primarily in the rare - faction part of the bipolar pulse , have contributions stemming from a non - ideal sensor calibration and reflections on the tank surfaces . however , a significant part of the discrepancies may lie in the beam pulse modelling or even the thermo - acoustic model itself . the prominent feature of the energy deposition of the laser beam is at the beam entry into the medium . overall , the geometry of this deposition is mainly a cylindric one with rotational symmetry around the @xmath63-axis , as for the proton beam , with coherent emission perpendicular to the beam axis ( direct signal ) . the signal from the discontinuity at the beam entry is emitted almost as from a point source ( beam entry signal ) . in contrast to the proton beam , the shapes of signals at different positions along the @xmath63-axis do not vary much , only the relative timing between the two signal components varies . therefore , fig . [ fig_laser_sim_meas ] shows only signals for a @xmath63-position in the middle of the water tank . here , both signal parts are well described by the simulation . the beam entry part of each signal is less well reproduced in the simulation due to its high frequency components where the sensor calibration is less well understood . + m for different sensor positions along @xmath80 ( reference at @xmath100 m ) . for more details [ fig_laser_sim_meas ] , width=325 ] to compare the characteristics of the signals in simulation and measurement in the following studies , their amplitudes and the point in time of the recording of the signal maximum were further studied . for the laser signal , the direct signal part is considered only . equation ( [ eq_pressure ] ) yields as velocity of propagation for the thermo - acoustic signal the speed of sound in the medium , here water . to verify the hydrodynamic origin of the measured signals the variation of the arrival times for different sensor positions perpendicular to the beam axis were analysed . figure [ fig_time_vs_distance ] shows the measured data and a linear fit for each beam type . the data is compatible with an acoustic sound propagation in water , as the fits yield a speed of sound compatible with pure water at the temperature used . for the proton beam @xmath101 ( @xmath102 ) was obtained for a water temperature of @xmath103c , where literature @xcite gives for pure water at normal pressure @xmath104 which is in complete agreement . for the laser measurements the water temperature of @xmath105c and thus the speed of sound were significantly higher , the observed @xmath106 ( @xmath107 ) is again in perfect agreement with the theoretical value of @xmath108 . the offset between proton and laser beam data in fig . [ fig_time_vs_distance ] is due to a differing delay time in between trigger time and arrival time of the different beams in the water and is irrelevant for the calculation of the speed of sound . position ) . the straight lines represent linear fits to the data points yielding @xmath109 and @xmath110 for the proton and laser beam , respectively . [ fig_time_vs_distance ] , width=325 ] the durations of the complete , unclipped signals vary with sensor positions mainly due to reflections , which were not simulated and can therefore not be compared . the comparison of simulated and measured signals for different sensor positions within the water tank , excluding the parts of the signals dominated by reflections , was shown in fig . [ fig_proton_sim_meas ] . the good agreement between model and measurement also manifests itself in the development of the signal amplitude with distance of sensors from the beam axis , shown in fig . [ fig_amp_vs_x_protons ] . to minimise systematic effects from reflections , only the amplitude of the leading maximum is analysed . though there are sizeable deviations , the overall shape of the curve is reproduced . the behaviour is again different for the two @xmath63-positions . the development at @xmath111 m follows the one expected from a cylindric source with a @xmath112 behaviour in the near field up to @xmath4@xmath113 m and a @xmath88 behaviour in the far field beyond that distance . at the smallest measured distances , the simulated behaviour deviates from the measured one . presumably this is due to simplifications made in the derivation of the model in sec . [ sec_model ] . at @xmath114 m the amplitude falls off more uniformly , this is again a combination of the point - like emission characteristic of the bragg peak interfering with the cylindric emission at @xmath115 m . ) for the proton beam experiment . points mark the measured amplitudes for two different @xmath63-positions and the lines the respective simulation results . [ fig_amp_vs_x_protons ] , width=325 ] the behaviour for the laser experiment is not as well reproduced by the simulation ( see fig . [ fig_amp_vs_x_laser ] ) . this is again attributed to the high - frequent signal components of the laser , where minor uncertainties in the simulation may result in big changes of the signal amplitude . especially the overlap of the direct with the beam entry window signal distort the signal shape . at distances exceeding 0.5 m the two signal parts can not be distinguished . ) for the laser beam experiment . points mark the measured amplitudes at @xmath116 m and the line the simulation result . [ fig_amp_vs_x_laser ] , width=325 ] assuming otherwise unchanged settings , the energy deposition density @xmath50 scales linearly with total deposited energy . thus the spill energy can be written as a pre - factor in eq . ( [ eq_pressure ] ) effecting the pressure and thus signal amplitude linearly . as shown in fig . [ fig_amplitude_vs_energy ] this behaviour was observed in the experiments yielding a zero - crossing of the pulse energy at @xmath117mpa for the proton beam and @xmath118mpa for the laser beam . both values are consistent with zero . the slope of the line depends on the energy deposition and the sensor positioning along the beam axis and can therefore not be compared for the two beams . as expected from the model , the signal duration and signal shape showed no significant dependence on energy . m. there is a ten percent systematic uncertainty in the absolute determination of the pulse energy . the lines represent linear fits to the data points yielding a zero - crossing of the amplitude compatible with no energy in a pulse . the insert shows the data for the proton beam.[fig_amplitude_vs_energy ] , width=325 ] the main feature of the thermo - acoustic model is its dependence on the temperature of the medium . figure [ fig_temp_laser ] shows the temperature dependence of the signal peak - to - peak amplitude for the laser beam , where a positive ( negative ) sign denotes a leading positive ( negative ) peak of the signal . the two data sets shown in the figure were recorded by two sensors simultaneously , which were positioned at @xmath119 perpendicular to the beam axis and at @xmath120 and @xmath121 along the beam axis , respectively . in the case of the proton beam setup , which will be discussed below , these hydrophone positions correspond roughly to the @xmath63-position of the bragg - peak and a @xmath63-position half way between the bragg - peak and the beam entry into the water , respectively . for comparability , the same positions and the same sensors were chosen for the laser and proton beam experiments . at @xmath122 . the insert shows a blow - up of the region around @xmath123c where the sign of the amplitude changes . [ fig_temp_laser],width=325 ] the laser beam signal shown in fig . [ fig_temp_laser ] changes its polarity around @xmath124 , as expected from the thermo - acoustic model . the theoretical expectation for the signal amplitude , which is proportional to @xmath125 and vanishes at @xmath126 for the given temperature and pressure , is fitted to the experimental data . in the fit , an overall scaling factor and a shift in temperature ( for the experimental uncertainty in the temperature measurement ) were left free as fit parameters . the fit yielded a zero - crossing of the amplitude at @xmath127 , where the error is dominated by the systematic uncertainty in the temperature setting . c , fitted with the model expectation as described in the text . a systematic deviation from the model expectation as the amplitude changes its sign is clearly visible . the amplitudes were normalised to @xmath128 at @xmath122 . [ temp_amp_protons_uncorr ] , width=325 ] . to allow for an easy comparison of the signal shapes , a point in time at the onset of the acoustic signals was chosen as zero time and for all signal amplitudes the corresponding offset was added or subtracted to yield a zero amplitude at that time . [ temp_comp_signals ] , width=325 ] analysing the proton data in the same fashion resulted in a fit that deviated from the model expectation , and a zero - crossing significantly different from @xmath129 at @xmath130 , see fig . [ temp_amp_protons_uncorr ] . the data strongly indicate the presence of a systematic effect near the zero - crossing of the signal amplitude . to understand this effect , the signal shapes near the temperature of @xmath129 were investigated ( fig . [ temp_comp_signals ] ) . a non - vanishing signal is clearly observable at @xmath129 and the signal inverts its polarity between @xmath129 and @xmath131 . in view of the results from the laser beam measurements and the obvious systematic nature of the deviation from the model visible in fig . [ temp_amp_protons_uncorr ] , we subtracted the residual signal at @xmath129 , which has an amplitude of @xmath132 of the @xmath122 signal , from all signals . thus a non - temperature dependent effect in addition to the thermo - acoustic signal was assumed . the resulting amplitudes shown in fig . [ fig_temp_proton ] are well described by the model prediction . c was subtracted at every temperature . the amplitudes were afterwards normalised to @xmath128 at @xmath122 . the insert shows a blow - up of the region around @xmath123c where the sign of the amplitude changes . [ fig_temp_proton ] , width=325 ] the production mechanism of the underlying signal at @xmath133 , which was only observed in the proton experiment , could not be unambiguously determined with the performed measurements . from the model point of view , the main simplification for the derivation of eq . ( [ eq_pressure ] ) was to neglect all non - isotropic terms and momentum transfer to the medium in the momentum density tensor @xmath134 by setting @xmath43 in eq . ( [ eq : ansatz ] ) . as discussed in sec . [ sec_model ] , dipole radiation could contribute significantly near the disappearance of the volume expansion coefficient for the case @xmath135 . also other non - thermo - acoustic signal production mechanism have been discussed in the literature which could give rise to an almost temperature independent signal , see e.g.@xcite . the obvious difference to the laser experiment are the charges involved both from the protons themselves and the ionisation of the water which could lead to an interaction with the polar water molecules . another difference are the massive protons compared to massless photons . residual signals at @xmath133 were found in previous works as well @xcite , as will be discussed in more detail in sec . [ subsec : comparison ] . for clarification further experiments are needed either with ionising neutral particles ( e.g. synchrotron radiation ) or with charged particles ( e.g. protons , @xmath136-particles ) with more sensors positioned around the bragg - peak . with such experiments it might be possible to distinguish between the effect of ionisation in the water and of net charge introduced by charged particles . with the analysis that has been described above , the signal production according to the thermo - acoustic model could be unambiguously confirmed . while the model has been confirmed in previous experiments , the simulations presented in this work constitute a new level of precision . the most puzzling feature , a residual signal at 4@xmath6c that was also observed in previous experiments , was investigated with high precision by scanning the relevant temperature region in steps of @xmath137c . the observed shift of the zero - crossing of the amplitude towards values higher than 4@xmath6c , caused by a leading rarefaction non - thermal residual signal at 4@xmath6c , is in qualitative agreement with @xcite . in @xcite , a residual signal at @xmath123c was also reported . since in that work the zero - crossing of the amplitude is observed at @xmath138c , i.e.a higher value than the expected @xmath123c , it can be assumed that the corresponding residual signal has a leading rarefaction . in @xcite , a residual signal is found at @xmath139c , however with a leading compression rather than rarefaction . the authors conclude that this may lead to a signal disappearance point _ below _ the expected value , in contrast to @xcite and the work presented in this article . for the measurements with a laser beam reported in @xcite , a residual signal was also observed at @xmath123c , albeit with a leading compression and a subsequent reduction of the temperature of the zero - crossing of the signal amplitude to about @xmath140c . this observation is in contrast to the laser experiment presented in this article . in conclusion , the works of all authors discussed here indicate a non - thermal residual signal for proton beams , albeit with varying results concerning the size of the effect and the shape of the underlying non - thermal signal . the results for the laser beam reported in @xcite differ from those described in the article at hands . it should be pointed out , however , that in @xcite and @xcite results are reported by the same authors for proton and for laser beams , respectively . a comparison of these two publications shows that a different behaviour near the temperature of @xmath123c was observed for the two types of beams . hence , to the best of our knowledge , there are currently no results in contradiction with the notion of different non - thermal effects in the interaction of proton vs. laser beams with water . the available data does not allow for a more detailed analysis of the correlation between experimental conditions and the temperature of the zero - crossing of the signal amplitude . efforts to detect neutrinos at ultra - high energies are at the frontier of research in the field of astroparticle physics . neutrinos are the only viable messengers at ultra - high energies beyond the local universe , i.e. distances well beyond several tens of megaparsecs . if successful , the investigation of these elusive particles will not only enhance the understanding of their own nature , but also provide important complementary information on the astrophysical phenomena and the environments that accelerate particles to such extreme energies . for acoustic particle detection , not only the technical aspects such as optimal design and detector layout are subject of research . but also the underlying physics processes the formation of hadronic cascades resulting from neutrino interactions in dense media have never been observed directly in detector experiments at these energies . producing ever more reliable extrapolation of reaction properties to ultra - high energies is an ongoing effort . with advancements in the simulations of cascades forming in water and improvements of detector simulation tools , the discrepancy between cascade parameters from independent simulations decreased : recent studies differ only slightly @xcite . at the same time it is necessary to gain a solid understanding of the sound signals generated from the energy depositions by particle cascades . for this purpose , laboratory measurements are required . this work , together with others @xcite has established the validity of the thermo - acoustic model with uncertainties at the @xmath141 level . the input to a model as discussed in this article is an energy deposition in water as it is also produced by a cascade that evolves from a neutrino interaction . in comparison with the uncertainty in the thermo - acoustic model , uncertainties due to the simulations of hadronic cascades and cascade - to - cascade variations are large @xcite , dominating the challenge to detect and identify sound signals resulting from neutrino interactions . it can hence be concluded that the current level of precision in modelling sound signals in the context of the thermo - acoustic model is fully sufficient for the understanding of acoustic neutrino signatures . the latter is necessary to improve the selection efficiency and background rejection for neutrino detection algorithms in potential future acoustic neutrino detectors . several experiments have been conducted @xcite to understand the acoustic background at the sites of potential future large - scale acoustic neutrino telescopes in sea water , fresh water and ice . the combination of simulation efforts , laboratory measurements and studies with in - situ test arrays will allow for a conclusion of the feasibility of acoustic neutrino detection . we have demonstrated that the sound generation mechanism of intense pulsed beams is well described by the thermo - acoustic model . in almost all aspects investigated , the signal properties are consistent with the model . the biggest uncertainties of the experiments are on the 10@xmath0 level . one discrepancy is the non - vanishing signal at 4@xmath49c for the proton beam experiment , which can be described with an additional non - temperature dependent signal with a @xmath132 contribution to the amplitude at 15@xmath49c . the model allows for calculations of the characteristics of sound pulses generated in the interaction of high energy particles in water with the input of the energy deposition of the resulting cascade . a possible application of this technique would be the detection of neutrinos with energies @xmath142 this work was supported by the german ministry for education and research ( bmbf ) by grants 05cn2we1/2 , 05cn5we1/7 and 05a08we1 . parts of the measurements were performed at the `` theodor svedberg laboratory '' in uppsala , sweden . the authors wish to thank all involved personnel and especially the acoustics groups of desy zeuthen and of uppsala university for their support .
|
the generation of hydrodynamic radiation in interactions of pulsed proton and laser beams with matter is explored .
the beams were directed into a water target and the resulting acoustic signals were recorded with pressure sensitive sensors .
measurements were performed with varying pulse energies , sensor positions , beam diameters and temperatures .
the obtained data are matched by simulation results based on the thermo - acoustic model with uncertainties at a level of 10@xmath0 .
the results imply that the primary mechanism for sound generation by the energy deposition of particles propagating in water is the local heating of the medium .
the heating results in a fast expansion or contraction and a pressure pulse of bipolar shape is emitted into the surrounding medium .
an interesting , widely discussed application of this effect could be the detection of ultra - high energetic cosmic neutrinos in future large - scale acoustic neutrino detectors .
for this application a validation of the sound generation mechanism to high accuracy , as achieved with the experiments discussed in this article , is of high importance .
cosmic neutrinos , acoustic neutrino detection , thermo - acoustic model , ultra - high energy cosmic rays , beam interaction
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
one - dimensional ( 1d ) interacting fermionic systems show remarkable physical properties and are promising elements for future nanoelectronics . the electron - electron interaction manifests itself in a particularly dramatic way in 1d systems , inducing a strongly correlated electronic state luttinger liquid ( ll ) @xcite . a paradigmatic experimental realization of quantum wires are carbon nanotubes @xcite ; for a recent review see ref . . further realizations encompass semiconductor , metallic and polymer nanowires , as well as quantum hall edges . there is currently a growing interest in non - equilibrium phenomena on nanoscales . a tunneling spectroscopy ( ts ) technique for non - equilibrium nanostructures was developed in ref . . employing a superconducting tunneling electrode allows one to explore not only the tunneling density of states ( tdos ) but also the energy distribution function . the energy relaxation found in this way provides information about inelastic scattering in the system . in a very recent experiment @xcite this ts method was applied to a carbon nanotube under strongly non - equilibrium conditions . in this paper , we develop a theory of ts of a ll out of equilibrium . specifically , we consider a ll conductor connected , via non - interacting leads , to reservoirs with different electrochemical potentials , @xmath0 and different temperatures @xmath1 , @xmath2 ( where the indices @xmath3 , @xmath4 stand for left- and right - movers ) . it is assumed that the coupling to the leads is adiabatic on the scale of the fermi wave length , so that no backscattering of electrons takes place . we model the leads as non - interacting 1d wires , so that the electron - electron interaction is turned on at the vicinity of the points @xmath5 , see fig . this model is quite generic to properly describe the problem at hand , independently of the actual geometry of the leads . note also that the 1d setup with strongly non - uniform interaction may be experimentally realized by using external screening gates . it is known that energy relaxation is absent in a uniform clean ll . within the golden - rule framework , the lack of energy relaxation for forward scattering processes results from 1d kinematic constraints that do not allow to satisfy the energy and momentum conservation laws simultaneously @xcite . on a more formal level , the conservation of energies of individual particles in a spatially uniform ll is protected by the integrability of the system , which implies an infinite number of conservation laws @xcite . inclusion of spatial dependence into the model violates these laws and leads to energy relaxation that takes place at the regions where the interaction varies in space @xcite . the fact that inhomogeneous interaction induces energy relaxation of electrons has been pointed out for the first time in ref . in the context of interacting quantum hall edges but a detailed analysis of this effect has been missing until now . on the other hand , one may expect this to be a dominant effect on the electron distribution function in experiments done on modern high - quality quantum wires ( such as ultraclean carbon nanotubes @xcite ) , under non - equilibrium conditions . there is thus a clear need in the theory of ts in non - equilibrium ll . it is worth noting that we assume the absence of backscattering due to impurities in the wire . when present , such impurities strongly affect the electronic properties of a ll wire : they induce diffusive dynamics at sufficiently high temperature @xmath6 and localization phenomena proliferating with lowering @xmath6 ( ref . ) , as well as inelastic processes @xcite . we also neglect the nonlinearity of the electron dispersion whose influence on spectral and kinetic properties of 1d electrons was recently studied in refs . , . the dashed line corresponds to the limit of a sharp variation of @xmath7 at the boundaries . ] within the ll model , the electron field is decoupled in a sum of right- and left - moving terms , @xmath8 , where @xmath9 is the fermi momentum . the hamiltonian of the system reads @xmath10 where @xmath11 is the electron velocity and @xmath12 is the spatially dependent electron - electron interaction constant . we will proceed by following the lines of the functional bosonization approach @xcite in the non - equilibrium ( keldysh ) formulation @xcite@xmath13@xcite . performing the hubbard - stratonovich transformation , one decouples the interaction term via a bosonic field @xmath14 and gets the action @xmath15=i\sum_{\eta = r , l}\psi^\dagger_\eta ( \partial_\eta-\phi)\psi_\eta-\frac{1}{2}\phi g^{-1}\phi\ , , \end{aligned}\ ] ] where @xmath16 and the fields are defined on the keldysh time contour . the information about physical observables is contained in keldysh green functions @xcite @xmath17 and @xmath18 ; see , in particular , appendix [ s9 ] where we express tunneling current in terms of functions @xmath19 and discuss how its measurement allows to determine @xmath19 experimentally . the green functions @xmath19 can be presented in the form @xmath20 e^{-{i\over 2 } \phi g^{-1}\phi } \nonumber\\ & \times & g^\gtrless_\eta[\phi](x , t;x',t ' ) , \label{green}\end{aligned}\ ] ] where we introduced the green function in a given field configuration , @xmath21 $ ] , and the sum of vacuum loops , @xmath22 $ ] . in 1d geometry the coupling between the fermionic and bosonic fields can be eliminated by a gauge transformation @xmath23 , if we require @xmath24 as a result , @xmath25 $ ] can be cast in the form @xmath26(x , t;x',t')&= & g^\gtrless_{\eta,0}(x - x';t - t ' ) e^{-i\eta ev(t - t')/2 } \nonumber \\ & \times & e^{\phi^\gtrless_\eta(x , t;x',t ' ) } \,.\end{aligned}\ ] ] here @xmath27 @xmath28 is the green function of free fermions , @xmath29 the coordinate @xmath30 labels the trajectory of a particle , and we use the convention that in formulas @xmath31 should be understood as @xmath32 for right / left moving electrons . it is convenient to perform a rotation in keldysh space , thus decomposing fields into classical and quantum components , @xmath33 , where the indices @xmath34 and @xmath35 refer to the fields on two branches of the keldysh contour . further , we introduce vector notations by combining @xmath36 and @xmath37 in a 2-vector @xmath38 . to proceed further , we resolve eq . ( [ diff_eq ] ) and express @xmath39 through @xmath14 as @xmath40 where @xmath41 is the green function of free bosons , @xmath42 its retarded and advanced components are given by @xmath43 the keldysh component of @xmath44 is given by @xmath45 , where @xmath46 is determined by the temperature @xmath47 of the reservoir from which the electrons moving in direction @xmath31 emerge , @xmath48 using eqs . ( [ phi ] ) and ( [ theta ] ) and performing a transformation to the coordinate space , we express the exponent @xmath49(x , t , x',t')$ ] through the bosonic field @xmath50 , @xmath51(x , t , x',t')= \int { d\omega\over 2\pi } dy \bm{\phi}^t_{-\omega } ( y)\bm{j}^{\gtrless}_{\eta,\omega}(y;x , t , x',t').\ ] ] the components of @xmath52 are found as @xmath53e^{-i\omega\xi_\eta}\ ! -\!\theta[\eta(\!x'\!-\!y\!)]e^{-i\omega\xi_\eta'}\right)\,,\nonumber \\ & & j^\gtrless_{2,\eta,\omega}(y)=-\frac{e^{i\eta \frac{\omega}{v}y}}{\sqrt{2}v } \left(e^{i\omega\xi_\eta}-e^{i\omega\xi_\eta'}\right)b_\eta^{(0)}(\omega ) \nonumber \\ & & \mp \frac{e^{i\eta\frac{\omega}{v}y}}{\sqrt{2}v } \left(\theta[\eta(y - x)]e^{-i\omega\xi_\eta } + \theta[\eta(y - x')]e^{-i\omega\xi_\eta'}\right),\label{j12}\end{aligned}\ ] ] where @xmath54 is the heviside @xmath55-function . the vacuum loop factor in eq . ( [ green ] ) is given by @xmath56 = \exp\left(-\frac{i}{2}\bm{\phi}^t \pi \bm{\phi}\right)\,,\end{aligned}\ ] ] where @xmath57 is the polarization operator , @xmath58 it can be decomposed into left and right moving parts , @xmath59 , with @xmath60 where @xmath61 . performing the averaging over the auxiliary field @xmath14 , we get @xmath62 where the effect of the interaction is represented by the `` debye - waller factor '' @xmath63 with @xmath64 here @xmath65 is the screened electron - electron interaction potential . its retarded component is given by @xmath66\,,\end{aligned}\ ] ] where the function @xmath67 is determined by the following differential equation @xmath68 which describes the plasmon propagation in a medium with spatially dependent sound velocity @xmath69 . the keldysh component of the interaction propagator is obtained as @xmath70 where @xmath71 at equilibrium , @xmath72 , this reduces to @xmath73b(\omega)\,,\end{aligned}\ ] ] in agreement with the fluctuation - dissipation theorem . so far we made no restriction on the way the interaction changes in space . let us consider first the case when the interaction turns on and off sharply on the scale set by the temperatures , @xmath74 . this limit can be modelled via a stepwise interaction as represented by the dashed line in fig . [ fig1 ] . equation ( [ plazmon ] ) for @xmath67 can be then straightforwardly solved by using the fact that the velocity @xmath75 is constant in each of three regions and employing the proper boundary conditions [ continuity of @xmath76 and of @xmath77 at @xmath78 . in the ts context , we are interested in the green functions @xmath19 with coinciding spatial arguments , @xmath79 . assuming @xmath80 to be in the interacting part of the wire ( and not too close to the boundaries ) and setting @xmath81 , we find @xmath82\,,\end{aligned}\ ] ] where @xmath83 is the conventional dimensionless parameter characterizing the interaction strength in a ll and @xmath84 the integral in eq . ( [ eq_f ] ) and in analogous formulas below is logarithmically divergent at large frequencies and require an ultraviolet regularization . specifically , these integrals are understood as regularized by a factor @xmath85 , where @xmath86 is an ultraviolet cutoff . deriving eq . ( [ eq_f ] ) , we have neglected terms of the form @xmath87 ( with non - zero integer @xmath88 ) that arise due to the fabry - perot - type interference of plasmon modes reflected at the boundaries . keeping these terms would lead to an additional oscillatory structure in energy @xcite with the scale @xmath89 . since we are interested in ts of long wires , we assume that this scale is much less than @xmath90 , so that oscillations are suppressed . substituting eq . ( [ eq_f ] ) into eq . ( [ eq_g ] ) , we finally get the green functions : @xmath91^{1+\alpha } \bigg[g^\gtrless_{l,0}(t)\bigg]^{\beta}e^{-i\eta evt/2}\,,\ ] ] where @xmath92 the green functions ( [ green_fun ] ) can be determined experimentally from ts measurements @xcite , see appendix [ s9 ] . their difference determines the tdos @xmath93 , @xmath94 while each of them separately ( or their sum ) contains also information about the distribution function , as discussed below . the results for the tdos have been found in ref . . next we consider the non - interacting parts of the wire , and discuss , e.g. , the right moving electrons . in the region i ( see fig . [ fig1 ] ) , @xmath95 , we find from eqs . ( [ f ] ) , ( [ j12 ] ) that @xmath96 , so that the green functions of the right movers are not modified by interaction . physically this is quite transparent : the right - moving electrons in this part of the system are just coming from the reservoir and are not yet `` aware '' of the interaction with the left - movers . the situation is distinctly different in the region iii , @xmath97 . assuming @xmath79 , we find @xmath98\,.\end{aligned}\ ] ] substituting eq . ( [ region3 ] ) into eq . ( [ eq_g ] ) , one gets @xmath99^{\cal t}\bigg[g_{l,0}^\gtrless(t ) \bigg]^{\cal r } e^{-i\eta evt/2},\end{aligned}\ ] ] where @xmath100 since @xmath101 in eq . ( [ region3 ] ) is real , the tdos is not affected by the interaction , @xmath102 , as expected . the modification of the functions @xmath103 as compared to that of incoming electrons , @xmath104 , implies therefore the change in the distribution function @xmath105 of right - movers . indeed , for non - interacting particles @xmath106 and @xmath107 $ ] . we thus see that the electrons ejected from the interacting part of the wire into the lead are affected by the interaction : their distribution function has changed . the left - moving electrons can be analyzed in the same way ; the corresponding results are obtained by replacing @xmath108 in eqs . ( [ green_fun ] ) , ( [ eq2 ] ) . clearly , the role of the regions i and iii is interchanged in this case . it is also worth mentioning that in the non - interacting parts of the wire the green functions are both galilean and translationally invariant , depending on coordinates and times via @xmath109 only . we turn now to generalization of these results for the case of an arbitrary shape of @xmath12 in the contact region between the interacting part of the wire and the non - interacting leads . the contact regions are in general characterized by some reflection coefficients @xmath110 for the plasmon amplitude , yielding reflection coefficients @xmath111 for the plasmon intensity ( @xmath112 for the left and right contact , respectively ) . the corresponding transmission coefficients are @xmath113 . it is instructive in this context to compare our present approach with that developed in ref . , where we analyzed the tunneling density of states and focussed on the case of smooth variation of @xmath12 in the contact regions . as we are going to show , the method of ref . can be generalized to the case of arbitrary contacts ( this was briefly discussed at the end of ref . ) and is also useful for the analysis of the electron distribution function . within that approach , the propagator of bosons is calculated in momentum space ( rather than in real space as in the above calculation ) . the keldysh component of the propagator is then characterized by distribution function functions @xmath114 and @xmath115 associated with poles at @xmath116 and @xmath117 and describing `` ghosts '' ( free electron - hole pairs ) and plasmons , respectively @xcite . while the distribution function of ghosts is simply determined by that of incoming electrons , the plasmons experience in general reflection at the boundaries . we have for the left boundary ( see fig . [ fig2 ] ) @xmath118 and similarly at the right boundary . here we have introduced the notation @xmath119 for plasmon distributions in the interacting region of the wire and @xmath120 for out - going channels . solving these equations , we find the plasmon distribution functions of right - movers in the interacting part of the wire , as well as in the outgoing channel ( in the right lead ) : @xmath121 the corresponding results for left movers are obtained by exchanging the indices r@xmath122l and 1@xmath1222 . in different parts of the wire . the distributions of incoming plasmons are determined by respective leads , @xmath123 ] the method of ref . allows us to express the exponents @xmath124 in terms of these distribution functions . for the interacting part of the wire , we get @xmath125\,.\end{aligned}\ ] ] the result for the tunneling into the non - interacting region iii of fig . [ fig1 ] can be obtained from eq . ( [ inf_theory_gen ] ) by using the distribution functions @xmath120 corresponding to this region and replacing the interaction constant @xmath126 by zero , @xmath127\,,\end{aligned}\ ] ] where @xmath128 is the total reflection coefficient on a double - step structure , @xmath129 . for the case of sharp boundaries the reflection and transmission coefficients are given by the fresnel law , @xmath130 and @xmath131 , so that eqs . ( [ inf_theory_gen ] ) and ( [ inf_theory_region3_gen ] ) reduce to the earlier results ( [ eq_f ] ) , ( [ region3 ] ) . the total reflection and transmission coefficients @xmath128 and @xmath132 take in this case the values ( [ tran_coeff ] ) ( which explains the notations introduced there ) . clearly , the general formulas ( [ inf_theory_gen ] ) and ( [ inf_theory_region3_gen ] ) can also be obtained in the framework of a real - space calculation that was presented above for sharp boundaries . to do this , one has to modify the boundary conditions for the green function in @xmath67 in eq . ( [ plazmon ] ) by including the appropriate reflection and transmission amplitudes @xmath110 and @xmath133 at two boundaries and then proceeding in the same way as in course of the derivation of eqs . ( [ eq_f ] ) and ( [ region3 ] ) . the two methods ( real space and k space ) are thus in full agreement with each other . the formal results obtained thus far can be implemented to obtain physical observables . consider first the non - interacting part of the setup , region iii of fig . the effect of the interaction there amounts to modification of the distribution function of outgoing particles ( right - movers ) , which has ( in time domain ) the form @xmath134 where @xmath135 is given by eq . ( [ inf_theory_region3_gen ] ) . this yields @xmath136 the way in which the electron distribution function is modified depends on the kinetics of the plasmons inside the interacting region . for adiabatic switching of interaction , there is essentially no plasmon scattering . therefore , the total reflection coefficient @xmath128 and , consequently , the exponent @xmath135 in the region iii vanish . in this case the fermions retain their distribution function : the right - movers going out into the right lead have the same distribution as the right - movers injected into the interacting region from the left lead . ( the same applies to the left - movers , of course . ) let us now discuss the opposite limit of strong reflection , @xmath137 . for a structure with a sharp boundary , this is the case provided the interaction is strong , @xmath138 . alternatively , this limit may be realized if the boundary regions are sufficiently extended and characterized by random @xmath7 such that plasmons with relevant frequencies are localized . regardless of the cause , in the limit @xmath137 the left- and right - moving electrons exchange their distribution functions , except for keeping their total flux ( i.e. the chemical potential ) . next , we consider the interacting part of the wire . analyzing the result ( [ inf_theory_gen ] ) , we see that two terms in square brackets have distinctly different physical origin . the second term , which is proportional to the local strength of the interaction @xmath126 at the measurement point is responsible for creation of the zero - bias anomaly ( zba ) as well as for its dephasing smearing , with the non - equilibrium dephasing rate @xcite @xmath139\,.\end{aligned}\ ] ] on the other hand , the first term in the integrand of ( [ inf_theory_gen ] ) , which is governed by the difference between the incoming and local distribution of plasmons , is fully analogous to the expression for @xmath140 in the non - interacting region , eq . ( [ inf_theory_region3_gen ] ) , and describes the modification of the distribution function inside the wire , @xmath141 ( 1-\cos\omega t)\right\ } \nonumber \\ & & = \frac{i}{2\pi}\frac{1}{t+i0 } \exp\left\{-\int_0^\infty { d\omega\over \omega } [ b_\eta^{\rm w}(\omega ) - 1 ] ( 1-\cos\omega t)\right\}\ , . \label{distr_func}\end{aligned}\ ] ] as is clear from eq . ( [ distr_func ] ) , the `` ghost '' term with @xmath142 essentially serves to cancel the bare distribution function @xmath143 , so that the distribution function @xmath144 is determined only by the plasmonic distribution @xmath145 in the wire . this is in fact a manifestation of a general relation between the functional and full bosonization approaches , as will be discussed in detail elsewhere @xcite . ( no interaction ) and @xmath146 ( with sharp boundaries ) . temperatures of the leads are @xmath147 and @xmath148 ; the bias voltage is @xmath149 . ] fourier transformation of our results into the energy representation can be done numerically ( for analytic calculation at equilibrium see appendix [ s8 ] ) ; representative results are shown in figs . [ fig3 ] , [ fig4 ] . in fig . [ fig3 ] we present distribution functions for non - interacting parts of the wire . temperatures are set to @xmath147 and @xmath148 ( in arbitrary units ) , the applied voltage is @xmath149 , and a sharp variation of the interaction at the boundaries ( as in sec . [ s3 ] ) is assumed . the distribution function of free fermions ( @xmath150 ) , plotted by a dashed line , is the same on both ends of the wire . for interacting electrons ( we choose the interaction parameter to be @xmath146 , which is in the range of characteristic values reported for carbon nanotubes , see , e.g. , ref . ) the distribution functions in two leads are different . in particular , the distribution function in the left lead ( region i in fig . [ fig1 ] ) has a sharp edge at the energy @xmath151 , which corresponds to cold right - moving electrons . in the right lead ( region iii ) , this edge is broadened due to interaction with hot left - moving electrons . the situation is opposite for left - moving particles . the distribution in the right lead has a broad edge at @xmath152 that corresponds to hot left - moving electrons . due to interaction inside the wire this edge in the region i sharpens . in fig . [ fig4 ] we present the results for the distribution functions of left- and right - moving quasiparticles in the central ( interacting ) part of the wire , eq . ( [ distr_func ] ) . for @xmath146 the plasmon reflection at the boundaries is strong . in a symmetric structure this leads to almost equal distribution functions of both types of carriers inside the wire . in the upper panel of fig . [ fig5 ] we show the results for tdos for @xmath153 . the minima of tdos are reached at energies @xmath154 . the broadening of the zba dips has two origins : smearing of the distribution function and dephasing . while the dephasing broadening [ cf . second term in eq . ( [ inf_theory_gen ] ) ] is the same for both chiral branches , the distribution functions [ cf . first term in eq . ( [ inf_theory_gen ] ) ] are in general different . a deeper minimum at @xmath151 reflects the fact that right - moving electrons in the wire have a much narrower distribution function . this is because at @xmath153 the energy relaxation at the boundaries is quite weak , so that the distribution functions of cold right - movers and hot left - movers are only slightly modified . the situation is different for @xmath146 , when distribution functions @xmath155 and @xmath156 are nearly identical ( up to a shift by @xmath157 ) , see fig . [ fig4 ] . as a result , the structure of the tdos also becomes symmetric . in fact , for the chosen value of the voltage , two broad zba dips merge together . ) , in the interacting part of the wire . all parameters are the same as in fig . ] ( normalized to its non - interacting value @xmath158 ) in the interacting region for @xmath153 , 0.4 , and 0.2 . the temperatures of leads and the voltage are the same as in fig . [ fig3 ] . ] we discuss now a relation between our results for the electron distribution function and previous findings on the electric and thermal conductance of a ll wire . in the absence of backscattering the number of left and right moving particles is separately preserved . as a result , the electric current is linear in the voltage @xmath159 , @xmath160 with unrenormalized landauer conductance @xmath161 , ref . . in our formalism , this relation immediately follows from eq . ( [ eq_n ] ) and the condition @xmath162 . this ensures that the modification of the distribution function of right ( or left ) movers by a spatially varying interaction does not affect the integral of the distribution function over energy , i.e. the total number of carriers of each type . we turn now to the thermal conductance . the energy current is easily found from the green functions of electrons in non - interacting parts of the wire , @xmath163\right|_{t = t'},\ ] ] which can be rewritten in terms of the electron distribution functions , @xmath164\,.\end{aligned}\ ] ] substituting the result ( [ eq_n ] ) , ( [ inf_theory_region3_gen ] ) for the distribution functions , we get the expression of the thermal current in terms of distribution functions of incoming electron - hole pairs , @xmath165\ , . \label{thermal_2}\ ] ] according to ( [ thermal_2 ] ) , the thermal conductance is affected by the interaction [ through the reflection coefficient @xmath166 , as was first found in ref . . note that due to the particle - hole symmetry of ll model , the applied voltage drops out of eq . ( [ thermal_2 ] ) . for the case of sufficiently sharp boundaries , when @xmath167 can be considered as @xmath168-independent for relevant frequencies , eq . ( [ thermal_1 ] ) reduces to @xmath169 deviation of the transmission coefficient @xmath167 from unity leads to the violation of the wiedemann - franz law @xcite . as is seen from our analysis , this deviation is a manifestation of a microscopic phenomenon : energy relaxation of electrons due to non - uniform interaction . the heat current ( [ thermal_2 ] ) can be equivalently represented in terms of plasmonic distributions in the wire @xmath170\ , . \label{thermal_4}\ ] ] this implies that the presentation of the heat current in the form ( [ thermal_1 ] ) is also valid in the interacting part of the wire , with the electronic distribution functions @xmath171 given by ( [ distr_func ] ) . thus , also in the interacting part of the wire , the energy current can be understood as carried by properly defined quasi - particle excitation . this is a remarkable result , which demonstrates that the concept of fermionic quasiparticles remains meaningful in a strongly interacting 1d system ( ll ) despite its non - fermi - liquid features . to summarize , we have developed a theory of tunneling spectroscopy of ll conductor connected to reservoirs away from equilibrium . in the specific setup considered here , each branch originates from a source which is at equilibrium . however , the right and the left sources have different temperatures and different chemical potentials . we have modeled the system as a ll with spatially non - uniform interaction , and calculated the single - electron green functions @xmath19 that carry information about the tdos and the fermionic distribution functions in different parts of the wire . the interaction affects the tunneling characteristics in three distinct ways . first , it induces a power - law zba in the tdos @xmath93 ( with two dips split by the voltage ) in the interacting part of the wire . second , it leads to broadening of zba singularities due to dephasing , with the dephasing rate governed by the interaction strength and the plasmon distribution inside the wire . both the zba and the dephasing effects are encoded in the second term of eq . ( [ inf_theory_gen ] ) . the third effect of the interaction which is specifically at the focus of the present work is the inelastic scattering of electrons , leading to their redistribution over energies . this effect takes place in those regions where the interaction strength varies in space ( near the wire boundaries in our model ) , inducing backscattering of plasmons ( but not of electrons ) . this leads to relaxation of the electron distribution functions : left and right moving fermions `` partly exchange '' their distributions , see eqs . ( [ n_r ] ) , ( [ distr_func ] ) and figs . [ fig3 ] , [ fig4 ] , [ fig5 ] . for slowly varying interaction , when the plasmons with relevant frequencies go through essentially without reflection , the energy relaxation of electrons is negligible . in the opposite limit , when the plasmons are almost entirely reflected ( due to strong and sharply switched interaction or , else , due to disordered boundary regions inducing the plasmon localization ) , the left- and right - movers essentially exchange their distribution functions ( but not their total density ) . we have also discussed a connection between these results and earlier findings on the thermal conductivity of ll structures . our results are important for the analysis of ts experiments on strongly correlated 1d structures ( in particular , carbon nanotubes @xcite ) out of equilibrium . in this connection , let us emphasize the following important point . what can actually be measured in experiment are green functions , @xmath17 and @xmath18 . the tdos @xmath93 in the interacting part of the wire , as well as the distribution function @xmath172 in the non - interacting regions are related to @xmath17 and @xmath18 in a simple way . on the other hand , in order to extract the distributions @xmath105 and @xmath173 from @xmath19 in the _ interacting _ part of the wire , a non - trivial deconvolution procedure is necessary . the broadening of ( split ) fermi - edge structures in @xmath19 in the interacting part of the wire is governed by both the distribution function and the dephasing . the dephasing contributes to the smearing of fermi - edge singularities also in higher - dimensional ( diffusive ) systems @xcite , and should be taken into account for the accurate interpretation of corresponding experiments@xcite . in the 1d case the role of dephasing becomes particularly dramatic ( if the interaction is sufficiently strong ) . this is very well illustrated by fig . [ fig5 ] : two fermi - edge singularities almost ( middle panel ) or even completely ( lower panel ) merge , despite the fact that the fermi edges in the distribution functions remain well separated ( fig . [ fig4 ] ) . a comment of a more general nature is in order here . our results illustrate the fact that there is no unique answer to the question : `` how much is a ll different from a fermi liquid ? '' on one hand , the strong , power - law zba in tdos of a ll clearly distinguishes it from the fermi liquid . in more formal terms , the single - particle residue @xmath174 , which is finite in the fermi liquid , vanishes in a power - law fashion at the fermi level of the ll . also the dephasing rate determining the broadening of zba , eq . ( [ tauphi ] ) , is linear in temperature , contrary to the fermi - liquid @xmath175 behavior . one could think that it makes little sense to speak about fermionic excitations in this situation , but this is not the case . first , the power - law vanishing of tdos has little importance ( like the value of @xmath174 in the fermi liquid ) for kinetic properties of the system . second , the dephasing rate ( [ tauphi ] ) is governed by processes with zero energy transfer and do not lead to any energy relaxation . as a result , the distribution function of fermionic excitations , @xmath171 , is a fully meaningful concept even in the case of a strong interaction . it stays preserved as long as the interaction is spatially constant ( or varies adiabatically slow with @xmath80 ) . furthermore , both the charge and the energy current in the interacting part of the wire can be understood as carried by these fermionic quasiparticles . from this point of view , the ll is a _ perfect _ fermi liquid . we conclude the paper by reviewing some future research prospects ; the work in those directions is currently underway . first , one may consider a more general non - equilibrium situation where the distribution functions `` injected '' into the interacting part of the wire are of non - equilibrium ( e.g. , double - step ) form by themselves @xcite , see setups b , c in fig.1 of ref . ; the first of these setups is close to the experimental situation of ref . . this requires a generalization of the bosonization technique that will be presented elsewhere @xcite . second , it is interesting to study correlations between outgoing left- and right - movers . in a general situation , one finds that their density matrices are not decoupled , i.e. they are entangled , which manifests itself , in particular , in current cross - correlations . third , one may study the effect of a random variation of the interaction strength @xmath7 in the wire . if the wire is sufficiently long , plasmons with not too low frequencies get localized . using our general results , one concludes that in the left ( right ) half of the wire both distributions @xmath155 , @xmath156 are determined by that of the left ( respectively , right ) reservoir , with a transition region which extends over the localization length of the middle section . to refine this picture , one has to include into consideration also plasmons with low frequencies ( that remain delocalized ) . also , including the spectral curvature will induce plasmon decay processes . ( in the context of thermal conductivity , this physics was discussed in ref . . ) finally , our results can be generalized to the case of chiral ll , where both branches move in the same direction , which is the situation characteristic for quantum - hall edge - state devices @xcite . we thank d. bagrets , n. birge , a. finkelstein , i. gornyi , d. maslov , y. nazarov , d. polyakov , and r. thomale for useful discussions . we are particularly grateful to the late yehoshua levinson for numerous illuminating discussions on the physics of non - equilibrium systems . this work was supported by us - israel bsf , isf of the israel academy of sciences , the minerva foundation , and dfg spp 1285 ( yg ) , ec transnational access program at the wis braun submicron center ( adm ) , german - israeli foundation under grant 965 , and einstein minerva center . the tunneling current between a probe and a quantum wire can be expressed in terms of the functions @xmath19 as @xmath176\,,\end{aligned}\ ] ] where the subscripts `` tp '' and `` w '' refer to the tunnel probe and the wire respectively , @xmath177 is a voltage between the tunneling probe and the wire , and @xmath178 is a tunneling matrix element in the coordinate representation . if electron tunneling is local in space , we have @xmath179 , where @xmath80 is a position of tunneling probe . since the tunneling probe is at equilibrium , one can use a standard relation between the green functions and distribution function @xmath180 of electrons in the probe , @xmath181.\end{aligned}\ ] ] differentiating the tunneling current with respect to voltage and substituting eq . ( [ app_a_eq3 ] ) into eq . ( [ app_a_eq1 ] ) , one finds @xmath182 \nonumber \\ & & -2\pi i\nu_{{\rm tp}}(\epsilon - eu)\nu_{{\rm w}}(\epsilon)\frac{\partial n_{{\rm tp}}(\epsilon - eu)}{\partial \epsilon}\bigg\ } \ , . \end{aligned}\ ] ] for a ll wire the green functions @xmath183 and the tdos @xmath184 represent a sum of contributions of both chiral branches . if the density of states in the tunneling probe ( @xmath185 ) is a constant ( as in a normal metal ) , the first term in eq . ( [ app_eq_2 ] ) drops out . in this case the result is proportional to the tdos in the wire . assuming that the tunneling probe is at zero temperature , one then finds @xmath186 on the other hand , if the density of states in the tunneling probe is strongly energy dependent ( as for superconducting electrodes ) , the first term in eq . ( [ app_eq_2 ] ) survives . unlike tdos ( which is determined by the difference @xmath187 ) , this term contains also the information about @xmath188 . therefore , measurement of the tunneling current with two different types of tunneling probes ( normal and superconducting ) allows one to find functions @xmath189 and @xmath190 separately . the idea to use superconducting electrodes for the tunneling spectroscopy was introduced in ref . and more recently employed in ref . . at thermal equilibrium the green functions in the energy domain can be calculated explicitly . using eq . ( [ inf_theory_gen ] ) and @xmath191 , we find the exponent @xmath192 for the green functions in interacting part of the wire , @xmath193 where we drop the chirality index @xmath31 , as it is immaterial for @xmath79 in equilibrium . using eq . ( [ eq_g ] ) and performing a fourier transform from the time into the energy domain , one finds @xmath194 after calculating an auxiliary integral @xmath195\bigr|^2 \,,\nonumber\end{aligned}\ ] ] one obtains @xmath196\bigr|^2 , \ ] ] where @xmath197 . similarly , one finds the function @xmath18 , @xmath198\bigr|^2.\ ] ] this yields the following asymptotic behavior of the green function at low temperatures ( @xmath199 ) , @xmath200 and high temperatures ( @xmath201 ) , @xmath202 \left(\frac{\pi t}{\lambda}\right)^\gamma\ , \,.\ ] ] using eqs . ( [ app_nu ] ) , ( [ app3 ] ) , and ( [ app4 ] ) , one obtains tdos at equilibrium , @xmath203\biggr|^2 \cosh\frac{\pi z}{2}.\end{aligned}\ ] ] equation ( [ app6 ] ) describes the well - known zba in tdos , @xmath204 , smeared at the scale @xmath205 . this smearing results from a combined effect of ( i ) the thermal broadening of the distribution function and ( ii ) the dephasing rate@xcite @xmath206 . m. bockrath , d.h . cobden , j. lu , a.g . rinzler , r.e . smalley , l. balents , and p.l . mceuen , nature ( london ) * 397 * , 598 ( 1999 ) ; z. yao , h.w.ch . postma , l. balents , and c. dekker , nature ( london ) * 402 * , 273 ( 1999 ) . h. c. fogedby , j. phys . c * 9 * , 3757 ( 1976 ) ; d. k. lee and y. chen , j. phys . a * 21 * , 4155 ( 1988 ) ; c.m . naon , m.c . von reichenbach , and m.l . trobo , nucl . phys . b * 435 * , 567 ( 1995 ) ; c.m . naon , m.j . salvay , and m.l . trobo , int . j. mod a * 19 * , 4953 ( 2004 ) ; i. v. yurkevich , in _ strongly correlated fermions and bosons in low - dimensional disordered systems _ , edited by i.v . lerner , b.l . altshuler , v.i . falko , and t. giamarchi ; a. grishin , i.v . yurkevich and i.v . lerner , phys . b. * 69 * , 165108 ( 2004 ) . for review of the keldysh technique see , e.g. , j. rammer and h. smith , rev . phys . * 58 * , 323 ( 1986 ) ; a. kamenev , in _ nanophysics : coherence and transport_(elsevier , 2005 ) , edited by h. bouchiat , y. gefen , g. montambaux , and j. dalibard , p. 177 . maslov and m. stone , phys . b * 52 * , r5539 ( 1995 ) ; v. ponomarenko , phys . rev . b * 52 * , r8666 ( 1995 ) ; i. safi and h.j . schulz , phys . b * 52 * , r17040 ( 1995 ) ; y. oreg and a.m. finkelstein , phys . rev . b * 54 * , r14265 ( 1996 ) . y. ji , y. chung , d. sprinzak , m. heiblum , d. mahalu , and h. shtrikman , nature * 422 * , 415 ( 2003 ) ; i. neder , f. marquardt , m. heiblum , d. mahalu , and v. umansky , nature phys . * 3 * , 534 ( 2007 ) ; i.p . levkivsky and e.v . sukhorukov , phys . b * 78 * , 045322 ( 2008 ) .
|
we develop a theory of tunneling spectroscopy of interacting electrons in a non - equilibrium quantum wire coupled to reservoirs .
the problem is modelled as an out - of - equilibrium luttinger liquid with spatially dependent interaction .
the interaction leads to the renormalization of the tunneling density of states , as well as to the redistribution of electrons over energies .
energy relaxation is controlled by plasmon scattering at the boundaries between regions with different interaction strength , and affects the distribution function of electrons in the wire as well as that of electrons emitted from the interacting regions into non - interacting electrodes .
# 1eq .
( [ # 1 ] ) # 1([#1 ] )
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
sgr a * , the supermassive black hole at the center of our galaxy , has been observed for several decades . monitoring stars orbiting around sgr a * has led to measurements of its mass and distance ( ghez et al . 2008 ; gillessen et al . 2009 ) . however , these measurements of mass and distance are strongly correlated . for purely astrometric measurements , mass and distance are related as @xmath3 , while for measurements of radial velocities mass and distance are related as @xmath4 . for combined data sets , the correlation between mass and distance behaves roughly as @xmath2 ( ghez et al . 2008 ; gillessen et al . this correlation between mass and distance constitutes a major source of uncertainty in our understanding of the properties of sgr a*. likewise , dynamical measurements of the masses of a number of nearby supermassive black holes have been obtained with often much greater uncertainties ( see , e.g. , gltekin et al . 2009 ) . another technique , vlbi , aims to image sgr a * directly . recent vlbi observations with an array consisting of the submillimeter telescope observatory ( smto ) in arizona , the james clerk maxwell telescope ( jcmt ) on mauna kea , and several of the dishes of the combined array for research in millimeter - wave astronomy ( carma ) in california resolved sgr a * on scales comparable to its event horizon and identified sub - horizon size structures ( doeleman et al . 2008 ; fish et al . images of accretion flows around black holes have the shadow of the compact object imprinted on them , which depends uniquely on its mass , spin , and inclination ( e.g. , falcke et al . 2000 ) as well as on possible deviations from the kerr metric ( johannsen & psaltis 2010 ) . based on such images and assuming the mass and distance obtained from the monitoring of stellar orbits , these vlbi observations inferred constraints on the inclination and spin of sgr a * ( broderick et al . 2009 , 2011 ) and placed limits on potential non - kerr signatures ( broderick et al . 2012 ) . in addition to the shadow , images of optically thin accretion flows around black holes carry a characteristic signature in the form of a bright ring ( johannsen & psaltis 2010 ) , which we refer to as the photon ring . light rays that approach the event horizon closely orbit around the black hole many times before they are detected by a distant observer , resulting in a bright ring due to their long optical path length through the accretion flow . the flux of such photons can account for a significant fraction of the total disk flux and produce higher order images ( cunningham 1976 ; laor , netzer , & piran 1990 ; viergutz 1993 ; bao , hadrava , & stgaard 1994 ; @xmath5 , fanton , & calvani 1998 ; agol & krolik 2000 ; beckwith & done 2005 ) . these photon rings are clearly visible in all time - dependent general - relativistic simulations of accretion flows that have been reported to date ( mo@xmath6cibrodzka et al . 2009 ; dexter , agol , & fragile 2009 ; shcherbakov & penna 2010 ) . johannsen & psaltis ( 2010 ) showed that a measurement of the ring diameter measures the ratio @xmath7 for the black hole , independent of its spin or deviation from the kerr metric . therefore , combining such a measurement with the observations of stars around sgr a * can reduce the correlation between mass and distance . in this paper , we explore the ability of this approach to refine the mass and distance measurements of sgr a*. we estimate the precision with which a thermal noise - limited vlbi array can infer the diameter of the ring of sgr a * and use a bayesian technique to simulate measurements of the mass and diameter of sgr a * in conjunction with parameters inferred from the existing data of the orbits of stars at comparable wavelengths . we show that , in this best - case scenario , the correlation between mass and distance is reduced significantly . in addition , we argue that the accretion flows of other nearby supermassive black holes are optically thin , allowing for vlbi observations of their respective photon rings . we assess the prospects of using this technique to infer the masses of these sources . the properties of photon rings are practically independent of the specific flow geometry and remain constant even if the accretion flow itself is highly variable ( johannsen & psaltis 2010 ) . the relative brightness as well as the constancy of these rings make them ideal targets for vlbi - imaging observations . for a kerr black hole with mass @xmath0 , the shape of a given photon ring has a diameter of @xmath8 which remains practically constant for all values of the spin and disk inclination ( johannsen & psaltis 2010 ) . in this expression , @xmath9 is the gravitational radius , and @xmath10 and @xmath11 are the gravitational constant and the speed of light , respectively . the angular diameter @xmath12 of the diameter of the photon ring of a black hole is given by the ratio of its diameter and distance , @xmath13 assuming the current mass and distance measurements of sgr a * , @xmath14 and @xmath15 ( gillessen et al . 2009 ) , the photon ring has an angular diameter of @xmath16 radio interferometers are limited by their intrinsic resolution as well as by interstellar scattering . in order to identify the range of wavelengths within which vlbi measurements of the photon ring of sgr a * are resolution - limited , we compare the blurring effects of interstellar scattering with the resolution of an interferometer . in figure [ f : openingangle ] we plot the minimum size of resolvable structures on the image of sgr a * using the interstellar scattering law of bower et al . we also estimate ( dashed line ) the resolution of a radio interferometer at a given wavelength @xmath17 by the expression @xmath18 with @xmath19 and a diameter @xmath20 , which is comparable to the baseline length between the jcmt on hawaii and the south pole telescope . this yields @xmath21 , where @xmath22 is the resolution in @xmath23arcsec and @xmath24 is the observed wavelength in mm . dotted lines mark the angular diameters corresponding to length scales of @xmath25 and @xmath26 at the assumed distance of sgr a*. as can be seen from this figure , at sub - mm wavelengths interstellar scattering becomes negligible and measurements are limited by the resolution . therefore , a measurement of the diameter of the photon ring of sgr a * will require vlbi observations at wavelengths @xmath27 . in the following , we estimate the observed flux of the photon ring of sgr a * assuming a schwarzschild black hole and employing a model of a geometrically thin advection - dominated accretion flow ( adaf ; narayan & yi 1994 , 1995 ; narayan , yi , & mahadevan 1995 ) . adafs model the accreting gas around a supermassive black hole as a quasi - spherically symmetric plasma consisting of thermal electrons and ions at different temperatures ( narayan & yi 1995a , b ) as well as of non - thermal electrons ( mahadevan 1999 ; zel , psaltis , & narayan 2000 ; yuan , quataert , & narayan 2003 ) . such an accretion flow is allowed to cool through comptonization and the emission of bremsstrahlung and synchrotron radiation , with the latter generating the predominant contribution to the observed spectrum of sgr a * at radio and wavelengths ( narayan et al . 1995 ) . for our estimate , we follow broderick et al . ( 2009 ) and assume an adaf model with a density of thermal electrons @xmath28 electron temperature @xmath29 and magnetic field @xmath30 where @xmath31 . for the coefficients we use ( broderick et al . 2011 ) @xmath32 and @xmath33 these coefficients lead to predictions of the spectrum , polarization , and image size for sgr a * that are in agreement with all current observations . we assume that all the emission at mm - wavelengths is due to thermal synchrotron radiation ( see narayan et al . while there may be small contributions of synchrotron emission from non - thermal electrons ( e.g. , mahadevan 1998 ; zel et al . 2000 ; yuan et al . 2003 ) or of other types of radiation at these wavelengths ( such as jets ; e.g. , falcke et al . 1993 ) , this assumption affects our analysis only marginally . in practice , the measured total flux incorporates all such contributions . following dolence et al . ( 2009 ) , we write the synchrotron emissivity as @xmath34 where @xmath35 @xmath36 @xmath37 and @xmath38 is the modified bessel function of the second kind of the second order . here , @xmath39 is the angle between the wave vector of the emitted photon and the magnetic field . for our estimate , we use the average @xmath40 . we assume that optical paths follow at least one loop along the circular photon orbit of a schwarzschild black hole located at radius @xmath41 . the emitted intensity of radiation is then given by the expression @xmath42 the observed intensity is related to the emitted intensity by the third power of the redshift factor , which we take to be the gravitational redshift of a photon observed at infinity emitted from @xmath41 ( i.e. , we neglect for this simple estimate the high velocity of the flow , which will serve to increase the intensity at infinity ) . this yields : @xmath43 where @xmath44 the photon ring has an approximate width of @xmath45 in the image plane ( johannsen & psaltis 2010 ) . then , the subtended solid angle is given by the expression @xmath46 this yields our estimate for the observed flux density of the photon ring : @xmath47 in figure [ ringflux ] , we plot the modeled ring flux density as a function of wavelength . we also plot a blackbody function evaluated at the same emission radius and temperature given by expression ( [ temperature ] ) as an upper flux density limit . the above estimate exceeds the blackbody flux density at wavelengths @xmath48 , and , thus , at these longer wavelengths , the emission becomes self - absorbed . therefore , we use the minimum of these two flux densities as an estimate of the ring flux density . at wavelengths @xmath49 , the modeled flux density of the photon ring of sgr a * is @xmath50 , about 1/15 of the total source flux density . since @xmath49 is also in the regime where vlbi observations are resolution - limited ( see figure [ f : openingangle ] ) , this range of wavelengths is optimal for such measurements . the ring diameter is determined solely by the mass via equation ( [ ringdiameter ] ) , and we can relate the black - hole mass and the angular diameter of the ring according to the expression @xmath51 therefore , for vlbi imaging , the mass is proportional to the distance , and we can reduce the correlation between mass and distance in combination with dynamical measurements . we now explore the prospect of combining dynamical measurements of sgr a * with vlbi imaging observations of the photon ring . we analyze the best - case scenario of a thermal noise - limited vlbi array in order to assess whether such a measurement is worthwhile . systematic limitations will degrade the vlbi measurement somewhat , as discussed below . we employ a bayesian analysis to estimate the probability distribution over the mass and distance of sgr a * from a measurement of the angular diameter of the photon ring in combination with the constraints obtained from stellar dynamics . we take the latter as our prior , @xmath52 , by converting into a likelihood the @xmath53 distribution with @xmath54 degrees of freedom obtained from the existing data set of the ephemerides of several s - stars ( gillessen et al . 2009 ) . we assume a gaussian posterior likelihood of obtaining a particular measurement of the angular diameter as @xmath55,\ ] ] where @xmath12 is given by equation ( [ openingangle ] ) and @xmath56 . we then use bayes theorem to write the likelihood of a particular mass and distance of the black hole given the data as @xmath57 where @xmath58 is the appropriate constant that normalizes the likelihood . the measurement uncertainty @xmath59 is the key parameter of the likelihood @xmath60 . in the following , we estimate the scaling of the measurement uncertainty with resolution . the spatial frequencies of interest are those beyond the first null of the bessel function describing the fourier transform of the ring structure in visibility space . for a given resolution and ring size , there are @xmath61 accessible half periods of the oscillation , where @xmath61 is given by @xmath62 the typical amplitude of a bessel function oscillation of the first few maxima is of order 0.3 of the peak . therefore , the signal we seek to measure has an amplitude of @xmath63 where @xmath64 is the observed flux density of sgr a * near 1 mm ( see , e.g. , broderick et al . 2009 ) . we estimate the expected signal to noise ratio on this measurement by scaling it from the current vlbi measurements . fish et al . ( 2011 ) measure the size of sgr a * with a signal to noise ratio of @xmath65 . near - future vlbi arrays will incorporate 5 or 6 stations . one of them , alma , will have the sensitivity of 50 of the current stations resulting in an overall increase in array sensitivity of a factor of up to roughly 9.3 . in addition , the scheduled increase of recording bandwidth will increase the sensitivity by a factor of @xmath66 . to account for the variation in system temperature for typical observing conditions and receiver performance we introduce a degradation in snr proportional to the observing wavelength squared , normalized to the 1.3 mm performance in fish et al . the total signal to noise ratio of this measurement will then be @xmath67 with the width of the distribution given by the expression @xmath68 in figure [ contours ] , we plot confidence contours of the joint probability distribution for the combination of future thermal noise - limited vlbi and current astrometric observations at three different wavelengths . the solid line marks the 95% confidence contour determined by the stellar ephemerides ( gillessen et al . a vlbi measurement at a wavelength of @xmath69 significantly improves the result from stellar orbits alone . at smaller wavelengths , the constraints on the mass and distance of sgr a * are similar . in the rightmost panel of figure 3 , we have extrapolated the distribution width @xmath59 given by expression ( [ distwidth ] ) to a nominal wavelength of @xmath70 . measurements at such short wavelengths will be limited by weather conditions and may have to rely on a smaller array with fewer telescopes . real observations will face more stringent limitations than those imposed by the interferometer thermal noise due to the complications of astrophysics and measurement systematics . the chief astrophysical limitation is the separation of the ring emission from the source structure in the uv - plane . in our estimate , we have used the location of the nulls as a benchmark for the uncertainties we expect from the vlbi measurement . in practice , however , the full visibility function has to be analyzed with a pattern matching technique that identifies the structure of the ring . such a technique has to extract the ring from a uv - plane that is only partially sampled by a given set of baselines . the physics of the accretion flow will also complicate things , as the structure of sgr a * may vary over the course of an observation . however , because the ring structure is persistent and only weakly altered by rapid changes in the accretion flow we expect that temporal averaging of the visibilities across multiple observing epochs will diminish the importance of such changes . the vlbi measurement itself must surmount systematic limitations to make the moderate dynamic range measurements proposed here . chief among these is the difficulty of calibrating the noise level at individual stations , which imposed a 5% uncertainty in fish et al . ( 2011 ) with the three - station array . in observations with the larger array considered here , there will be many more internal cross - checks available to improve the relative calibration of stations ( the absolute calibration is not important ) . in particular , the use of three phased interferometers ( mauna kea , carma , alma ) that simultaneously record conventional interferometric data will permit scan - by - scan cross calibration of the amplitude scale of the array . furthermore , the larger arrays will be able to make use of closure phases and closure amplitudes that are immune to calibration errors as part of the ring detection , although we have ignored such procedures here because of the difficulty of simply parameterizing the improvement they can permit . other effects , such as the coherence of the reference systems between stations ( reported as @xmath71% in fish et al . 2011 ) can be more carefully measured and corrected to prevent them from imposing fundamental limitations to the ring detection . besides sgr a * , there exist other nearby supermassive black holes , for which a combination of dynamical measurements and vlbi observations could be feasible . since these supermassive black holes are located in host galaxies other than the milky way , observations are much less affected by interstellar scattering . as an example , broderick & loeb ( 2009 ) and takahashi & mineshige ( 2011 ) analyzed the prospects of imaging the shadow of m87 with vlbi observations at several different wavelengths . in figure [ smbhs ] , we plot the angular diameter of the photon rings against the distances of a collection of nearby supermassive black holes . sgr a * is closest to us and has the largest angular diameter , closely followed by m87 and m31 due to their large black hole masses . the top dashed line indicates the resolution of a telescope array with a baseline equal to the diameter of the earth ( from equation ( [ resolution ] ) ) at a wavelength of 1 mm . for comparison , we also show the resolution of a future space telescope located at 30% the distance to the moon ( comparable to the orbit of _ chandra _ ) at a wavelength of 0.5 mm corresponding to an angular diameter of about @xmath72 as well as of the proposed _ millimetron _ mission ( wild et al . 2009 ) at a distance of @xmath73 and a wavelength of @xmath74 . lcccccccc sgr a * & 53 & 0.008 & @xmath75 & 32.48 & 2.4 & 1 & ... & @xmath76 + ngc 4486 ( m87 ) & 22 & 17.0 & @xmath77 & 39.83 & 0.897 & 2 & -5.15 & @xmath76 + ngc 0224 ( m31 ) & 19 & 0.8 & @xmath78 & 32.14 & @xmath79 & 3 & -8.90 & + ngc 4649 ( m60 ) & 13 & 16.5 & @xmath80 & 37.45 & @xmath81 & 4 & -7.84 & @xmath82 + ngc 3115 & 9.6 & 10.2 & @xmath83 & ... & ... & & -7.03 & + ic 1459 & 9.2 & 30.9 & @xmath84 & 39.76 & 0.264 & 5 & ... & + ngc 4374 ( m84 ) & 9.1 & 17.0 & @xmath85 & 38.77 & 0.129 & 6 & -6.29 & + ngc 5128 ( cen a ) & 7.0 & 4.4 & @xmath86 & 39.85 & 6.9 & 7 & ... & + ngc 4594 ( m104 ) & 5.7 & 10.3 & @xmath87 & 37.89 & 0.25 & 8 & -4.68 & + ic 4296 & 2.5 & 54.4 & @xmath88 & 38.59 & 0.155 & 9 & ... & + ngc 1399 & 2.5 & 21.1 & @xmath89 & ... & @xmath90 & 3 & ... & @xmath82 + ngc 4342 & 2.1 & 18.0 & @xmath91 & ... & ... & & ... & + ngc 3031 ( m81 ) & 2.0 & 4.1 & @xmath92 & 36.97 & 0.1812 & 10 & -5.29 & @xmath93 + ngc 4261 & 1.7 & 33.4 & @xmath94 & 39.32 & 0.059 & 11 & -5.21 & + ngc 3585 & 1.6 & 21.2 & @xmath95 & ... & ... & & ... & + ngc 3998 & 1.6 & 14.9 & @xmath96 & 38.03 & @xmath97 & 8 & -4.43 + ngc 4697 & 1.6 & 12.4 & @xmath98 & ... & @xmath97 & 8 & ... & + ngc 4026 & 1.4 & 15.6 & @xmath99 & ... & ... & & ... & + ngc 3379 ( m105 ) & 1.1 & 11.7 & @xmath100 & 35.81 & ... & & -7.57 & + ngc 3245 & 1.0 & 22.1 & @xmath101 & 36.98 & ... & & -5.83 & + ngc 5845 & 1.0 & 28.7 & @xmath102 & ... & ... & & ... & + ngc 3377 & 1.0 & 11.7 & @xmath103 & ... & ... & & -6.16 & + in order to be able to resolve the photon ring with vlbi , the key question is whether the accretion flow of the target supermassive black hole is optically thin . in some cases , the spectra of these sources peak at wavelengths near @xmath104 , similar to the spectrum of sgr a * , suggesting that the emission comes from an adaf ( di matteo et al . 2000 ; doi et al . 2005 ) . naively , we would expect an approximately linear scaling of the electron density of an adaf with the ratio @xmath105 of the black hole , where @xmath106 is its mass accretion rate . the details of such a relation depend on a variety of factors , such as the temperature profile , the emissivity , and the radiative efficiency . however , most of the nearby supermassive black holes have very low radiative efficiencies ( ho 2009 ) . therefore , it is plausible that the accretion flows of these nearby supermassive black holes become optically thin at wavelengths that are comparable to 1 mm making them accessible to vlbi observations . such observations are best carried out at wavelengths near the flux peak , where the accretion flow is becoming optically thin . at wavelengths @xmath107 , the emitted flux is likely to be too low to be detected with a vlbi array , while at wavelengths @xmath108 , the accretion flow is optically thick . in table 1 , we summarize the angular diameters , distances , masses , radio luminosities @xmath109 , flux densities @xmath110 near 1 mm , ratios of the bolometric luminosity to the eddington luminosity @xmath111 , and peak wavelengths @xmath112 for supermassive black holes , whose photon rings have an angular diameter of at least @xmath72 . in addition to sgr a * , the black holes in the centers of m87 , m31 , and m60 are good potential targets for vlbi observations , because of the large angular diameters of their respective photon rings and , in the case of m87 and m60 , the measured peak in the synchrotron part of their spectra near 1 mm . m87 has a high measured flux density at 86 ghz ( lee et al . 2008 ) and should be readily observable at wavelengths close to 1 mm . in the case of m60 , however , di matteo et al . ( 1999 ) report an upper limit on the flux density of @xmath113 . no similar flux density measurement of m31 has been reported to date . we estimate the flux density at 1 mm of m31 from a simple power law fit of from the nasa extragalactic database . this flux density is relatively small , and m31 as well as m60 may be too faint to be observable with vlbi . other sources , such as centaurus a , are luminous enough to be detectable at wavelengths near 1 mm . with increasing vlbi resolution , even their photon rings may become observable . in the following , we assess the improvement on the mass measurements of the two supermassive black holes ( sgr a * and m87 ) whose photon rings have the largest angular diameters . we assume fixed distances of @xmath15 ( gillessen et al . 2009 ) and @xmath114 ( gltekin et al . 2009 ) for sgr a * and m87 , respectively . for sgr a * , we estimate an error of the combined mass measurement from the existing stellar ephemerides and our simulated vlbi data of only @xmath115 ( see figure [ contours ] ) . for m87 , we estimate the smallest relative error that thermal noise - limited vlbi imaging observations of the ring can achieve from the signal to noise ratio for observations of sgr a * given by equation ( [ snr ] ) , which we scale with the angular diameter of the black hole to obtain the expression @xmath116 note , however , that m87 has a much larger mass and that the dynamical timescales of its accretion flow is much longer . therefore , vlbi imaging observations of its photon ring will be much less affected by the variability of the accretion flow as in the case of sgr a*. in table 2 , we compare the relative errors of the mass measurements of sgr a * and m87 ( gillessen et al . 2009 ; gltekin et al . 2009 ) with our estimate of the error of vlbi observations of the respective photon rings at several different wavelengths . in the case of sgr a * , imaging its photon ring improves the error by a factor of about two . in the case of m87 , imaging the photon ring at a wavelength of @xmath117 would lead to a result that is similar to the current mass measurement . lcccc + source & @xmath118 & @xmath119 & @xmath119 & @xmath119 + & & ( 1.0 mm ) & ( 0.8 mm ) & ( 0.5 mm ) + sgr a * & 0.09 & 0.05 & 0.05 & 0.05 + m87 & 0.29 & 1.75 & 0.42 & 0.30 + as we pointed out in section 2 for the case of sgr a * , these errors require further refinement by in - depth imaging simulations . in addition , the morphology of vlbi emission can be complicated by the presence of a jet ( for m87 , see broderick & loeb 2009 ; dexter et al . 2011 ) . as in the case of sgr a * , the combination of the results from both the dynamical and vlbi imaging observations of m87 would further reduce the error in the masses . the relative errors of mass measurements of both techniques likewise depend on the error in the measured distances to these sources . these errors , in turn , depend on uncertainties in the hubble constant , peculiar motions of the gas in host galaxies , as well as assumptions on the proper motion of the milky way ( see , e.g. , hodge 1981 ; jacoby et al . the details of these effects on the relative errors of the mass are beyond the scope of our analysis . in this paper , we investigated the prospects of measuring the mass and distance of sgr a * as well as the masses of several other nearby supermassive black holes with a combination of dynamical observations and vlbi imaging of the respective photon rings of these sources . in order to resolve the photon ring of a black hole , its accretion flow must be optically thin . we argued that the wavelengths at which the accretion flows of these sources become optically thin should be roughly comparable to the location of the peak in the synchrotron emission of sgr a * and identified several supermassive black holes as optimal targets . we explored the prospects of imaging the photon ring of sgr a * as well as of other nearby supermassive black holes with near - future vlbi arrays . we estimated the signal to noise ratio with which such arrays can image the photon ring in the best - case scenario if the vlbi observations are limited by thermal noise . based on our estimate , we simulated confidence contours of a mass measurement of sgr a * using existing data of stellar ephemerides . we showed that the combination of both techniques can indeed reduce the correlation between mass and distance significantly resulting in relative errors of the mass and distance of only a few percent . we also identified several sources of uncertainty that have to be taken into account for an actual detection of the photon ring of sgr a*. the uncertainties of measurements based on stellar orbits will be further reduced by the continued monitoring and by the expected improvement in astrometry possible with the second generation instrument _ gravity _ for the _ very large telescope interferometer _ ( eisenhauer et al . further improvements of the vlbi sensitivity will be achieved by the _ event horizon telescope _ , a planned global array of telescopes ( doeleman et al . 2009a , b ; fish et al . 2009 ) . we estimated the improvement of the mass measurement of m87 using vlbi techniques . such observations are promising at wavelengths near @xmath117 because of the large size of its photon ring . for m31 and m60 , the supermassive black holes with the largest photon rings besides sgr a * and m87 , the flux densities may be too low to be detectable with vlbi . as the resolution of vlbi arrays increases , additional sources will become observable . angular resolution of @xmath120arcsec requires longer baselines and/or shorter observing wavelengths . however , the atmosphere precludes regular vlbi observations at wavelengths shorter than @xmath121 at even the best sites . a measurement of the photon - ring diameter is likely to yield useful results only if the observations extend well beyond the first null in the ring s visibility function . even at a wavelength of @xmath122 , space - vlbi observations will be required to reach this point for all except the first four entries in table 1 . vlbi between earthbound antennas and a satellite has been achieved at 6 cm wavelength using the japanese _ halca _ satellite ( hirabayashi et al . 1998 ) . the recently launched _ radioastron _ ( kardashev 2009 ) will extend such observations to 1.2 cm wavelength and baselines as large as @xmath123 , for a resolution of @xmath124 @xmath23arcsec . a future explorer - class space mission designed to observe at 1 mm wavelength or shorter , where source opacity and scattering effects will be far less important , could provide the angular resolution needed to study a far larger sample of sources . such a capability may be provided by the russian - european _ millimetron _ mission , which plans to deploy a 12 m antenna with vlbi capabilities to 0.4 mm and maximum baseline @xmath125 km ( wild et al . 2009 ) . we thank d. zaritzky for useful comments . tj and dp were supported by the nsf career award nsf 0746549 . this work was supported at mit haystack observatory by nsf grants ast@xmath126 , ast@xmath127 , and ast@xmath128 . agol , e. , & krolik , j. h. 2000 , apj 528 , 161 bao , g. , hadrava , p. , & stgaard , e. 1994 , apj , 435 , 55 beckwith , k. , & done , c. 2005 , mnras 359 , 1217 bendo , g. j. , et al . 2006 , apj , 645 , 134 bower , g. c. , goss , w. m. , falcke , h. , backer , d. c. , & lithwick , y. 2006 , apj , 648 , l127 broderick , a. e. , johannsen , t. , loeb , a. , & psaltis , d. , 2012 , in preparation broderick , a. e. , fish , v. l. , doeleman , s. s. , & loeb , a. 2011 , , 735 , 110 broderick , a. e. , fish , v. l. , doeleman , s. s. , & loeb , a. 2009 , , 697 , 45 broderick , a. e. , & loeb , a. 2009 , , 697 , 1164 @xmath5 , a. , fanton , c. , & calvani , m. 1998 , new astronomy , 3 , 647 cunningham , c. 1976 , apj , 208 , 534 dexter , j. , agol , e. , & fragile , p. c. 2009 , apj , 703 , l142 dexter , j. , mckinney , j. c. , & agol , e. 2011 , arxiv:1109.6011 di matteo , t. , fabian , a. c. , carilli , l. c. , & ivison , j. r. 1999 , adv . space res . , 23 , 1075 di matteo , t. , quataert , e. , allen , s. w. , narayan , r. , & fabian , a. c. 2000 , mnras , 311 , 507 doeleman , s. s. , fish , v. l. , broderick , a. e. , loeb , a. , & rogers , a. e. e. 2009a , apj , 695 , 59 doeleman , s. s. , et al . 2009b , astro2010 : the astronomy and astrophysics decadal survey , science white papers , no . 68 doeleman , s. s. et al . 2008 , nature , 455 , 78 doi , a. , kameno , s. , kohno , k. , nakanishi , k. , & inoue , m. 2005 , mnras , 363 , 692 dolence , j. c. , gammie , c. f. , mo@xmath129cibrodzka , m. , & leung , p. k. 2009 , apjs , 184 , 387 eisenhauer , f. , et al . 2011 , msngr . , 143 , 16 falcke , h. , melia , f. , & agol , e. 2000 , apj 528 , l13 falcke , h. , mannheim , k. , & biermann , p. l. 1993 , a&a , 278 , l1 fish , v. l. , et al . 2011 , apj , 727 , 36 fish , v. l. , broderick , a. e. , doeleman , s. s. , & loeb , a. 2009 , apj , 692 , l14 ghez , a. m. et al . 2008 , apj , 689 , 1044 gillessen , s. , eisenhauer , f. , trippe , s. , alexander , t. , genzel , r. , martins , f. , & ott , t. 2009 , apj , 692 , 1075 gltekin , k. , cackett , e. m. , miller , j. m. , di matteo , t. , markoff , s. , & richstone , d. o. 2009 , apj , 706 , 404 hirabayashi , h. , et al . 1998 , science , 281 , 1825 ho , l. c. 2009 , apj , 699 , 626 hodge , p. w. 1981 , a&aa , 19 , 357 israel , f. p. , raban , d. , booth , r. s. , & rantakyr , f. t. 2008 , a&a , 483 , 741 jacobi , g. h. , et al . 1992 , pasp , 104 , 599 johannsen , t. , & psaltis , d. 2010 , apj , 718 , 446 kardashev , n. s. 2009 , phys . u. , 52 , 1127 laor , a. , netzer , h. , & piran , t. 1990 , mnras , 242 , 560 lee , s .- s . , et al . 2008 , apj , 136 , 159 leeuw , l. l. , sansom , a. e. , robson , e. i. , haas , m. , & kuno , n. 2004 , apj , 612 , 837 mahadevan , r. 1998 , , 394 , 651 middelberg , e. , roy , a. l. , walker , r. c. , & falcke , h. 2005 , a&a , 433 , 897 mo@xmath6cibrodzka , m. , gammie , c. f. , dolence , j. c. , shiokawa , h. , & leung , p. k. 2009 , apj , 706 , 497 narayan , r. , & yi , i. 1995 , apj , 444 , 231 narayan , r. , yi , i. , & mahadevan , r. 1995 , nature , 374 , 623 narayan , r. , & yi , i. 1994 , , 428 , l13 zel , f. , psaltis , d. , & narayan , r. 2000 , apj , 541 , 234 pellegrini , s. , venturi , t. , comastri , a. , fabbiano , g. , fiore , f. , vignali , c. , morganti , r. , & trinchieri , g. 2003 , apj , 585 , 677 sadler , e. m. , et al . 2008 , mnras , 385 , 1656 schdel , r. , krips , m. , markoff , s. , neri , r. , & eckart , a. 2007 , a&a , 463 , 551 shcherbakov , r. v. , & penna , r. f. 2011 , asp conf . ser . , 439 , 372 takahashi , r. , & mineshige , s. 2011 , apj , 729 , 86 viergutz , s. u. 1993 , astron . , 272 , 355 wild , w. , et al . 2009 , ex . a. , 23 , 221 yuan , f. , quataert , e. , & narayan , r. 2003 , , 598 , 301
|
dynamical mass measurements to date have allowed determinations of the mass @xmath0 and the distance @xmath1 of a number of nearby supermassive black holes . in the case of sgr
a * , these measurements are limited by a strong correlation between the mass and distance scaling roughly as @xmath2 .
future very - long baseline interferometric ( vlbi ) observations will image a bright and narrow ring surrounding the shadow of a supermassive black hole , if its accretion flow is optically thin . in this paper
, we explore the prospects of reducing the correlation between mass and distance with the combination of dynamical measurements and vlbi imaging of the ring of sgr a*. we estimate the signal to noise ratio of near - future vlbi arrays that consist of five to six stations , and we simulate measurements of the mass and distance of sgr a * using the expected size of the ring image and existing stellar ephemerides .
we demonstrate that , in this best - case scenario , vlbi observations at 1 mm can improve the error on the mass by a factor of about two compared to the results from the monitoring of stellar orbits alone .
we identify the additional sources of uncertainty that such imaging observations have to take into account .
in addition , we calculate the angular diameters of the bright rings of other nearby supermassive black holes and identify the optimal targets besides sgr a * that could be imaged by a ground - based vlbi array or future space - vlbi missions allowing for refined mass measurements .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
sensitivity of diffractive lepto and photoproduction to the gluon density in the proton gives an excellent tool to test these structure functions . intensive experimental study of diffractive processes were performed in desy @xcite . theoretical analysis shows that the cross sections of diffractive hadron production are expressed in terms of skewed parton distributions ( spd ) @xcite . the diffractive charm @xmath0 production and @xmath2 production are determined by the gluon spd @xmath3 because the charm component in the proton is small . for light hadron production effects of the quark spd should be important for not small @xmath4 . in the polarized case , the spin - dependent gluon distributions can be investigated . in future , there will be an excellent possibility of studying spin effects with transversely polarized target at hermes and compass . in this report , we consider double spin asymmetries for longitudinally polarized leptons and transversely polarized protons in diffractive vector @xmath0 production at high energies ( see @xcite ) which is expressed in terms of the polarized cross sections @xmath5 at small @xmath4 the diffractive contribution to the asymmetry @xmath1 determined by the pomeron exchange should be important . in the qcd- based models the pomeron is usually regarded as a two - gluon state . within the two - gluon exchange model the two - gluon coupling with the proton can be written as follows : @xmath6 here the structure proportional to @xmath7 determines the spin - non - flip contribution . the term @xmath8 leads to the transverse spin - flip at the vertex . it has been found within the model approaches @xcite that the ratio @xmath9 and has a weak energy dependence ( weak @xmath4 dependence ) . the weak energy dependence of spin asymmetries in exclusive reactions is not in contradiction with the experiment @xcite . the diffractive @xmath0 production in the lepton - proton reaction is determined by the photon - two - gluon fusion . the spin - average and spin - dependent cross section can be written in the form @xmath10 here @xmath11 is a normalization coefficient , the functions @xmath12 are determined by a sum of all graphs integrated over the gluon momenta @xmath13 and @xmath14 . we calculate here the imaginary parts of the photon - two - gluon fusion amplitudes which are expressed directly in terms of the functions @xmath15 and @xmath16 from ( [ ver ] ) . the function @xmath17 has the form @xmath18 here @xmath19 and @xmath20 with @xmath21 . the gluon structures @xmath22(@xmath23 ) are connected with the @xmath24(@xmath25 ) spd ( see ( [ bqq],[kqq ] ) ) . thus , the functions @xmath15 and @xmath16 are the nonintegrated gluon distributions . the hard part @xmath26 in ( [ np ] ) can be calculated perturbatively when @xmath27 is not small , about @xmath28 or larger . the spin - dependent contribution @xmath29 has two terms proportional to the scalar products ( @xmath30 ) and ( @xmath31 ) @xcite . we consider here only the term proportional to @xmath32 which can be written as follows : @xmath33.\ ] ] the other term @xmath34 has been discussed in @xcite . the asymmetry is determined by the ratio @xmath35 . at small @xmath4 the gluon structure functions have large imaginary part . in this case the asymmetry can be approximated as @xmath36 in numerical calculations we use a simple parameterization of the spd as a product of the form factor and the ordinary gluon distribution @xcite . in our estimations we use the value @xmath37 . we analyze the case when the @xmath38 asymmetry has a maximal value ( the momentum @xmath39 is parallel to the target polarization @xmath40 ) . the predicted @xmath1 asymmetry in diffractive light @xmath0 production at @xmath41 is shown in fig . this asymmetry is not small for @xmath42 . the @xmath38 asymmetry has a strong mass dependence . for heavy quark production this asymmetry becomes negative , fig . it is interesting to have predictions for a light quark production at the slac and hermes low energy range @xmath43 . here the `` window '' for perturbative calculation is quite small . really , it can be found in this case that the maximum value of the transverse momentum is limited by @xmath44 because we have the restriction @xmath45 from ( [ sigma ] ) . in target experiments , it is usually difficult to detect the final hadron and determine the momentum transfer . in this case , we estimate the asymmetry integrated over momentum transfer @xmath46 @xmath47 ; @xmath48 . the predicted integrated asymmetry is about 3% . note that we have calculated here only the gluon contribution to the asymmetry . at hermes energies the contribution of the quark spd to the @xmath1 asymmetry should be important . to conclude , we would like to emphasize that the diffraction contribution to the @xmath1 asymmetry is found to be proportional to the ratio of @xmath49 structure functions . the predicted coefficient @xmath50 in ( [ cltqq ] ) is not small , about 0.3 - 0.5 . this shows the possibility of studying the transverse distribution @xmath51 in future experiments with a transversely polarized target ( hermes , compass and future erhic facilities ) . these results could be applicable to reactions with heavy quarks . for processes with light hadron production , our predictions can be used in the small @xmath4 region ( @xmath52 e.g. ) where the contribution of the quark spd is expected to be small . the recoil particle detector is needed to distinguish the diffractive events . really , in the case when the recoil detector is absent , the diffractive events are detected together with nondiffractive ones . the measured asymmetry in this case looks like @xmath53 here @xmath54 and @xmath55 . the ratio @xmath56 integrated over @xmath4 has been found at hera to be about 0.200.30 @xcite . in this case , the diffractive contribution to asymmetry will be smaller by the factor 3 - 5 . 99 h1 collaboration , s. aid et al . b472 * , 3 ( 1996 ) ; + zeus collab . , j. breitweg et al . , z. phys , * c75 * , 215 ( 1997 ) . zeus collab . , j. breitweg et al . , eur . j. * c5 * , 41 ( 1998 ) ; + h1 collab . , c. adloff et al . , eur . j. * c6 * 421 ( 1999 ) . radyushkin , phys.rev , d * 56 * , 5524 ( 1997 ) . x. ji , phys.rev . d * 55 * 7114 ( 1997 ) . goloskokov , hep - ph/0112268 to appear in euro . goloskokov , s.p . kuleshov , o.v . selyugin , z. phys . * c50 * , 455 ( 1991 ) . goloskokov , p. kroll , phys . d * 60 * , 014019 ( 1999 ) . h1 collaboration , aid s. et al , _ z. phys . _ * c69 * , 27 ( 1995 ) .
|
we consider double spin asymmetries for longitudinally polarized leptons and transversely polarized protons in diffractive @xmath0 production which is connected with @xmath1 asymmetry .
the predicted asymmetry is large and can be used to obtain the information on the polarized skewed gluon distributions in the proton .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the mechanism of heavy meson decays is one of the most interesting and challenging fields in particle physics , it involves both strong and weak interactions . nowadays strong interaction in the non - perturbative region is still an unsolved problem . compared with hadronic decays , leptonic decay is simpler . strong interaction only occurs within the initial particle . pure - leptonic decay of heavy meson can be used to determine the decay constant , which describes the possibility - amplitude for the quark - untiquark emerging at the same point . the pure - leptonic decay is helicity suppressed . the decay branching ratio of a pseudoscalar meson @xmath3 with quark content @xmath4 within the standard model is @xmath5 where @xmath6 is the fermi coupling constant , @xmath7 the cabibbo - kobayashi - maskawa ( ckm ) matrix element , @xmath8 the life time of the meson @xmath3 , and @xmath9 and @xmath10 the masses of the meson @xmath3 and lepton @xmath11 , respectively . the decay rate is proportional to the lepton mass squared @xmath12 is the consequence of the helicity suppression . however , the presence of one photon in the final state can compensate the helicity suppression . as a result , the radiative leptonic decay can be as large as , or even larger than the pure - leptonic decay mode . it thus opens a window for detecting the dynamics of strong interaction in the heavy meson or studying the effect of strong interaction in the decay . the radiative leptonic decay rates of the charged @xmath13 and @xmath14 mesons have been studied with various methods in the literature . in ref . @xcite , @xmath13 and @xmath15 are calculated in a non - relativistic quark model , the branching ratios of the order of @xmath16 for @xmath17 and @xmath18 for @xmath19 are found . in ref . @xcite with perturbative qcd approach , it is found that the branching ratio of @xmath20 is of the order of @xmath21 and @xmath22 of the order of @xmath16 , while the branching ratio of @xmath23 is at the order of @xmath18 . on the other hand , a smaller branching ratio is obtained for @xmath24 within the light front quark model @xcite . smaller result for @xmath25 is also obtained in ref . @xcite within the non - relativistic constituent quark model , which gives that the branching ratio of @xmath26 is of the order of @xmath18 and @xmath27 of the order of @xmath28 . the problem of factorization in qcd for @xmath29 is studied in ref . @xcite . in this work , we study the radiative leptonic decays of the charged @xmath13 , @xmath14 and @xmath30 mesons to @xmath31 including both the short and long - distance contributions . the short - distance contribution is considered at tree level . the wave function of the heavy meson used here is obtained in the relativistic potential model previously @xcite . the long - distance contribution is estimated by using the idea of the vector meson dominance ( vmd ) @xcite followed by the transition of the vector meson to a photon . we find that the long - distance contribution can enhance the decay rates seriously . the remaining part of this paper is organized as follows . in sec.ii , we present the short - distance amplitude . in sec.iii , the long - distance contribution is considered . the numerical results and discussion are given in sec.iv . sec.v is a brief summary . we use @xmath3 to denote the pseudoscalar meson which is composed of a heavy anti - quark @xmath32 and a light quark @xmath33 , such as @xmath13 and @xmath14 mesons . there are four feynman diagrams contributing to the radiative decays @xmath34 at tree level , which are shown in fig . [ fig : feynsd ] . however the contribution of fig . [ fig : feynsd ] ( d ) is suppressed by a factor of @xmath35 , it can be neglected for simplicity . the effective hamiltonians corresponding to the other three diagrams in fig . [ fig : feynsd ] can be written as : where @xmath37 is defined as @xmath38 , and @xmath7 represents for the ckm matrix elements . @xmath39 and @xmath40 are the electric charges of the quarks @xmath41 and @xmath33 , respectively . @xmath42 is the electro - magnetic field . @xmath43 , @xmath44 and @xmath45 can be divided into two terms for convenience , according to the numerator of the fermion propagator . for example , @xmath43 can be written as @xmath46 where @xmath47 with @xmath48 the amplitude of the radiative leptonic decay can be obtained by inserting the operator of the effective hamiltonian between the initial and final particle states . for example , the contribution of fig . [ fig : feynsd ] ( a ) is @xmath49 the matrix elements @xmath50 and @xmath51 only depend on the momenta @xmath52 and @xmath53 . according to their lorentz structure , they can be decomposed as a linear combination of two terms of @xmath52 and @xmath53 @xmath54 \label{eq : six}. \end{split}\ ] ] the coefficients @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 , @xmath61 and @xmath62 are all dimensionless constants . the terms of @xmath56 , @xmath59 and @xmath60 do not contribute to the decay amplitude @xmath63 when substituting the above decomposition into eq.[eq : five ] . therefore these terms can be dropped . the coefficients can be obtained by the treatment in the following . multiplying @xmath64 with @xmath65 , we can obtain @xmath55 as @xmath66 similarly , multiplying @xmath67 with @xmath68 , we have @xmath69 multiplying @xmath67 with @xmath70 and @xmath71 , and using @xmath72 , one can get @xmath73\frac{\left(1-{\gamma } _ 5\right)}{p_q \cdot p_{\gamma}}q\mid p > \label{eq : nine}. \end{split}\ ] ] finally @xmath74 can obtained by multiplying @xmath67 with @xmath75 : @xmath76 the amplitude @xmath77 can be treated in the same way with some coefficients defined as follows @xmath78 \label{eq:11}. \end{split}\ ] ] using the matrix element @xmath79 , the amplitude @xmath80 can be treated simlarly . finally , the total amplitude can be expressed as @xmath81\left[\left(p_p{\cdot}p_{\gamma}\right)\varepsilon _ { \gamma}^{{\mu}*}-\left(p_p{\cdot}\varepsilon\right)p_{\gamma}^{\mu}\right ] \right.\\ & \left.-2\left(q_q\frac{p_p{\cdot}\varepsilon}{p_p{\cdot}p_{\gamma}}\right)a_{a2}m_p p_p^{\mu}\right\},\\ & \mathcal{a}_b = \frac{- i e g_f v_{qq}}{2\sqrt{2}}\left(u_l p_{\text { l$\mu $ } } v_{\bar{\nu}}\right)\left\ { -\frac{q_qm_pa_{b1}}{p_p{\cdot}p_{\gamma}}i \epsilon ^{\alpha \beta \mu \sigma}p_{\gamma \beta } \varepsilon ^*_{\gamma \alpha}p_{p\sigma}\right.\\ & \left.+\left[\frac{q_q m_d a_{b1}}{p_p\cdot p_{\gamma}}+\frac{2 q_q e_{b2 } m_p^3}{\left(p_p\cdot p_{\gamma}\right)^2}\right]\left[\left(p_p{\cdot}p_{\gamma}\right)\varepsilon _ { \gamma}^{{\mu}*}-\left(p_p{\cdot}\varepsilon\right)p_{\gamma}^{\mu}\right ] \right.\\ & \left.+2\left(q_q\frac{p_p{\cdot}\varepsilon}{p_p{\cdot}p_{\gamma}}\right)a_{a2}m_p p_p^{\mu}\right\},\\ & \mathcal{a}_c = \frac{- i e g_f v_{qq}}{2\sqrt{2}}\left(u_l p_{\text { l$\mu $ } } v_{\bar{\nu}}\right)\left\ { \frac{m_pa_{a2}}{p_l{\cdot}p_{\gamma}}i \epsilon ^{\alpha \beta \mu \sigma}p_{\gamma \beta } \varepsilon^ * _ { \gamma \alpha}p_{p\sigma}\right.\\ & \left.+\frac{m_p a_{a2}}{p_l\cdot p_{\gamma}}\left[\left(p_p{\cdot}p_{\gamma}\right)\varepsilon _ { \gamma}^{{\mu}*}-\left(p_p{\cdot}\varepsilon\right)p_{\gamma}^{\mu}\right ] \right.\\ & \left . + \frac{p_l{\cdot}\varepsilon}{p_l{\cdot}p_{\gamma}}a_{a2}m_p p_p^{\mu}\right\ } \label{eq : add1}. \end{split}\ ] ] the above equations show that the contribution of each diagram in fig . [ fig : feynsd ] is not gauge invariant separately , but the sum of them is indeed gauge invariant , which is given in the following @xmath82 \right.\\ & \left . + 2\left[\left(q_q - q_q\right)\frac{p_p{\cdot}\varepsilon}{p_p{\cdot}p_{\gamma}}+\frac{p_l{\cdot}\varepsilon}{p_l{\cdot}p_{\gamma}}\right]a_{a2}m_d p_p^{\mu}\right\}\\ & \times \left(u_l p_{\text { l$\mu $ } } v_{\bar{\nu}}\right ) . \label{eq:12 } \end{split}\ ] ] this equation clearly shows that the sum of the contributions of all the diagrams in fig . [ fig : feynsd ] is gauge invariant . in eq.([eq:12 ] ) the factors @xmath83 and @xmath42 are @xmath84 next we shall calculate the coefficients @xmath55 , @xmath57 , @xmath61 and @xmath74 . the pseudoscalar meson state can be written in terms of the quark - antiquark creation and annihilation operators @xmath85\mid 0 > , \label{eq:14 } \end{split}\ ] ] where @xmath86 is the color index , the factor @xmath87 the normalization factor for color indices , and @xmath88 the 3-momentum of the quarks in the rest frame of the heavy meson . the wave function @xmath89 has been calculated in the relativistic potential model previously , the numerical solution of the wave function can be fitted in the exponential form @xcite @xmath90 the numerical solutions of the parameter @xmath91 for @xmath14 , @xmath30 and @xmath13 mesons are quoted from ref . @xcite recently @xmath92 with the meson state given in eq . ( [ eq:14 ] ) , the matrix element @xmath93 can be calculated straightforwardly , the result is @xmath94\sqrt{\frac{m_q m_q}{p_q^0 p_q^0 } } , \label{eq:16 } \end{split}\ ] ] where @xmath95 is the dirac spinner for the pseudoscalar meson , it can be obtained in the dirac - representation as @xmath96 in this section , we estimate the contributions of long - distance physics . according to the spirt of vector meson dominance model @xcite , we consider the resonance process @xmath97 , where the intermediate vector resonance @xmath83 can be @xmath98 , @xmath99 and @xmath100 . the contribution comes from the semileptonic intermediate @xmath101 , followed by the vector resonance turning to an on - shell photon @xmath102 , which is shown in fig . [ fig : ld ] . the amplitude of the long - distance contribution can be written as for the decays of @xmath14 and @xmath13 mesons , @xmath83 represents for @xmath98 and @xmath99 mesons , while for @xmath30 meson decay , @xmath83 represents for @xmath100 meson . @xmath104 is the relative phase between the long and short - distance contributions . the matrix element @xmath105 is used to define the decay constant of the vector meson @xmath106 where the factor @xmath107 s are @xmath108 the vector s decay constant @xmath109 can be derived from the decay rate of @xmath110 . after a short calculation , the decay constants can be related to the vector meson s leptonic decay widths @xmath111 where @xmath112 is the electromagnetic fine structure constant , @xmath113 , @xmath114 and @xmath115 are the charges of the quarks @xmath116 , @xmath117 and @xmath118 , respectively . the hadronic matrix element @xmath119 in eq . ( [ eq:19 ] ) can be decomposed according to its lorentz structure as @xcite @xmath120 where @xmath121 , and @xmath83 , @xmath122 , @xmath123 , @xmath124 and @xmath125 are form factors . to obtain the amplitude gauge invariant , we take the trick used in ref.@xcite in treating the long - distance contribution in the precess @xmath126 via the resonance @xmath127 . with the lorentz decomposition of the hadronic matrix element in eq.([eq:23 ] ) , the product of the two matrix elements in eq.([eq:19 ] ) can be calculated to be @xmath128\right.\\ & \left .- i\left(p_p{\cdot}\varepsilon _ { \gamma}\right)\left[\frac{\left(m_p+m_v\right)p_v^{\mu}}{p_p{\cdot}p_v}a_1\left(q^2\right)\right.\right.\\ & \left.\left.-\frac{\left(p_p+p_v\right)^{\mu}}{m_p+m_v}a_2\left(q^2\right)+\frac{2 m_v q^{\mu}}{q^2}\left(a_3\left(q^2\right)-a_0\left(q^2\right)\right)\right]\right\ } \label{eq:24}. \end{split}\ ] ] in the rest - frame of the heavy meson @xmath3 , the product of the four - momentum of the meson @xmath3 and the polarization vector of the photon satifies @xmath129 . then the last term in the above equation can be dropped . with @xmath130 , we obtaine @xmath131\right\ } , \label{eq:25 } \end{split}\ ] ] where : @xmath132 the numerical calculation is performed in the center - of - mass frame of the heavy meson , and the momentum of the photon is taken as @xmath133 . for the input parameters , the masses of the quarks are taken as @xmath134 which are taken to be consistent with that used to derive the wave function of the heavy meson in the relativistic potential model in ref . @xcite . the infrared divergence appears as the energy of the real photon goes to soft limit or the momentum of the photon is parallel to the momentum of a massless lepton . this divergence can be canceled when the decay width of the radiative leptonic decay is added with the pure leptonic decay width , in which one - loop radiative corrections are included @xcite . the radiative leptonic decay with the energy of the photon lower than the experimental resolution can not be distinguished from the pure leptonic decay . only photons with the energy larger than the experimental resolution can be distinguished . therefore the radiative leptonic decay width depends on the photon energy resolution . the photon energy resolution can be a few mev in experiment @xcite . the dependence of the decay width on the resolution @xmath136 is shown in table [ tab : irdiv ] and fig . [ fig : irdiv ] . for example , if taking @xmath137 , the decay width and branching ratio of @xmath138 are @xmath139 in the following all the numerical calculation is performed by taking the resolution @xmath137 . the radiative leptonic decays of @xmath13 , @xmath14 and @xmath141 are studied in this work . the short - distance contribution is calculated by using the wave functions of the heavy mesons obtained in the relativistic potential model , more details about the quark - momentum distribution are included in this work . in addition to the short - distance contribution , the long - distance contribution is also estimated in the vector meson dominance model . the study shows that the long - distance contributions can seriously affect the decay rates . the branching ratio of @xmath142 can be enhanced to the order of @xmath16 , which should only be at the order of @xmath18 if only considering the short - distance contribution . this work is supported in part by the national natural science foundation of china under contracts nos . 10575108 , 10975077 , 10735080 , 11125525 and by the fundamental research funds for the central universities no . 65030021 . d. atwood , g. eilam , a. soni , mod . a*11 * ( 1996 ) 1061 . korchemsky , d. pirjol , t .- yan , phys . rev . d*61 * ( 2000 ) 114510 . geng , c .- c . lih , w .- m . zhang , mod . phys .lett . a*15 * ( 2000 ) 2087 . song , phys . lett . b*562 * ( 2003 ) 75 . s. descotes - genon and c.t . sachrajda , nucl . phys . b*650 * ( 2003 ) 356 . . yang , eur . phys . j. c*72 * ( 2012 ) 1880 ( arxiv:1104.3819 ) . j. j. sakurai , ann . of phys . * 11 * ( 1960 ) 1 . y. gell - mann and f. zachariasen , phys . rev . * 124 * ( 1961 ) 953 . j. j. sakurai and d. schildknecht , phys . lett . b*40 * ( 1972 ) 121 . feynman , photon - hadron interaction ( benjamin , new york , 1972 ) . t. h. bauer et al . , rev . * 50 * ( 1978 ) 261 . p. colangelo , f. de fazio , m. ladisa , g. nardulli , p. santorelli , a. tricarico , eur . j. c*8 * , ( 1999 ) 81 . m. wirbel , b. stech , and m. bauer , z. phys . c - particles and fields 29 ( 1985 ) 637 . d. fakirov , b. stech , nucl.phys . b*133 * ( 1978 ) 315 . n. g. deshpande , x .- he , and j. trampetic , phys . lett . b*367 * ( 1996 ) 362 . k. nakamura et al . ( particle data group ) , j. phys . g*37 * ( 2010 ) 075021 . chang et al . rev . d*60 * ( 1999 ) 114013 . alice collaboration ( k. aamodt et al . ) , jinst ( 2008)3:s08002 . p. ball , phys . rev . d*48 * ( 1993 ) 3190 . p. ball , r. zwicky , phys . rev . d*71 * ( 2005 ) 014029 . du , j .- w . li , m .- z . yang , eur . j. c*37 * ( 2004 ) 173 .
|
in this work we study the radiative leptonic decays of @xmath0 , @xmath1 and @xmath2 , including both the short - distance and long - distance contributions .
the short - distance contribution is calculated by using the relativistic quark model , where the bound state wave function we used is that obtained in the relativistic potential model .
the long - distance contribution is estimated by using vector meson dominance model
. 0.25 cm
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
rovibrationally excited h@xmath0 molecules have been observed in many astrophysical objects ( for recent studies , see weintraub et al . 1998 ; van dishoeck et al . 1998 ; shupe et al . 1998 ; bujarrabal et al . 1998 ; stanke et al . 1998 ) . the rovibrational levels of the molecule may be populated by ultraviolet pumping , by x - ray pumping , by the formation mechanism , and by collisional excitation in shock - heated gas ( dalgarno 1995 ) . the excited level populations are then modified by collisions followed by quadrupole emissions . the main colliding partners apart from h@xmath0 are h and he . although he is only one tenth as abundant as h , collisions with he may have a significant influence in many astronomical environments depending on the density , temperature and the initial rotational and vibrational excitation of the molecule . collisions with he and h@xmath0 are particularly important when most of the hydrogen is in molecular form , as in dense molecular clouds . to interpret observations of the radiation emitted by the gas , the collision cross sections and corresponding rate coefficients characterizing the collisions must be known . emissions from excited rovibrational levels of the molecule provide important clues regarding the physical state of the gas , dissociation , excitation and formation properties of h@xmath0 . here we investigate the collisional relaxation of vibrationally excited h@xmath0 by he . rovibrational transitions in h@xmath0 induced by collisions with he atoms have been the subject of a large number of theoretical calculations in the past ( alexander 1976 , 1977 ; alexander and mcguire 1976 ; dove et al . 1980 ; eastes and secrest 1972 ; krauss and mies 1965 ; mcguire and kouri 1974 ; raczkowski et al . 1978 ) and continue to attract experimental ( audibert et al . 1976 ; michaut et al . 1998 ) and theoretical attention ( flower et al . 1998 ; dubernet & tuckey 1999 ; balakrishnan et al . 1999 ) . recent theoretical calculations are motivated by the availability of more accurate representations of the interaction potentials and the possibility of performing quantum mechanical calculations with few approximations . the potential energy surface determined by muchnick and russek ( 1994 ) was used by flower et al . ( 1998 ) and by balakrishnan et al . ( 1999 ) in recent quantum mechanical calculations of rovibrational transition rate coefficients for temperatures ranging from 100 to 5000k . flower et al . presented their results for vibrational levels @xmath3 and 2 of ortho- and para - h@xmath0 . balakrishnan et al . ( 1999 ) reported similar results for @xmath4 and 1 . though both authors have adopted similar close - coupling approaches for the scattering calculations , flower et al . used a harmonic oscillator approximation for h@xmath0 vibrational wave functions in evaluating the matrix elements of the potential while the calculations of balakrishnan et al . made use of the h@xmath0 potential of schwenke ( 1988 ) and the corresponding numerically determined wave functions . the results of the two calculations agreed well for pure rotational transitions but some discrepancies were seen for rovibrational transitions . we believe this may be due to the different choice of vibrational wave functions . the sensitivity of the rate coefficients to the choice of the h@xmath0 wave function was noted previously and differences could be significant for excited vibrational levels . we find this to be the case for transitions involving @xmath5 . thus , in this article , we report rate coefficients for transitions from @xmath6 to 6 initial states of h@xmath0 induced by collisions with he atoms using numerically exact quantum mechanical calculations . we also report results of quasiclassical trajectory ( qct ) calculations and examine the suitability of classical mechanical calculations in predicting rovibrational transitions in h@xmath0 . the quantum mechanical calculations were performed using the nonreactive scattering program molscat developed by hutson and green ( 1994 ) with the he - h@xmath0 interaction potential of muchnick and russek ( 1994 ) and the h@xmath0 potential of schwenke ( 1988 ) . we refer to our earlier paper ( balakrishnan , forrey & dalgarno , 1999 ) for details of the numerical implementation . different basis sets were used in the calculations for transitions from different initial vibrational levels . we use the notation [ @xmath7@xmath8(@xmath9@xmath10 ) to represent the basis set where the quantities within the square brackets give the range of vibrational levels and those in braces give the range of rotational levels coupled in each of the vibrational levels . for transitions from @xmath11 and 4 we used , respectively , the basis sets [ 03](011 ) & [ 4](03 ) , [ 03](011 ) & [ 4](09 ) and [ 35](011 ) & [ 1,6](011 ) . for @xmath12 and 6 of para h@xmath0 we used , respectively , [ 46](014 ) & [ 3,7](08 ) and [ 57](014 ) & [ 4,8](08 ) . during the calculations , we found that the @xmath13 transitions are weak with cross sections that are typically orders of magnitude smaller than for the @xmath14 transitions . thus , for @xmath12 and 6 of ortho - h@xmath0 , we have only included the @xmath14 vibrational levels with @xmath15=013 in the basis set to reduce the computational effort . the basis sets were chosen as a compromise between numerical efficiency and accuracy and could introduce some truncation errors for transitions to levels which lie at the outer edge of the basis set . our convergence tests show that truncation errors are small . rovibrational transition cross sections @xmath16 where the pairs of numbers @xmath17 and @xmath18 respectively denote the initial and final rovibrational quantum numbers , were computed for kinetic energies ranging from 10@xmath1 to 3 ev . sufficient total angular momentum partial waves were included in the calculations to secure convergence of the cross sections . the quasiclassical calculations were carried out using the standard classical trajectory method as described by lepp , buch and dalgarno ( 1995 ) in which the procedure of blais and truhlar ( 1976 ) was adopted for the final state analysis . because rovibrational transitions are rare at low velocities , useful results could be obtained only for collisions at energies above 0.1 ev . the results are averages over 10000 trajectories . the quantum mechanical calculations were performed using molscat ( hutson & green , 1994 ) suitably adapted for the present system with the potential represented by a legendre polynomial expansion in which we retained nonvanishing terms of orders 0 to 10 , inclusive . calculation of rovibrational transition rate coefficients over a wide range of temperatures requires the determination of scattering cross sections at the energies spanned by the boltzmann distribution at each temperature . this is a computationally demanding problem especially when quantum mechanical calculations are required . for many systems , qct calculations offer a good compromise between accuracy and computational effort . however , the validity of classical mechanics is in question especially for lighter systems and at lower temperatures where quantum mechanical effects such as tunneling are important . due to the small masses of the atoms involved , the present system offers an excellent opportunity to test the reliability of qct calculations in predicting rovibrational transitions . though such attempts have been made in the past ( dove et al . 1980 ) the classical mechanical and quantum mechanical calculations were done in different energy regimes and a one - to - one comparison was not possible . we carried out quantum mechanical and qct calculations of rovibrational transition cross sections for the present system over a wide range of energies . in figure 1 we compare the results for pure rotational de - excitation transitions ( @xmath19 ) with @xmath20 in @xmath4 to 5 . there are striking similarities and differences between the quantum mechanical and qct results . they agree quite well at higher energies and have a similar energy dependence with both calculations predicting the same maximum for the cross sections before they fall off . in contrast , the agreement becomes less satisfactory at lower energies with the qct cross sections rapidly decreasing to zero as the energy is decreased . the quantum mechanical results pass through minima and subsequently increase with further decrease of the kinetic energy . this is a purely quantum mechanical effect that has important consequences for the low temperature rate coefficients ( balakrishnan et al . the quantum mechanical cross sections eventually conform to an inverse velocity dependence as the relative translational energy is decreased to zero and the corresponding rate coefficients are finite in the limit of zero temperature , in accordance with wigner s threshold law . the results illustrate that qct calculations may be used reliably to calculate rotational transitions for energies higher than 0.5 ev , but at lower energies , quantum mechanical calculation must be employed . the energy regime for the validity of qct results is more restricted for rovibrational transitions as illustrated in figure 2 in which we show cross sections for rovibrational transitions @xmath21 with @xmath22 in @xmath23 to 5 . the results indicate that the qct method is inadequate to calculate rovibrational transition cross sections at impact energies below 1 ev . the qct results exhibit a sharp fall below a threshold near an energy of 1 ev whereas the quantum mechanical results vary smoothly . similar results hold for other transitions . the higher collision energies required for the validity of the qct method for rovibrational transitions compared to pure rotational transitions is due in part to the much smaller cross sections for rovibrational transitions making the results more sensitive to the details of the dynamics . rate coefficients @xmath24 were calculated for the temperature range @xmath25 to 4000 k by averaging the cross sections over a boltzmann distribution of relative velocities . flower et al . ( 1998 ) have reported rate coefficients for rovibrational transitions from @xmath26 for @xmath27 and 4500 k. a comparison of some of the de - excitation rate coefficients from the @xmath6 level calculated in this paper and those reported by flower _ is given in table 1 for @xmath28 k. the agreement is good for rovibrational transitions involving @xmath29 and 4 but larger differences , by factors between 2 and 4 , are seen for transitions involving larger values of @xmath30 . our rate coefficients are generally greater than those of flower _ et al . _ ( 1998 ) for transitions where the discrepancy is large . our rate coefficient calculations extend those of flower _ et al . _ ( 1998 ) to include the @xmath31 and 6 vibrational levels . the preferential formation of h@xmath0 in these vibrational levels has been discussed by dalgarno ( 1995 ) . in tables 2 and 3 , we present our comprehensive results for rovibrational de - excitation transitions from different rotational levels in para- and ortho - h@xmath0 from the @xmath6 level as functions of temperature . the corresponding excitation rate coefficients may be obtained from detailed balance @xmath32 } k_{vj , v'j'}(t)\ ] ] where @xmath33 is the boltzmann constant and @xmath34 is the rovibrational energy of the molecule in the @xmath35 level . similar results for @xmath23 to 6 are presented in tables 4 - 11 . tables 2 - 11 reveal some interesting aspects of energy transfer . it can be seen that for temperatures less than 1000 k , rate coefficients for rovibrational transitions involving @xmath36 predominate over other transitions where changes in both @xmath37 and @xmath15 occur . this is clearly seen for ( @xmath38 ) transitions with @xmath39 for @xmath40 and the effect becomes stronger with increasing rotational excitation but less important as @xmath41 increases . this is an example of quasi - resonant scattering ( stewart et al . 1988 ; forrey et al . a detailed study of this process in the limit of zero temperature and its correspondence with classical mechanics has been carried out recently ( forrey et al . 1999 ) . the efficient conversion of vibrational energy into rotational energy may produce a significant population of high rotational levels in environments where molecular hydrogen is subjected to an intense flux of x - rays or ultraviolet photons . this work was supported by the national science foundation ( nsf ) , division of astronomy . mv was supported by the nsf through the research experience for undergraduates program at the smithsonian astrophysical observatory . hutson , j.m . , & green , s. 1994 , molscat version 14 distributed by collaborative computational project 6 ( daresbury laboratory : uk engineering and physical sciences research council ) . krauss , m. , & mies , f.h . 1965 , j. chem . phys . , 42 , 2703 . lepp , s. , buch , v. , & dalgarno , a 1995 , apjs , 98 , 345 . mcguire , p. , & kouri . , d. j. , 1974 , j. chem . , 60 , 2488 . michaut , x. , saint - loup , r. , berger , h. , dubernet , m. l. , joubert , p. , & bonamy , j. 1998 , j. chem . 109 , 951 . muchnick , p. , & russek , a. 1994 , j. chem . phys . , 100 , 4336 . raczkowski , a.w . , lester , w.a . jr . , & miller , w.h . 1978 , j. chem . phys . , 69 , 2692 . schwenke , d. w. , 1988 , j. chem . phys . , 89 , 2076 .
|
we present quantum mechanical and quasiclassical trajectory calculations of cross sections for rovibrational transitions in ortho- and para - h@xmath0 induced by collisions with he atoms .
cross sections were obtained for kinetic energies between 10@xmath1 and 3 ev , and the corresponding rate coefficients were calculated for the temperature range 100@xmath24000 k. comparisons are made with previous calculations .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
symmetry braking phase transitions in the early universe have several cosmological consequences and provide an important link between particle physics and cosmology . in particular , different types of topological objects may have been formed by the vacuum phase transitions after planck time kibble , v - s . these include domain walls , cosmic strings and monopoles . a global monopole is a spherical symmetric gravitational topological defect created by a phase transition of a system comprised by self - coupling scalar field , @xmath0 , whose original global @xmath1 symmetry is spontaneously broken to @xmath2 . the matter fields play the role of an order parameter which outside the monopole s core acquires a non - vanishing value . the global monopole was first introduced by sokolov and starobinsky soko77 . a few years later , the gravitational effects associated with a global monopole have been considered in ref . @xcite , where the authors have found that for points far from the monopole s center , the geometry is similar to the black - hole with a solid angle deficit . neglecting the mass term we get the point - like global monopole spacetime with the metric tensor given by the following line element @xmath3where the parameter @xmath4 is smaller than unity and depends on the energy scale where the symmetry is broken . it is of interest to note that the effective metric produced in superfluid @xmath5 by a monopole is described by the line element ( [ mmetric ] ) with the negative angle deficit , @xmath6 , which corresponds to the negative mass of the topological object @xcite . the quasiparticles in this model are chiral and massless fermions . in quantum field theory the non - trivial topology of the global monopole spacetime induces non - zero vacuum expectation values for physical observables . the quantum effects due to the point - like global monopole spacetime on the matter fields have been considered for massless scalar m - l and fermionic @xcite fields , respectively . the influence of the non - zero temperature on these polarization effects has been discussed in ref . @xcite for scalar and fermionic fields . moreover , the calculation of quantum effects on massless scalar field in a higher dimensional global monopole spacetime has also been developed in ref . the quantum effects of a scalar field induced by a composite topological defect consisting a cosmic string on a @xmath7-dimensional brane and a @xmath8-dimensional global monopole in the transverse extra dimensions are investigated in ref . the combined vacuum polarization effects by the non - trivial geometry of a global monopole and boundary conditions imposed on the matter fields are investigated as well . in this direction , the total casimir energy associated with massive scalar field inside a spherical region in the global monopole background has been analyzed in refs . @xcite by using the zeta function regularization procedure . scalar casimir densities induced by spherical boundaries have been calculated in refs . @xcite to higher dimensional global monopole spacetime by making use of the generalized abel - plana summation formula @xcite . more recently , using also this formalism , a similar analysis for spinor fields obeying mit bag boundary conditions has been developed in refs . @xcite . in general , the quantum analysis of matter fields in global monopole spacetime consider this object as been a point - like one . because of this fact the renormalized vacuum expectation value of the energy - momentum tensor presents singularities at the monopole s center . of course , this kind of problem can not be expected in a realistic model . so a procedure to cure this divergence is to consider the global monopole as having a non - trivial inner structure . in fact , in a realistic point of view , the global monopoles have a characteristic core radius determined by the symmetry braking energy scale where the symmetry of the system is spontaneously broken . a simplified model for the monopole core is presented in ref . @xcite . in this model the region inside the core is described by the de sitter geometry . the vacuum polarization effects associated with a massless scalar field in the region outside the core of this model are investigated in ref . in particular , it has been shown that long - range effects can take place due to the non - trivial core structure . recently the quantum analysis of a scalar field in a higher - dimensional global monopole spacetime considering a general spherically symmetric model for the core , has been considered in ref . @xcite . in the four - dimensional version of this model the induced electrostatic self - energy and self - force for a charged particle are investigated in ref . continuing in this direction , in the present paper we analyze the effects of global monopole core on properties of the fermionic quantum vacuum . the most important quantities characterizing these properties are the vacuum expectation values of the energy - momentum tensor and the fermionic condensate . though the corresponding operators are local , due to the global nature of the vacuum , the vacuum expectation values describe the global properties of the bulk and carry an important information about the structure of the defect core . in addition to describing the physical structure of the quantum field at a given point , the energy - momentum tensor acts as the source of gravity in the einstein equations . it therefore plays an important role in modelling a self - consistent dynamics involving the gravitational field . the problem under consideration is also of separate interest as an example with gravitational polarization of the fermionic vacuum , where all calculations can be performed in a closed form . the corresponding results specify the conditions under which we can ignore the details of the interior structure and approximate the effect of the global monopole by the idealized model . the exactly solvable models of this type are useful for testing the validity of various approximations used to deal with more complicated geometries , in particular in black - hole spacetimes . the plan of this paper is as follows . section [ sec : eigfunc ] presents the geometry of the problem and the eigenfunctions for a massive spinor field . by using these eigenfunctions , in section [ sec : extreg ] we evaluate the vacuum expectation values of the energy - momentum tensor and the fermionic condensate in the region outside the monopole core . in section sec : flowerpot we consider the special case of the flower - pot model for the core and the vacuum expectation values are investigated in both exterior and interior regions . various limiting cases are considered . the main results of the paper are summarized in section [ sec : conc ] . in appendix sec : app1 we discuss the contribution of possible bound states into the vacuum expectation values of the energy - momentum tensor and show that in the flower - pot model there are no bound states . throughout we use the system of units with @xmath9 . in this section we analyze the eigenfunctions for a massive fermionic field in background of the global monopole geometry considering a nontrivial inner structure to the latter . the explicit expression for the metric tensor in the region inside the monopole core is unknown and here we consider a general model for a four - dimensional global monopole with a core of radius @xmath10 , assuming that the geometry of the spacetime is described by two distinct metric tensors in the regions inside and outside the core . adopting this model we can learn under which conditions we can ignore the specific details for its core and consider the monopole as being a point - like object . in spherical coordinates we will consider the corresponding line element in the interior region , @xmath11 , with the form @xmath12 in the region outside , @xmath13 , the metric tensor is given by the line element ( [ mmetric ] ) , where the parameter @xmath14 codifies the presence of the global monopole . for an idealized point - like global monopole the geometry is described by line element ( [ mmetric ] ) for all values of the radial coordinate and it has singularity at the origin . in eq . ( [ metricinside ] ) the functions @xmath15 , @xmath16 , @xmath17 are continuous at the core boundary , consequently they satisfy the conditions @xmath18if there is no surface energy - momentum tensor on the bounding surface @xmath19 , the radial derivatives of these functions are continuous as well . when the surface energy - momentum tensor is present the junctions in the radial derivatives of the components of the metric tensor are expressed in terms of the surface energy density and stresses . introducing a new radial coordinate @xmath20 with the core center at @xmath21 , the angular part of line element ( [ metricinside ] ) is written in the standard minkowskian form . however , with this choice , in general , we will obtain non - standard form of the angular part in the exterior line element ( [ mmetric ] ) . in the model under consideration we will assume that inside the core the spacetime geometry is regular . in particular , from the regularity of the interior geometry at the core center one has the conditions @xmath22 and @xmath23 for @xmath24 . we are interested in quantum effects of a spinor field propagating on background of the spacetime described by line elements ( [ mmetric ] ) and ( [ metricinside ] ) . the dynamics of the massive spinor field in curved spacetime is governed by the dirac equation @xmath25with the covariant derivative operator @xmath26here @xmath27 are the dirac matrices in curved spacetime , and @xmath28 is the spin connection given in terms of the flat @xmath29 matrices by the relation @xmath30 in the equations above @xmath31 is the vierbein satisfying the condition @xmath32 . in the region inside the monopole core we choose the basis tetrad @xmath33where the rows of the matrix are specified by the index @xmath34 and the columns by the index @xmath35 . by using eq . ( [ e ] ) , for the non - vanishing components of the spin connection we find @xmath36where the prime denotes derivative with respect to the radial coordinate , @xmath37 and @xmath38 are the standard unit vectors along the three spatial directions in spherical coordinates , and @xmath39with @xmath40 being the pauli matrices . from the obtained results , for the combination entering in the dirac equation we have @xmath41after some intermediate steps , the dirac equation reads @xmath42with @xmath43 being the standard angular momentum operator . to find the vacuum expectation value ( vev ) of the energy - momentum tensor we need the corresponding eigenfunctions . for the problem under consideration there are two types of eigenfunctions with different parities which we will distinguish by the index @xmath44 . these functions are specified by the total angular momentum @xmath45 and its projection @xmath46 . assuming the time dependence in the form @xmath47 , the four - component spinor fields specified by the set of quantum numbers @xmath48 with @xmath49 , can be written in terms of two - component ones as @xmath50where @xmath51and @xmath52 are the standard spinor spherical harmonics ( see , for instance , ref . the latter are eigenfunctions of the operator @xmath53 as shown below : @xmath54note that we have the relation @xmath55 substituting the function @xmath56 into the dirac equation above , and using for the flat dirac matrices the representation given in ref . bere82 , for the radial functions we obtain a set of two coupled linear differential equations : @xmath57 in the region @xmath13 for the radial functions we have the solutions@xmath58 , \ ; \label{fext } \\ g_{\beta } ( r ) & = & g_{\beta } ^{\mathrm{(ex)}}(r)=-\frac{1}{\sqrt{r}}\frac{% n_{\sigma } k}{\omega + m}\left [ c_{1}j_{\nu _ { \sigma } + n_{\sigma } } ( kr)+c_{2}y_{\nu _ { \sigma } + n_{\sigma } } ( kr)\right ] , \label{gext}\end{aligned}\]]where @xmath59 and @xmath60 are the bessel and neumann functions of the order @xmath61by taking into account relation ( [ relsphhar ] ) , the corresponding eigenfunctions are written in the form@xmath62 in eqs . ( [ fext ] ) and ( [ gext ] ) , the integration constants @xmath63 and @xmath64 are determined from the matching condition with the interior solution . the regular solution to eqs . ( [ f1 ] ) , ( [ g1 ] ) in the interior region , @xmath65 , we will denote by@xmath66near the core center , @xmath24 , these functions behave like @xmath67 and @xmath68 , where the radial coordinate @xmath69 is introduced in the paragraph after formula ( [ uvbound ] ) . note that the functions @xmath70 and @xmath71 are solutions of the same equation and , hence , @xmath72the interior eigenfunctions have the form@xmath73from the continuity condition of the eigenfunctions on the surface @xmath19 , for the coefficients @xmath63 and @xmath64 in eq . ( [ fext ] ) one finds@xmath74 , \label{c1 } \\ c_{2 } & = & \frac{\pi } { 2}n_{\sigma } ka^{3/2}\left [ r_{1,n_{\sigma } } ( a , k)j_{\nu _ { \sigma } + n_{\sigma } } ( ka)-r_{2,n_{\sigma } } ( a , k)j_{\nu _ { \sigma } } ( ka)\right ] . \label{c2}\end{aligned}\]]note that from eqs . ( [ f1 ] ) , ( [ g1 ] ) , for the values of the interior radial functions on the boundary of the core we have the following relations@xmath75where @xmath76 , @xmath77 , @xmath78 , and we have introduced the notation@xmath79 substituting the expressions for the coefficients @xmath63 and @xmath64 into the formulae for the radial eigenfunctions in the exterior region , one finds@xmath80with the notation@xmath81here and in what follows , for a function @xmath82 with @xmath83 we use the tilded notation defined by the formula@xmath84 \notag \\ & = & kaf_{\nu _ { \sigma } } ^{\prime } ( ka)-\left [ a\frac{r_{1,n_{\sigma } } ^{\prime } ( a , k)}{r_{1,n_{\sigma } } ( a , k)}+\frac{1}{2}au_{a}^{\prime } + ah_{a}^{\prime } -\frac{1}{2}\right ] f_{\nu _ { \sigma } } ( ka ) . \label{fnutilde}\end{aligned}\]]in deriving the second form we have used the relation ( [ r21rel1 ] ) and the recurrence formula for the cylindrical functions . the eigenfunctions are normalized by the condition@xmath85where @xmath29 is the determinant for the spatial metric and @xmath86 is understood as the kronecker delta symbol for the discrete components of the collective index @xmath87 and as the dirac delta function for the continuous ones . as the normalization integral is divergent for @xmath88 , the main contribution comes from large values @xmath89 . we may replace the cylindrical functions with the argument @xmath90 in eq . ( [ psiint ] ) by the corresponding asymptotic expressions . in this way one finds@xmath91 ^{-1}. \label{r1a}\]]this relation determines the normalization coefficient for the interior region . in addition to the modes with real @xmath92 , modes with purely imaginary @xmath93 can be present . these modes correspond to the bound states . the eigenfunctions for the bound states and their normalization are discussed in appendix [ sec : app1 ] . in this section we consider the vevs for the energy - momentum tensor and the fermionic condensate in the region outside the global monopole core . we expand the field operator in terms of the complete set of eigenfunctions @xmath94 : @xmath95 , \label{operatorexp}\]]where @xmath96 is the annihilation operator for particles , and @xmath97 is the creation operator for antiparticles , @xmath98 , for @xmath99 and @xmath100 , for @xmath101 . in order to find the vev for the operator of the energy - momentum tensor , we substitute the expansion ( operatorexp ) and the analog expansion for the operator @xmath102 into the corresponding expression for spinor fields , @xmath103\ . \label{emtform}\]]by making use of the standard anticommutation relations for the annihilation and creation operators , for the vev one finds the following mode - sum formula @xmath104where @xmath105 is the amplitude for the corresponding vacuum state . substituting the eigenfunctions ( [ psiint ] ) into the mode - sum formula ( [ modesum ] ) , using the formula @xmath106and relation ( [ r1a ] ) , the vev of the energy - momentum tensor is presented in the form ( no summation over @xmath35 ) @xmath107 } { \tilde{j}_{\nu _ { \sigma } } ^{2}(x)+\tilde{y}_{\nu _ { \sigma } } ^{2}(x)% } \ . \label{qext}\]]in formula ( [ qext ] ) we use the notations @xmath108 & = & -x\left [ ( \sqrt{x^{2}+m^{2}a^{2}}-ma)g_{\nu _ { \sigma } , 0}^{2}(x , y)\right . \ \notag \\ & & \left . + ( \sqrt{x^{2}+m^{2}a^{2}}+ma)g_{\nu _ { \sigma } , n_{\sigma } } ^{2}(x , y)% \right ] , \label{fnueps } \\ f_{\sigma \nu _ { \sigma } } ^{(1)}\left [ x , g_{\nu _ { \sigma } , p}(x , y)\right ] & = & % \frac{x^{3}}{\sqrt{x^{2}+m^{2}a^{2}}}\left [ g_{\nu _ { \sigma } , 0}^{2}(x , y)+g_{\nu _ { \sigma } , n_{\sigma } } ^{2}(x , y)\right ] \ -2f_{\sigma \nu _ { \sigma } } ^{(2)}\left [ x , g_{\nu _ { \sigma } , p}(x , y)\right ] , \label{fnup } \\ f_{\sigma \nu _ { \sigma } } ^{(\mu ) } \left [ x , g_{\nu _ { \sigma } , p}(x , y)\right ] & = & \frac{x^{3}(2\nu _ { \sigma } + n_{\sigma } ) } { 2y\sqrt{x^{2}+m^{2}a^{2}}}% g_{\nu _ { \sigma } , 0}(x , y)g_{\nu _ { \sigma } , n_{\sigma } } ( x , y),\;\mu = 2,3 , \label{fnupperp}\end{aligned}\]]and the function @xmath109 is defined by eq . ( [ gnumu2 ] ) . the expression on the right of eq . ( [ qext ] ) is divergent . it may be regularized introducing a cutoff function @xmath110 with the cutting parameter @xmath35 which makes the divergent expressions finite and satisfies the condition @xmath111 for @xmath112 . after the renormalization the cutoff function is removed by taking the limit @xmath113 . in the discussion below we will implicitly assume that the corresponding expressions are regularized . the parts in the vevs induced by the non - trivial structure of the core are finite and do not depend on the regularization scheme used . to find the part in the vev of the energy - momentum tensor induced by the non - trivial core structure , we subtract the corresponding components for the point - like monopole geometry . the latter are given by the expressions which are obtained from eq . ( [ qext ] ) replacing the integrand by @xmath114 $ ] ( see ref . @xcite ) . in order to evaluate the corresponding difference we use the relation @xmath115 } { \tilde{j}_{\nu _ { \sigma } } ^{2}(x)+\tilde{y}_{\nu _ { \sigma } } ^{2}(x)% } -f_{\sigma \nu _ { \sigma } } ^{(\mu ) } \left [ x , j_{\nu _ { \sigma } } ( xr / a)\right ] = -\frac{1}{2}\sum_{s=1,2}\frac{\tilde{j}_{\nu _ { \sigma } } ( x)}{\tilde{h}_{\nu _ { \sigma } } ^{(s)}(x)}f_{\sigma \nu _ { \sigma } } ^{(\mu ) } \left [ x , h_{\nu _ { \sigma } } ^{(s)}(xr / a)\right ] \ , \label{relf}\]]where @xmath116 , @xmath117 , are the hankel functions . this allows to present the vacuum energy - momentum tensor components in the form@xmath118where @xmath119 is the vev for the point - like monopole and the part ( no summation over @xmath35 ) @xmath120 \ , \label{qrbout}\]]is induced by the non - trivial core structure . in formula ( [ qrbout ] ) , the integrand of the @xmath121 ( @xmath122 ) term exponentially decreases in the upper ( lower ) half of the complex plane @xmath123 . consequently , we rotate the integration contour on the right of this formula by the angle @xmath124 for @xmath125 and by the angle @xmath126 for @xmath122 . this leads to the representation@xmath127 , \label{tmunuc2}\end{aligned}\]]where @xmath128 . first of all let us consider the part of the integral over the interval @xmath129 . by using the standard properties of the hankel functions it can be easily seen that in this interval one has @xmath130 = e^{-i\pi \lambda } f_{\sigma \nu _ { \sigma } } ^{(\mu ) } \left [ e^{-\pi i/2}x , h_{\nu _ { \sigma } } ^{(2)}(e^{-\pi i/2}xr / a)\right ] . \label{fnusigrel1}\]]further , from equations ( [ f1 ] ) and ( [ g1 ] ) it follows that the interior solution can be written as @xmath131 . on the base of this observation for the combination entering into the tilted notations in eq . ( [ tmunuc2 ] ) one finds@xmath132now , it can be seen that for these values @xmath123 one has@xmath133combining relations ( [ fnusigrel1 ] ) and ( [ reljh ] ) , we see that the part of the integral in eq . ( [ tmunuc2 ] ) over the interval @xmath134 vanishes . to simplify the part of the integral over the interval @xmath135 , we note that for the functions with different parities the following relation takes place:@xmath130 = -e^{-i\pi \lambda } f_{\sigma ^{\prime } \nu _ { \sigma ^{\prime } } } ^{(\mu ) } \left [ e^{-\pi i/2}x , h_{\nu _ { \sigma ^{\prime } } } ^{(2)}(e^{-\pi i/2}xr / a)\right ] , \label{frel3}\]]where @xmath136 , and @xmath137 . further we note that the functions @xmath138 and @xmath139 satisfy the same equation and , hence , @xmath140 . by using relations ( [ r21rel1 ] ) and ( [ r21rel2 ] ) , now it can be seen that@xmath141from this formula , by taking into account that @xmath142 , we find the relation@xmath143combining formulae ( [ frel3 ] ) and ( [ reljh3 ] ) , we see that the different parities give the same contribution to the vev of the energy - momentum tensor . by taking into account this result and introducing the modified bessel functions , for the core - induced part in the vev we find ( no summation over @xmath144)@xmath145 . \label{tmunu6}\]]here for a given function @xmath146 we use the notations@xmath147 & = & \left ( \frac{m^{2}}{% x^{2}}-1\right ) \left [ \left ( 1+\frac{i\eta _ { s}m}{\sqrt{x^{2}-m^{2}}}% \right ) f_{\nu } ^{2}(y)\right . \notag \\ & & \left . -\left ( 1-\frac{i\eta _ { s}m}{\sqrt{x^{2}-m^{2}}}\right ) f_{\nu + 1}^{2}(y)\right ] , \label{fnu0 } \\ f_{\nu } ^{(1,\eta _ { s})}\left [ x , f_{\nu } ( y)\right ] & = & f_{\nu } ^{2}(y)-f_{\nu + 1}^{2}(y)-\lambda _ { f}\frac{2\nu + 1}{y}f_{\nu } ( y)f_{\nu + 1}(y ) , \label{fnu1 } \\ f_{\nu } ^{(\mu , \eta _ { s})}\left [ x , f_{\nu } ( y)\right ] & = & \lambda _ { f}\frac{% 2\nu + 1}{2y}f_{\nu } ( y)f_{\nu + 1}(y),\;\mu = 2,3 , \label{fnu2}\end{aligned}\]]with @xmath148 ( the function @xmath149 $ ] with @xmath150 will be used below ) , and the barred notation is defined by the formula@xmath151 f(x ) . \label{fetasbar}\]]it can be checked that the core - induced part in the vev of the energy - momentum tensor obeys the continuity equation @xmath152 , which for the geometry under consideration takes the form@xmath153 in addition , for a massless spinor field this part is traceless and the trace anomaly is contained in the point - like monopole part only . the fermionic condensate can be found from the formula for the vev of the energy - momentum tensor by making use of the relation @xmath154 . the condensate is presented in the form of the sum of idealized point - like monopole and core - induced parts:@xmath155by taking into account formula ( [ tmunu6 ] ) for the components of the energy - momentum tensor , for the part coming from the non - trivial core structure one finds@xmath156 . \label{condens1}\end{aligned}\]]this formula may also be derived directly from the mode - sum formula @xmath157 with the eigenfunctions ( [ psiex ] ) . in deriving formulae ( [ tmunu6 ] ) and ( [ condens1 ] ) we have assumed that no bound states exist . in appendix [ sec : app1 ] we show that these formulae are valid also in the case when the bound states are present . for @xmath13 the core - induced parts in the vevs of the energy - momentum tensor and the fermionic condensate , given by eqs . ( [ tmunu6 ] ) and ( condens1 ) , are finite and the renormalization is necessary for the point - like monopole parts only . of course , we could expect this result as in the region @xmath13 the local geometry is the same in both models and , hence , the divergences are the same as well . as it has been already mentioned , if there is no surface energy - momentum tensor on the bounding surface @xmath19 , then the radial derivatives of the metric are continuous and , hence , in eqs . ( [ fnutilde ] ) and ( fetasbar ) one has @xmath158 , @xmath159 . in models with an additional infinitely thin spherical shell located at @xmath19 , these quantities are related to the components of the corresponding surface energy - momentum tensor @xmath160 . denoting by @xmath161 the normal to the shell normalized by the condition @xmath162 and assuming that it points into the bulk on both sides , from the israel matching conditions one has@xmath163 in this formula the curly brackets denote summation over each side of the shell , @xmath164 is the induced metric on the shell , @xmath165 its extrinsic curvature and @xmath166 . for the region @xmath167 one has @xmath168 and the non - zero components of the extrinsic curvature are given by the formulae@xmath169the corresponding expressions for the region @xmath170 are obtained by taking in these formulae @xmath171 , @xmath172 and changing the signs for the components of the extrinsic curvature tensor . now from the matching conditions ( [ matchcond ] ) we find@xmath173note that the combination in the square brackets in eqs . ( [ fnutilde ] ) and ( [ fetasbar ] ) is related to the surface energy - momentum tensor by the formula@xmath174where @xmath175 is the trace of the surface energy - momentum tensor . in this section , as an application of the general results given above we consider a simple example of the core model assuming that the spacetime inside it is flat . the corresponding model for the cosmic string core was considered in refs . @xcite and for the global monopole core in ref . @xcite . following these papers we will refer to this model as flower - pot model . taking in the region inside the core @xmath171 , from the zero curvature condition one finds @xmath176 . the value of the constant is found from the continuity condition for the function @xmath17 at the boundary which gives @xmath177 . hence , in the flower - pot model the interior line element has the form@xmath178 ^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2 } ) . \label{intflow}\]]in terms of the radial coordinate @xmath89 the origin is located at @xmath179 . from the matching conditions ( [ matchcond2 ] ) we find the corresponding surface energy - momentum tensor with the non - zero components and the trace given by@xmath180the corresponding surface energy density is positive for the global monopole with @xmath181 . we will consider the vevs in the exterior and interior regions separately . in the region inside the core the radial eigenfunctions regular at the origin are the functions@xmath182where@xmath183and @xmath184 is the standard minkowskian radial coordinate , @xmath185 . note that for an interior minkowskian observer the radius of the core is @xmath186 . the normalization coefficient @xmath187 is found from the condition ( [ r1a]):@xmath188with the tilted notation for the cylindrical functions@xmath189 f_{\nu _ { \sigma } } ( x ) . \label{ftildeflow}\]]note that @xmath190 for @xmath191 . in this case for the barred notation in eq . ( [ fetasbar ] ) one has @xmath192 @xmath193 . hence , in the flower - pot model the part in the energy - momentum tensor due the non - trivial structure of the core is given by the formula @xmath194 , \label{tmunu8}\end{aligned}\]]where we have introduced the notation@xmath195 g(x ) . \label{cfg}\]]in formula ( [ tmunu8 ] ) , @xmath196 = f_{\nu } ^{(\mu , \eta _ { s})}\left [ x , f_{\nu } ( y)\right ] $ ] for @xmath197 , and @xmath198 = \left ( \frac{% m^{2}}{x^{2}}-1\right ) \left [ f_{l/\alpha -1/2}^{2}(y)-f_{l/\alpha + 1/2}^{2}(y)\right ] . \label{gnu0}\]]note that in terms of notation ( [ cfg ] ) one has @xmath199for @xmath191 we have @xmath200 and the vevs vanish . in the similar way , for the fermionic condensate from eq . ( [ condens1 ] ) we find@xmath201 . \label{condfp}\end{aligned}\]]for a massless fermionic field the core - induced part in the condensate vanishes . from formulae ( [ tmunu8 ] ) and ( [ condfp ] ) it can be seen that for fixed values of @xmath89 and @xmath202 , in the limit @xmath203 the core - induced parts vanish as @xmath204 . note that by using the recurrence relations for the modified bessel functions , the functions @xmath205 in these formulae can also be written in the form@xmath206with @xmath207 , @xmath208 and @xmath150 , @xmath148 . in particular , from formula ( [ cif ] ) with @xmath209 it follows that @xmath210 . as we will show in appendix sec : app1 , this means that there are no bound states in the flower - pot model . the core - induced part in the vevs of the energy - momentum tensor and the fermionic condensate given by formulae ( [ tmunu8 ] ) and ( [ condfp ] ) , are finite for @xmath13 and diverge on the core boundary . surface divergences are well - known in quantum field theory with boundaries and are investigated for various boundary geometries and fields . to find the corresponding asymptotic behavior for the points near the boundary , we note that in this region the main contribution comes from large values of @xmath211 . by using the uniform asymptotic expansions for the modified bessel functions for large values of the order ( see , for instance , @xcite ) , to the leading order we find@xmath212as the parts corresponding to the geometry of the point - like global monopole are finite at @xmath19 , from here we conclude that near the core the vevs are dominated by the core - induced parts . at large distances from the core , @xmath213 , in the case of a massless fermionic field we introduce a new integration variable @xmath214 and expand the integrand over @xmath215 . the main contribution comes from the lowest mode @xmath216 and to the leading order one has@xmath217 . \label{tmunulargedist}\]]note that for a massless field the integrand does not depend on @xmath89 . the integrals in eq . ( [ tmunulargedist ] ) may be evaluated by using the formula for the integrals involving the product of the macdonald functions ( see ref . @xcite ) . for a massive field , assuming @xmath218 , the main contribution into the integral over @xmath123 comes from the lower limit of the integration . replacing the functions @xmath219 by the corresponding asymptotic formulae for large values of the argument , to the leading order we obtain the following estimates@xmath220as we see , in this limit the core - induced energy density and the radial stress are suppressed by the factor @xmath221 with respect to the azimuthal stress . for @xmath222 the solid angle for the exterior geometry is small and the corresponding scalar curvature is large . in this limit we replace the modified bessel function containing in the index @xmath223 by the corresponding uniform asymptotic expansions for large values of the order and the functions containing in the argument @xmath224 by the expansions for small values of the argument . in this way it can be seen that the main contribution into the vevs comes from the mode @xmath225 and the vevs are suppressed by the factor @xmath226 $ ] . in figure [ fig1 ] we have plotted the dependence of the core - induced energy density ( full curves ) and radial stress ( dashed curves ) for a massless fermionic field as functions on the scaled radial coordinate @xmath227 for @xmath228 and @xmath229 . the azimuthal stresses are found from the zero trace condition . in figure [ fig2 ] the same quantities evaluated for @xmath230 are presented as functions on the parameter @xmath14 . now let us consider the vacuum polarization effects inside the core for the flower - pot model . the corresponding eigenfunctions have the form given by eq . ( [ psibetin ] ) where the functions @xmath70 and @xmath231 are defined by formulae ( [ rlflow ] ) . substituting the eigenfunctions into the mode - sum formula , for the corresponding energy - momentum tensor one finds ( no summation over @xmath35)@xmath232 } { j_{\lambda _ { \sigma } } ^{2}(\alpha x)\left [ \tilde{j}_{\nu _ { \sigma } } ^{2}(x)+\tilde{y}_{\nu _ { \sigma } } ^{2}(x)\right ] } . \label{tmunuinfl}\]]to find the renormalized vev of the energy - momentum tensor we need to evaluate the difference between the vev given by eq . ( [ tmunuinfl ] ) and the corresponding vev for the minkowski bulk:@xmath233the appropriate form for the minkowskian part is obtained from eq . ( qext ) taking @xmath191 , replacing the integrand by @xmath234 $ ] and @xmath235 . as a result for the subtracted vev one finds@xmath236 \notag \\ & & \times \left [ \frac{j_{\lambda _ { \sigma } } ^{-2}(\alpha x)/\alpha } { \tilde{j% } _ { \nu _ { \sigma } } ^{2}(x)+\tilde{y}_{\nu _ { \sigma } } ^{2}(x)}-\frac{\pi ^{2}}{% 4}\right ] . \label{tmunusub2}\end{aligned}\]]the integral in this formula is slowly convergent and the integrand is highly oscillatory . in order to transform the expression for the subtracted vev of the energy - momentum tensor into more convenient form , we note that the following identity takes place@xmath237substituting eq . ( [ identflow2new ] ) into formula ( [ tmunusub2 ] ) , we rotate the integration contour in the complex plane @xmath123 by the angle @xmath124 for @xmath121 and by the angle @xmath126 for @xmath122 . under the condition @xmath238 the contributions from the semicircles with the radius tending to infinity vanish . the integrals over the segments @xmath239 and @xmath240 of the imaginary axis cancel out and after introducing the modified bessel functions we can see that different parities give the same contribution . consequently , the renormalized vev for the energy - momentum tensor can be presented in the form ( no summation over @xmath35)@xmath241 , \label{tmunusub3}\end{aligned}\]]where the functions @xmath196 $ ] are the same as in eq . ( [ tmunu8 ] ) with @xmath150 in eqs . ( [ fnu1 ] ) and ( [ fnu2 ] ) . note that similar to eq . ( [ cif ] ) , the function in the numerator of the integrand is also presented in the form@xmath242for @xmath191 one has @xmath243 and , as we could expect , the vev of the energy - momentum tensor vanishes . it can be explicitly checked that the components of the tensor given by formula ( [ tmunusub3 ] ) satisfy the continuity equation ( [ conteq ] ) and this tensor is traceless for a massless field . in the way similar to that for the exterior region , for the renormalized fermionic condensate inside the core we find@xmath244 . \label{condint}\end{aligned}\ ] ] the vevs given by formulae ( [ tmunusub3 ] ) and ( [ condint ] ) are finite for @xmath245 and diverge on the core boundary . for the points near the boundary the main contribution comes from large @xmath211 and we replace the modified bessel functions by the corresponding uniform expansions for large values of the order . in this way it can be seen that the leading terms in the asymptotic expansion with respect to the distance from the boundary are given by the formulae@xmath246comparing these results with the corresponding formulae for the exterior region , we see that near the core boundary the energy density and azimuthal stresses have the same signs inside and outside the core , whereas the radial stresses have opposite signs . near the core center the contribution of the summand with a given @xmath211 behaves like @xmath247 and the main contribution comes from the lowest mode @xmath225 with the leading term ( no summation over @xmath35)@xmath248where we have introduced notations@xmath249 in the case of a massless field , formula ( [ nearcenter ] ) also gives the behavior of the vacuum energy - momentum tensor in the limit when the core radius is large and @xmath69 is fixed , @xmath250 . in the same limit , for a massive field , assuming @xmath251 , we replace in eq . ( tmunusub3 ) the modified bessel functions containing in the argument @xmath252 by the corresponding asymptotic expressions for large values of the argument . by taking into account that the main contribution into the integral comes from the lower limit of the integration , to the leading order we have ( no summation over @xmath35)@xmath253 , \label{t00radlarge } \\ \langle t_{\mu } ^{\nu } \rangle _ { \mathrm{ren } } & \approx & \frac{\delta _ { \mu } ^{\nu } ( 1-\alpha ) } { 8\pi \alpha ^{4}a^{3}\tilde{r}}\sqrt{\pi \alpha am}% e^{-2\alpha am}\sum_{l=1}^{\infty } l^{2}g_{l-1/2}^{(\mu ) } \left [ m , i_{l-1/2}(m\tilde{r})\right ] , \label{tmnradlarge}\end{aligned}\]]with @xmath254 . in this case the energy density is suppressed with respect to the vacuum stresses by an additional factor @xmath255 . for small values of the parameter @xmath14 assuming that the core radius @xmath256 for an internal minkowskian observer is fixed , we replace the functions @xmath257 in eq . ( [ tmunusub3 ] ) by the corresponding uniform asymptotic expansion for large values of the order . the leading term is obtained by making use of the replacements@xmath258^{\prime } + \sqrt{l^{2}+y^{2}}+1/2 , \label{replace}\ ] ] in the integrand of eq . ( [ tmunusub3 ] ) with @xmath259 and @xmath260 . as a result , in this limit the vevs behave like @xmath261 . due to the factor @xmath262 in the volume element the corresponding global quantities , such as total energy vanish as @xmath14 . in figure [ fig3 ] we have plotted the dependence of the renormalized vacuum energy density ( full curves ) and radial stress ( dashed curves ) for a massless fermionic field as functions on @xmath263 for @xmath264 and @xmath229 . in figure [ fig4 ] the same quantities evaluated for @xmath265 are presented as functions on the parameter @xmath266 . in the present paper we have considered the polarization of the fermionic vacuum by the gravitational field of the global monopole wit a non - trivial core structure . the previous investigations in this direction are concerned with the idealized point - like model , where the curvature has singularity at the origin . in a realistic point of view , the global monopole has a characteristic core radius determined by the symmetry braking energy scale . for a general spherically symmetric static model of the core with finite thickness , we have evaluated the vevs of the energy - momentum tensor and the fermionic condensate for a massive spinor field . these quantities are among the most important characteristics of the vacuum properties , which carry an information about the core structure . in the region outside the core we have presented the vevs as a sum of two contributions . the first one corresponds to the geometry of a point - like global monopole and the second one is induced by the non - trivial structure of the monopole core . in the general spherically symmetric static model for the core , we have derived closed analytic expressions for the core - induced parts given by formula ( tmunu6 ) for the energy - momentum tensor and by formula ( [ condens1 ] ) for the fermionic condensate , where the properties of the core are codified by the coefficient in the square brackets in the notation ( [ fetasbar ] ) . in appendix [ sec : app1 ] we show that the formulae for the core - induced parts in the vevs of the energy - momentum tensor and fermionic condensate are valid also in the case when bound states are present . for points away from the core boundary these parts are finite and the renormalization is reduced to that for the point - like monopole geometry . of course , we could expect this result as in the model under consideration the exterior geometry is the same as that for the point - like global monopole . for the points on the boundary the vevs contain surface divergences well - known in quantum field theory with boundaries . as an example of the application of the general results , in section sec : flowerpot we have considered a simple model with a flat spacetime inside the core . the corresponding model for the cosmic string core is known in literature as flower - pot model and here we use the same terminology for the global monopole . to have matching between the exterior and interior metrics , in this model we need the surface energy - momentum tensor located on the boundary of the core and having components given by eq . ( surfemtflow ) . in the model with solid angle deficit ( @xmath181 ) the corresponding surface energy density is positive . the core - induced parts of the exterior vevs in the flower - pot model are obtained from the general results taking as the interior radial functions in the eigenmodes the functions ( [ rlflow ] ) . these parts are given by formula ( [ tmunu8 ] ) for the energy - momentum tensor and by formula ( [ condfp ] ) for the fermionic condensate . we have investigated the core - induced parts in various asymptotic regions of the parameters . in the limit when the core radius tends to zero , @xmath203 , for fixed values @xmath89 and @xmath202 , these parts behave like @xmath204 . for points near the core boundary the leading terms in the corresponding asymptotic expansions are given by formulae ( nearcore ) , ( [ nearcorecond ] ) . in this region the total vevs are dominated by the core - induced parts . at large distances from the core these parts tend to zero as @xmath267 for a massless field and are exponentially suppressed by the factor @xmath268 for a massive field . we have also investigated the limit of strong gravitational fields corresponding to small values of the parameter @xmath14 . in this limit the main contribution into the vevs comes from the lowest mode @xmath225 and the vevs are suppressed by the factor @xmath226 $ ] . for the flower - pot model we have also investigated the vevs of the energy - momentum tensor and the fermionic condensate inside the core . though the corresponding spacetime geometry is monkowskian , the non - trivial topology of the exterior region induces vacuum polarization effects in this region as well . the renormalization is achieved by the subtraction from the mode - sums the corresponding quantities for the minkowski spacetime . by making use of identity ( [ identflow2new ] ) , after an appropriate deformation of the integration contour , we have presented the renormalized vev of the energy - momentum tensor in the form ( [ tmunusub3 ] ) and the fermionic condensate in the form ( [ condint ] ) . these quantities are finite for strictly interior points and diverge on the boundary of the core with the leading divergences given by formulae ( [ nearint ] ) , ( nearboundint ) . in particular , near the core boundary the energy density and azimuthal stresses have the same signs inside and outside the core , whereas the radial stresses have opposite signs . near the core center the main contribution comes from the mode @xmath225 and the vevs tend to a finite limiting value with isotropic vacuum stresses . although the exact behavior for the fermionic field is unknown for a realistic model of the global monopole spacetime , the flower - pot model considered here presents some expected results as , for example , finite vacuum polarization effects at the monopole s center . for large values of the core radius the renormalized vev of the energy - momentum tensor inside the core vanishes as @xmath269 for a massless field and as @xmath270 for a massive one . in the limit @xmath271 , assuming that the core radius @xmath186 for an internal minkowskian observer is fixed , the vacuum densities in the interior region behave as @xmath261 . note that in this paper we have considered quantum vacuum effects in a prescribed background , i.e. the gravitational back - reaction of quantum effects is not taken into account . this back - reaction could have important effects on the dynamical evolution of the bulk model . we do not consider this extension of the theory , but note that the results presented here constitute the starting point for such investigations . aas was supported by pve / capes program and in part by the armenian ministry of education and science grant no . 0124 . erbm thanks conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) and fapesq - pb / cnpq ( pronex ) for partial financial support . in this appendix we consider the changes in the procedure described in the main text when bound states are present . for these states the quantity @xmath92 is purely imaginary , @xmath272 , and the corresponding exterior eigenfunctions have the form@xmath273with @xmath274 . to have a stable ground state we will assume that @xmath275 . from the continuity of the eigenfunctions at @xmath19 one has@xmath276excluding from these relations the normalization coefficient @xmath277 we see that for possible bound states @xmath29 is a solutions of the equation@xmath278with the barred notation from eq . ( [ fetasbar ] ) . the coefficient @xmath277 in eq . ( [ psibound ] ) is found from the normalization condition ( [ normcond ] ) . to derive a formula for the normalization integral , we rewrite equations ( [ f1 ] ) , ( [ g1 ] ) in terms of the functions @xmath279 and @xmath280 and differentiate both equations with respect to @xmath281 . further we multiply the first equation by @xmath282 , the second one by @xmath283 and add them . combining the resulting equation with eqs . ( [ f1 ] ) , ( [ g1 ] ) , it can be seen that the following relation takes place@xmath284integrating this relation we obtain the formula@xmath285=e^{u+2h } \left [ g_{\beta } ( r)\frac{\partial f_{\beta } ( r)}{\partial \omega } -f_{\beta } ( r)\frac{\partial g_{\beta } ( r)}{\partial \omega } \right ] , \label{normint}\]]where @xmath286 is the value of the radial coordinate corresponding to the center of the core . by using this formula , in the case of bound states for the normalization integral in eq . ( [ normcond ] ) one finds@xmath287=e^{u+2h}\left [ g_{\beta } ( r)\frac{\partial f_{\beta } ( r)}{\partial \omega } -f_{\beta } ( r)\frac{\partial g_{\beta } ( r)}{\partial \omega } \right ] _ { r = a+}^{r = a-}. \label{normint}\]]the expression on the right can be further simplified by using the continuity of the eigenfunctions on the boundary of the core . in this way for the normalization coefficient one finds the formula@xmath288 as a result , for the contribution of the bound state with @xmath272 to the vev of the energy - momentum tensor we have the formula ( no summation over @xmath35)@xmath289 , \label{tmunubound}\]]where @xmath290 and we have introduced the notations@xmath291 & = & \left ( 1-\frac{m^{2}}{\gamma ^{2}}% \right ) \left [ \left ( \frac{m}{\sqrt{m^{2}-\gamma ^{2}}}-1\right ) k_{\nu _ { \sigma } } ^{2}(y ) \right . \nonumber \\ & & \left . + \left ( \frac{m}{\sqrt{m^{2}-\gamma ^{2}}}+1\right ) k_{\nu _ { \sigma } + n_{\sigma } } ^{2}(y)\right ] , \nonumber \\ b^{(1)}[\gamma , k_{\nu _ { \sigma } } ( y ) ] & = & k_{\nu _ { \sigma } } ^{2}(y)+\frac{% 2ln_{\sigma } } { \alpha y}k_{\nu _ { \sigma } } ( y)k_{\nu _ { \sigma } + n_{\sigma } } ( y)-k_{\nu _ { \sigma } + n_{\sigma } } ^{2}(y ) , \label{bmu } \\ b^{(\mu ) } [ \gamma , k_{\nu _ { \sigma } } ( y ) ] & = & -\frac{ln_{\sigma } } { \alpha y}% k_{\nu _ { \sigma } } ( y)k_{\nu _ { \sigma } + n_{\sigma } } ( y),\;\mu = 2,3 . \nonumber\end{aligned}\]]in deriving eq . ( [ tmunubound ] ) we have used the relation @xmath292 , which directly follows from the wronskian relation for the modified bessel functions in combination with eq . ( [ appbseq ] ) . in the case when several bound states are present the sum of their separate contributions should be taken . the vev of the energy - momentum tensor is the sum of the part coming from the modes with real @xmath92 given by eq . ( [ qrbout ] ) and of the part coming from the bound states given by eq . ( [ tmunubound ] ) . in order to transform the first part we again rotate the integration contour in eq . ( [ qrbout ] ) by the angle @xmath124 for @xmath121 and by the angle @xmath126 for @xmath122 . but now we should take into account that the integrand has poles at @xmath293 which are zeroes of the functions @xmath294 in accordance with eq . ( [ appbseq ] ) . rotating the integration contour we will assume that the pole @xmath295 , @xmath117 , on the imaginary axis is avoided by the semicircle @xmath296 in the right half plane with small radius @xmath297 and with the center at this pole . the integration over these semicircles will give an additional contribution@xmath298 . \label{poles}\]]by evaluating the integrals in this formula it can be seen that this term cancels the contribution ( [ tmunubound ] ) coming from the corresponding bound state . hence , we conclude that the formulae given above for the core - induced parts in the vevs are valid in the case of the presence of bound states as well . in the flower - pot model the equation ( [ appbseq ] ) for the bound states takes the form@xmath299 in section [ sec : flowerpot ] we have shown that the function on the left of this equation is always negative and , hence , in the flower - pot model no bound states exist .
|
we study the vacuum polarization effects associated with a massive fermionic field in a spacetime produced by a global monopole considering a nontrivial inner structure for it . in the general case of the spherically symmetric static core with finite support we evaluate the vacuum expectation values of the energy - momentum tensor and the fermionic condensate in the region outside the core .
these quantities are presented as the sum of point - like global monopole and core - induced contributions .
the asymptotic behavior of the core - induced vacuum densities are investigated at large distances from the core , near the core and for small values of the solid angle corresponding to strong gravitational fields . as an application of general results the flower - pot model for the monopole s core
is considered and the expectation values inside the core are evaluated .
pacs number(s ) : 03.07.+k , 98.80.cq , 11.27.+d
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
theoretical work on gravitational lensing is traditionally done in a quasi - newtonian approximation formalism , see , e.g. , schneider , ehlers and falco @xcite or petters , levine and wambsganss @xcite , which is based , among other things , on the approximative assumptions that the gravitational field is weak and that the bending angles are small . under these assumptions , lensing is described in terms of a `` lens equation '' that determines a `` lens map '' from a `` deflector plane '' to a `` source plane '' , thereby relating image positions on the observer s sky to source positions . although for all practical purposes up to now this formalism has proven to be very successful , there are two motivations for doing gravitational lens theory beyond the quasi - newtonian approximation . first , from a methodological point of view it is desirable to investigate qualitative features of lensing , such as criteria for multiple imaging or for the formation of einstein rings , in a formalism without approximations , as far as possible , to be sure that these features are not just reflections of the approximations . second , lensing phenomena where strong gravitational fields and large bending angles are involved are no longer as far away from observability as they have been a few years ago . in particular , the discovery that there is a black hole at the center of our galaxy @xcite , and probably at the center of most galaxies , has brought the matter of lensing in strong gravitational fields with large bending angles closer to practical astrophysical interest . if a light ray comes sufficiently close to a black hole , the bending angle is not small ; in principle , it may even become arbitrarily large , corresponding to the light ray making arbitrarily many turns around the black hole . unboundedly large bending angles also occur e.g. with wormholes ; the latter are more exotic than black holes , in the sense that up to now there is no clear evidence for their existence , but nonetheless considered as hypothetical candidates for lensing by many authors . if one wants to drop the assumptions of weak fields and small angles , gravitational lensing has to be based on the lightlike geodesic equation in a general - relativistic spacetime , without approximations . in this paper we will discuss this issue for the special case of a spherically symmetric and static spacetime . in view of applications , this includes spherical non - rotating stars and black holes , and also more exotic objects such as wormholes and monopoles with the desired symmetries . the main goal of this paper is to demonstrate that in this case lensing without approximations can be studied , quite conveniently , in terms of a lens equation that is not less explicit than the lens equation of the quasi - newtonian formalism . lensing without weak - field or small - angle approximations was pioneered by darwin @xcite and by atkinson @xcite . whereas darwin s work is restricted to the schwarzschild spacetime throughout , atkinson derives all relevant formulas for an unspecified spherically symmetric and static spacetime before specializing to the schwarzschild spacetime in schwarzschild and in isotropic coordinates . all important features of schwarzschild lensing are clearly explained in both papers . in particular , they discuss the occurrence of infinitely many images , corresponding to light rays making arbitrarily many turns around the center and coming closer and closer to the light sphere at @xmath2 . however , they do not derive anything like a lens equation . the notion of a lens equation without weak - field or small - angle approximations was brought forward much later by frittelli and newman @xcite . it is based on the idea of parametrizing the light cone of an arbitrary observation event in a particular way . for a general discussion of this idea and of the resulting `` exact gravitational lens map '' in arbitrary spacetimes the reader may consult ehlers , frittelli and newman @xcite or perlick @xcite . here we are interested only in the special case of a spherically symmetric and static spacetime . then the geodesic equation is completely integrable and the exact lens equation of frittelli and newman can be written quite explicitly . one can evaluate this equation from the spacetime perspective , as has been demonstrated by frittelli , kling and newman @xcite for the case of the schwarzschild spacetime , thereby getting a good idea of the geometry of the light cone . here we will use an alternative representation , using the symmetry for reducing the dimension of the problem . after fixing two radius values @xmath0 and @xmath1 , lensing for an observation event somewhere at @xmath0 and static light sources distributed at @xmath1 is coded in a lens equation , explicitly given in terms of integrals over the metric coefficients , that relates two angles to each other . this representation results in a particularly convenient method of visualizing all relevant lensing properties , as will be demonstrated with two examples . the lens equation discussed in this paper should be compared with the lens equation for spherically symmetric and static spacetimes that was introduced by virbhadra , narasimha and chitre @xcite and then , in a modified form , by virbhadra and ellis @xcite . the virbhadra - ellis lens equation has found considerable interest . it was applied to the schwarzschild spacetime @xcite and later also to other spherically symmetric and static spacetimes , e.g. to a boson star by dabrowski and schunck @xcite , to a fermion star by bili , nikoli and viollier @xcite , to spacetimes with naked singularities by virbhadra and ellis @xcite , to the reissner - nordstrm spacetime by eiroa , romero and torres @xcite and to a gibbons - maeda - garfinkle - horowitz - strominger black hole by bhadra @xcite . in the last two papers , the authors concentrate on light rays that make several turns around the center and they use analytical methods developed by bozza @xcite . the virbhadra - ellis lens equation takes an intermediary position between the exact lens equation and the quasi - newtonian approximation . it makes no assumptions as to the smallness of bending angles , but it does make approximative assumptions as to the position of light sources and observer . for the virbhadra - ellis lens equation to be valid the spacetime must be asymptotically flat for @xmath3 and both observer and light sources must be at positions where @xmath4 is large ; moreover , one has to restrict to light sources close to the radial line opposite to the observer position , i.e. , to the case that there is only a small misalignment . ( the question of how one can free oneself from the latter assumption was addressed by dabrowski and schunck @xcite and by bozza @xcite . ) the lens equation to be discussed in the present paper is not restricted to the asymptotically flat case , and it makes no restriction on the position of light sources or observer . we consider an arbitrary spherically symmetric and static spacetime . for our purpose it will be advantageous to write the metric in the form @xmath5 here @xmath6 and @xmath7 are the standard coordinates on the sphere , @xmath8 ranges over @xmath9 and @xmath4 ranges over an open interval @xmath10\ , r_{\mathrm{min } } \ , , \ , r_{\mathrm{max } } \ , [ \;$ ] where @xmath11 . we assume that the functions @xmath12 , @xmath13 , and @xmath14 are strictly positive and ( at least piecewise ) differentiable on the interval @xmath10\ , r_{\mathrm{min } } \ , , \ , r_{\mathrm{max } } \ , [ \;$ ] . as the lightlike geodesics are not affected by the conformal factor @xmath15 ( apart from their parametrizations ) , the lens equation will depend on the metric coefficients @xmath16 and @xmath17 only . we will see below that many qualitative features of the lens equation are determined by the coefficient @xmath17 alone . for introducing our lens equation we have to fix two radius values @xmath0 and @xmath1 between @xmath18 and @xmath19 . the index @xmath20 stands for `` observer '' , the index @xmath13 stands for `` source '' . we think of an observer at @xmath21 , @xmath22 , @xmath23 . it is our goal to determine the appearance , on the observer s sky , of static light sources distributed on the sphere @xmath24 . by symmetry , we may restrict to the plane @xmath23 . we consider past - oriented lightlike geodesics that start at time @xmath25 at the observer and terminate , at some time @xmath26 which depends on the geodesic , somewhere on the sphere @xmath27 . to each of those light rays we assign the angle @xmath28 , measured at the observer between the ray s tangent and the direction of @xmath29 , and the angle @xmath30 , swept out by the azimuth coordinate along the ray on its way from the observer to the source , see figure [ fig : lensmap ] . the desired lens equation is an equation of the form @xmath31 which relates image positions on the observer s sky , given by @xmath28 , to source positions in the spacetime , given by @xmath30 modulo @xmath32 . we restrict @xmath28 to values between @xmath33 and @xmath34 ; then @xmath35 can be viewed as a colatitude coordinate on the observer s celestial sphere . by symmetry , @xmath36 must be equivalent to @xmath37 . for a given angle @xmath28 , neither existence nor uniqueness of an angle @xmath30 with @xmath31 is guaranteed . existence fails if the respective light ray never meets the sphere @xmath27 ; uniqueness fails if it meets this sphere several times . in the latter case the observer sees two or more images of light sources at @xmath1 at the same point on the sky , one behind the other . we will refer to images which are covered by other images as to `` hidden images '' . the lens equation can be solved for @xmath30 , thereby giving a _ lens map _ @xmath38 , only if hidden images do not exist ( or are willfully ignored ) . to work out the lens equation we have to calculate the lightlike geodesics in the plane @xmath23 of the metric ( [ eq : g ] ) , which is an elementary exercise . as a conformal factor has no influence on the lightlike geodesics ( apart from their parametrization ) , they are solutions of the euler - lagrange equations of the lagrangian @xmath39 , i.e. @xmath40 @xmath41 @xmath42 where an overdot denotes differentiation with respect to the curve parameter @xmath43 . as an aside we mention that , by ( [ eq : elr ] ) , a circular light ray exists at radius @xmath44 if and only if @xmath45 . comparing this condition with the equivalent but less convenient eq . ( 33 ) in atkinson s article @xcite shows that it is advantageous to write the metric in the form ( [ eq : g ] ) . the relevance of circular light rays in view of lensing was discussed by hasse and perlick @xcite , also see claudel , virbhadra and ellis @xcite for related results . to get the past - oriented light ray that starts at time @xmath25 at the observer in the direction determined by the angle @xmath28 we have to impose the initial conditions @xmath46 @xmath47 @xmath48 for each @xmath28 , the initial value problem ( [ eq : elr ] ) , ( [ eq : elphi ] ) , ( [ eq : elt]),([eq : inr ] ) , ( [ eq : inphi ] ) , ( [ eq : int ] ) has a unique maximal solution @xmath49 where @xmath43 ranges from 0 up to some @xmath50 . every image on the oberver s sky of a light source at @xmath1 corresponds to a pair @xmath51 such that @xmath52 with some parameter value @xmath53 \ , 0 , s_{\mathrm{max } } ( \theta ) \ , [ \:$ ] . in other words , we get the desired lens equation @xmath31 if we eliminate @xmath54 from the two equations ( [ eq : el ] ) . we get an explicit expression for the lens equation , and for the travel time @xmath54 , by writing the functions @xmath55 and @xmath56 in terms of integrals . from the constant of motion @xmath57 we find , with the help of ( [ eq : elphi ] ) , ( [ eq : elt]),([eq : inphi ] ) , ( [ eq : int ] ) , @xmath58 if @xmath59 does not change sign , integration of ( [ eq : rs ] ) yields @xmath60 with @xmath61 known , @xmath62 is determined by integrating ( [ eq : elphi ] ) with ( [ eq : inphi ] ) , @xmath63 ( [ eq : intphis ] ) can be rewritten as an integral over @xmath4 , with @xmath64 substituted from ( [ eq : rs ] ) . this gives us the lens equation in the form @xmath65 if @xmath59 changes sign , ( [ eq : intrs ] ) has to be replaced by a piecewise integration . similarly , the substitution from the @xmath43-integration in ( [ eq : intphis ] ) to an @xmath4-integration must be done piecewise . in this case , the lens equation is not of the form ( [ eq : phi ] ) ; in particular , it is not guaranteed that the lens equation can be solved for @xmath30 . in any case , we get exact integral expressions for the lens equation , and for the travel time @xmath54 , from which all relevant lensing features can be determined in a way that is not less explicit than the quasi - newtonian approximation formalism . this will be demonstrated by two examples in section [ sec : examples ] . in subsection [ subsec : mono ] we treat a particularly simple example where the metric coefficients @xmath17 and @xmath16 are analytic and the integral ( [ eq : intrs ] ) can be explicitly calculated in terms of elementary functions . in this case it suffices to calculate ( [ eq : intrs ] ) for arbitrarily small @xmath43 to get the whole function @xmath66 by maximal analytic extension ; i.e. , in this case it is not necessary to determine the points where @xmath59 changes sign and to perform a piecewise integration . in the first part of this section we want to discuss for which values of @xmath28 the lens equation @xmath67 admits a solution . in other words , we want to determine which part of the observer s sky is covered by the light sources distributed at @xmath24 . we restrict to the case @xmath68 . ( the results for the case @xmath69 follow immediately from our discussion ; we just have to make a coordinate transformation @xmath70 and , correspondingly , to change @xmath28 into @xmath71 . the case @xmath72 can be treated by a limit procedure . ) for a light ray with one end - point at @xmath0 and the other at @xmath1 the right - hand side of ( [ eq : rs ] ) must be non - negative for all @xmath4 between @xmath0 and @xmath1 . this condition restricts the possible values of @xmath28 by @xmath73 where @xmath74 note that our assumptions guarantee that this infimum is strictly positiv , @xmath75 . furthermore , a light ray with @xmath76 can arrive at @xmath1 only if it passes through a minimal radius value @xmath77 . as ( [ eq : rs ] ) requires @xmath78 , this can be true only if @xmath79 where @xmath80 @xmath81 . so in general the light sources at @xmath1 cover on the observer s sky a disk of angular radius @xmath82 around the pole @xmath83 and , if @xmath84 , in addition a ring of angular width @xmath85 around the pole @xmath86 , see figure [ fig : sky ] . the two domains join if @xmath87 . we see that the allowed values of @xmath28 are determined by the metric coefficient @xmath14 alone . we will now demonstrate that @xmath14 alone also determines the occurrence or non - occurrence of hidden images . hidden images occur if a light ray from @xmath0 intersects the sphere @xmath24 at least two times ; between these two intersections it must pass through a maximal radius @xmath88 which , by ( [ eq : rs ] ) , has to satisfy @xmath89 . such a radius @xmath90 exists for all @xmath28 with @xmath91 where @xmath92 as @xmath28 is restricted by @xmath73 , hidden images can not occur if @xmath93 . the latter condition is satisfied in asymptotically flat spacetimes , where @xmath94 for @xmath95 , if we choose @xmath1 sufficiently large . this is the reason why in the more special situation of the virbhadra - ellis lens equation @xcite hidden images can not occur . in the rest of this section we discuss the question of multiple imaging and the occurrence of einstein rings . for a light source at @xmath24 , @xmath96 , @xmath23 with @xmath97 , images on the observer s sky are in one - to - one correspondence with solutions @xmath28 of the equation @xmath98 with @xmath99 . we call the integer @xmath100 the `` winding number '' of the corresponding light ray . an image with @xmath101 is called `` primary '' and an image with @xmath102 is called `` secondary '' . images with other values of @xmath100 correspond to light rays that make at least one full turn and have been termed `` relativistic '' by virbhadra and ellis @xcite . note that different images of a light source may have the same winding number . if we send @xmath103 to 0 or to @xmath104 , solutions @xmath28 of equation ( [ eq : mult ] ) with @xmath105 come in pairs @xmath106 . by spherical symmetry , every such pair gives rise to an einstein ring . there are as many einstein rings as the equation @xmath107 admits solutions with positive integers @xmath108 . even integers @xmath108 correspond to einstein rings of the source at @xmath109 , and odd integers @xmath108 correspond to einstein rings of the source at @xmath110 . to each solution @xmath51 of the lens equation we can assign redshift , travel time , apparent brightness and image distortion . the general redshift formula for static metrics ( see , e.g. , straumann @xcite , p. 97 ) specified to metrics of the form ( [ eq : g ] ) says that the redshift @xmath111 is given by @xmath112 if the observer s worldline is a @xmath8-line at @xmath21 and the source s worldline is a @xmath8-line at @xmath27 . in our situation @xmath0 and @xmath1 are fixed , so the redshift is a constant . recall that @xmath51 is a solution of the lens equation if and only if there is a parameter @xmath54 such that the equations ( [ eq : el ] ) hold . this assigns a travel time @xmath54 to each solution @xmath51 of the lens equation . if there are no hidden images , the equation @xmath113 gives @xmath54 as a single - valued function of @xmath28 . quite generally , determination of the angular diameter distance requires solving the sachs equations for the optical scalars along lightlike geodesics , see e.g. schneider , ehlers and falco @xcite . for the schwarzschild metric , this has been explicitly worked out by dwivedi and kantowski @xcite . their method easily carries over to arbitrary spherically symmetric and static spacetimes as was demonstrated by dyer @xcite . in what follows we give a reformulation of these results in terms of our lens equation . to that end we fix a solution @xmath114 of the lens equation and thereby a ( past - oriented ) light ray from the observer at @xmath0 to a light source at @xmath1 . around this ray , we consider an infinitesimally thin bundle of neighboring rays , with vertex at the observer . the angular diameter distance is defined as the square - root of the ratio between the cross - sectional area of this bundle at the light source and the opening solid angle at the observer . owing to the symmetry of our situation there are two preferred spatial directions perpendicular to the ray : a radial direction ( along a meridian on the observer s sky ) and a tangential direction ( along a circle of equal latitude on the observer s sky ) . therefore , the angular diameter distance naturally comes about as a product of a radial part and a tangential part . to calculate the radial part , we consider the infinitesimally neighboring ray which corresponds to an infinitesimally neighboring solution @xmath115 of the lens equation , i.e. @xmath116 and @xmath117 satisfy @xmath118 we define the _ radial angular diameter distance _ as @xmath119 with @xmath120 given by figure [ fig : lensmap ] , i.e. , @xmath121 measures , in the direction perpendicular to the original ray , how far the neighboring ray is away . by ( [ eq : elphi ] ) , ( [ eq : inphi ] ) and ( [ eq : rs ] ) , @xmath120 must satisfy @xmath122 with @xmath120 given by ( [ eq : psi ] ) and @xmath123 given by ( [ eq : dphidtheta ] ) , @xmath124 is determined by ( [ eq : dr ] ) for every solution @xmath114 of the lens equation . note that @xmath124 is singular at those solutions of the lens equation where @xmath125 has a zero . if the lens equation can be solved for @xmath30 , we may view @xmath124 as a ( single - valued ) function of @xmath28 . to calculate the tangential part we consider an infinitesimally neighboring light ray that results by applying a rotation around the axis @xmath22 , @xmath23 . such rotations are generated by the killing vector field @xmath126 . at points with @xmath23 , this killing vector field takes the form @xmath127 . hence , if we rotate by an infinitesimal angle @xmath128 , the neighboring ray intersects the sphere @xmath24 at a distance @xmath129 from the original ray . relating this distance to the angle @xmath130 between the two rays at the observer gives the _ tangential angular diameter distance _ @xmath131 by this equation , @xmath132 is uniquely determined for each solution @xmath51 of the lens equation . again , @xmath132 may be viewed as a function of @xmath28 if the lens equation can be solved for @xmath30 . @xmath124 and @xmath132 together give the ( averaged ) _ angular diameter distance _ or _ area distance _ @xmath133 note that both @xmath124 and @xmath132 may be negative . images with @xmath134 are said to have _ even parity _ and images with @xmath135 are said to have _ odd parity_. images with odd parity show the neighborhood of the light source side - inverted in comparison to images with even parity . a solution @xmath51 of the lens equation is called a _ radial critical point _ if @xmath136 and a _ tangential critical point _ if @xmath137 . the latter condition is equivalent to @xmath138 and @xmath105 , i.e. , to the occurrence of an einstein ring . note that ( radial and tangential ) critical points come in pairs , @xmath114 and @xmath139 . every such pair corresponds to a circle of equal latitude on the observer s sky which may be called a ( radial or tangential ) _ critical circle _ , as in the quasi - newtonian approximation formalism , see schneider , ehlers and falco @xcite , p. 233 in the quasi - newtonian formalism one usually introduces the inverse magnification factors @xmath140 and @xmath141 as substitutes for @xmath124 and @xmath142 . in our situation , where there is no flat background metric , not even asymptotically , the magnification factors can not be defined in a reasonable way , but working with @xmath124 and @xmath142 is completely satisfactory . in arbitrary spacetimes , the angular diameter distance @xmath143 is related to the ( uncorrected ) _ luminosity distance _ @xmath144 by the universal formula @xmath145 , see , e.g. , schneider , ehlers and falco @xcite , eq . ( 3.80 ) . with the redshift @xmath111 given by ( [ eq : z ] ) and the angular diameter distance @xmath143 given by ( [ eq : d ] ) , we find @xmath146 for an isotropically radiating light source with bolometric luminosity @xmath147 , the total flux at the observer is @xmath148 , see again schneider , ehlers and falco @xcite , eq . ( 3.79 ) . hence , if we distribute standard candles at @xmath24 , their apparent brightness on the observer s sky is proportional to @xmath149 . @xmath150 and @xmath132 immediately give the apparent distortion of images . for the sake of illustration , we may think of small spheres , with infinitesimal diameter @xmath151 , distributed with their centers at @xmath24 . by definition of @xmath124 and @xmath132 , each solution @xmath152 of the lens equation corresponds to an elliptical image of such a sphere on the observer s sky , with the radial ( meridional ) diameter of the ellipse equal to @xmath153 and with the tangential ( latitudinal ) diameter of the ellipse equal to @xmath154 . thus , we may use the _ ellipticity _ @xmath155 as a measure for image distortion . we consider the metric @xmath156 where @xmath157 is a positive constant . for @xmath158 , this is just minkowski spacetime in spherical coordinates . for @xmath159 , there is a deficit solid angle and a singularity at @xmath160 ; the plane @xmath161 , @xmath23 has the geometry of a cone . similarly , for @xmath162 there is a surplus solid angle and a singularity at @xmath163 . for @xmath164 , the metric is non - flat . the einstein tensor has non - vanishing components @xmath165 , so the weak energy condition is satisfied ( without a cosmological constant ) if and only if @xmath166 . in that case it was shown by barriola and vilenkin @xcite that the metric may be viewed as a model for the spacetime around a monopole resulting from breaking a global @xmath167 symmetry . to within the weak - field approximation , basic features of lensing by such a monopole were discussed in the original paper by barriola and vilenkin @xcite and also by durrer @xcite . in what follows we give a detailed account in terms of our exact lens equation . note that the virbhadra - ellis lens equation @xcite is not applicable to this case , at least not without modification , because for @xmath164 the spacetime is not asymptotically flat in the usual sense . comparison of ( [ eq : monog ] ) with ( [ eq : g ] ) shows that the metric coefficients are given by @xmath168 on the interval @xmath169 . with these metric coefficients , the integrals ( [ eq : intrs ] ) and ( [ eq : intphis ] ) can be calculated in an elementary fashion , yielding the solution to the initial value problem in the form @xmath170 @xmath171 for @xmath172 , @xmath43 ranges from 0 to @xmath173 , so @xmath174 ranges from 0 to @xmath175 . eliminating @xmath54 from the two equations ( [ eq : el ] ) gives the lens equation , @xmath176 which is to be considered on the domain @xmath177 we restrict to the case that the integer @xmath178 defined by @xmath179 is odd . the lens equation is plotted for the case @xmath180 in figure [ fig : mono1 ] and for the case @xmath69 in figure [ fig : mono2 ] . ( for producing the pictures we have chosen @xmath157 such that @xmath181 . ) in either case we find that there are @xmath178 einstein rings . for a light source at @xmath182 with @xmath183 there are @xmath178 images if @xmath184 ( non - shaded regions in figures [ fig : mono1 ] and [ fig : mono2 ] ) and @xmath185 images otherwise ( shaded regions in figures [ fig : mono1 ] and [ fig : mono2 ] ) . in the following we concentrate on the case @xmath180 . then the lens equation can be solved for @xmath30 , giving a lens map @xmath186 on the domain @xmath187 , i.e. , in the notation of figure [ fig : sky ] we have @xmath87 and @xmath188 . in the case @xmath158 ( flat spacetime ) , the lens map can be continuously extended into the point @xmath189 ( and by periodicity onto the full circle , @xmath190 mod @xmath32 ) . for @xmath164 the ray with @xmath189 can not pass through the singularity at @xmath163 , so the lens map is not to be extended beyond the open interval @xmath191-\pi,\pi \ , [ \ : \,$ ] . on this interval , @xmath30 increases monotonously from @xmath192 to @xmath193 , see figure [ fig : mono1 ] . thus , the range of the lens map is independent of @xmath0 and @xmath1 . the ratio @xmath194 influences the shape of the graph of the lens map in the following way . for @xmath195 it becomes a straight line , @xmath196 . for @xmath197 it becomes a broken straight line , @xmath198 for @xmath199 and @xmath200 for @xmath201 . note that linearity of the lens map implies that the angular distance on the observer s sky between consecutive images is the same for all light sources . as @xmath30 is a ( single - valued ) function of @xmath28 , so are all observables . by evaluating the formulas derived in section [ sec : obs ] for the case at hand we find @xmath202 @xmath203 @xmath204 which gives @xmath205 and @xmath206 as functions of @xmath28 via ( [ eq : dlum ] ) and ( [ eq : epsilon ] ) . the observables are plotted in figures [ fig : mono3 ] , [ fig : mono4 ] and [ fig : mono5 ] . the metric @xmath207 where @xmath208 is a positive constant , is an example for a traversable wormhole of the morris - thorne class , see box 2 in morris and thorne @xcite . it was investigated , already in the 1970s , by ellis @xcite who called it a `` drainhole '' . lensing in the ellis spacetime was discussed , in a scattering formalism assuming that observer and light source are at infinity , by chetouani and clment @xcite . in the following we give a detailed account of lensing in this spacetime with the help of our lens equation . by comparison of ( [ eq : wormg ] ) with ( [ eq : g ] ) we find @xmath209 where the radius coordinate @xmath4 ranges from @xmath210 to @xmath211 . ( we do _ not _ identify the region where @xmath4 is positive with the region where @xmath4 is negative . ) the function @xmath14 has a minimum at @xmath163 , thereby indicating the existence of circular light rays at the neck of the wormhole . in the following we consider the case that observer and light sources are on different sides of the neck of the wormhole , @xmath212 . as a first step , we determine for which angles @xmath28 the lens equation admits a solution , recall figure [ fig : sky ] . in the case at hand , the angles @xmath82 and @xmath213 defined by ( [ eq : ci ] ) and ( [ eq : cii ] ) are given by @xmath214 and @xmath215 . hence , the lens equation admits a solution for all angles @xmath28 with @xmath216 and @xmath217 light sources distributed at @xmath24 illuminate a disk of angular radius @xmath218 on the observer s sky . the apparent rim of the disk corresponds to light rays that spiral asymptotically towards @xmath163 . as the constant @xmath219 defined by ( [ eq : c ] ) satisfies @xmath220 , there are no hidden images , i.e. , the lens equation can be solved for @xmath30 . with @xmath28 restricted by ( [ eq : wormdelta ] ) we read from ( [ eq : rs ] ) that @xmath59 has no zeros along a ray that starts at @xmath221 and passes through @xmath24 . hence , ( [ eq : intrs ] ) gives us @xmath55 and thereby the travel time in terms of an elliptic integral , @xmath222 similarly , ( [ eq : phi ] ) gives us @xmath30 as a ( single - valued ) function of @xmath28 in terms of an elliptic integral and thereby the lens equation , @xmath223 see figure [ fig : worm1 ] . as @xmath30 increases monotonously from @xmath224 to @xmath173 on the domain @xmath225- \delta _ i , \ , \delta _ i\ , [ \,$ ] , there are infinitely many einstein rings whose angular radii converge to @xmath82 . if we fix a light source at @xmath96 with @xmath97 , figure [ fig : worm1 ] gives us infinitely many images which can be characterized in the following way . for every @xmath99 there is a unique @xmath226 - \delta _ i , \ , \delta _ i \ , [ \:$ ] such that @xmath227 , and @xmath228 for @xmath229 . as @xmath30 is a ( single - valued ) function of @xmath28 , so are all observables . by evaluating the formulas derived in section [ sec : obs ] for the case at hand we find @xmath230 @xmath231 @xmath232 @xmath233 which gives @xmath205 and @xmath206 as functions of @xmath28 via ( [ eq : dlum ] ) and ( [ eq : epsilon ] ) . the observables are plotted in figures [ fig : worm2 ] , [ fig : worm3 ] , [ fig : worm4 ] and [ fig : worm5 ] . as there are infinitely many einstein rings whose angular radii converge to @xmath82 , the tangential angular diameter distance must have infinitely many zeros that converge to @xmath82 . this is difficult to show in a picture unless one transforms @xmath28 into a new coordinate @xmath234 that goes to infinity for @xmath235 . if one chooses a logarithmic transformation formula , as has been done in figure [ fig : worm3 ] , one sees that in terms of the new coordinate @xmath234 the einstein rings become equidistant . this feature is not particular to the ellis wormhole ; @xmath236 as a function of @xmath28 always diverges logarithmically when a circular light ray at a radius @xmath44 with @xmath237 and @xmath238 is approached . the proof can be taken from bozza @xcite . one may also treat the case that observer and light sources are on the same side of the neck of the wormhole . if the observer is closer to the neck than the light sources , @xmath239 or @xmath240 , the results are quite similar to the case above . the only difference is in the fact that the light sources appear as a disk of radius _ bigger _ than @xmath241 , i.e. , the disk covers more than one hemisphere . if the observer is farther from the neck than the light sources , @xmath242 or @xmath243 , there are hidden images . i.e. , one does not get a single - valued lens map @xmath244 . the qualitative features of lensing by an ellis wormhole are very similar to the qualitative features of lensing by a schwarzschild black hole . the radii @xmath245 , @xmath246 , @xmath247 in the ellis case correspond respectively to the radii @xmath248 , @xmath249 , @xmath247 in the schwarzschild case . as a matter of fact , we encounter these same features whenever the function @xmath14 has one minimum , @xmath45 and @xmath250 , and no other extrema on the considered interval . the lens equation and the formulas for redshift , travel time and radial angular diameter distance used in this paper refer to lightlike geodesics of the @xmath251-dimensional metric @xmath252 , independently of whether this metric results from restricting a spherically symmetric and static spacetime to the equatorial plane . therefore , these results apply equally well to the plane @xmath253 of a cylindrically symmetric and static spacetime and , of course , to genuinely @xmath251-dimensional spacetimes with the assumed symmetries such as the btz black hole . ( for lightlike and timelike geodesics in the metric of the btz black hole see cruz , martnez and pea @xcite . ) e.g. , the metric @xmath254 results not only by restricting the spacetime of a barriola - vilenkin monopole to the plane @xmath23 , as discussed in subsection [ subsec : mono ] , but also by restricting the cylindrically symmetric and static metric @xmath255 to the plane @xmath256 . the latter metric is well - known to describe the spacetime around a static string , see vilenkin @xcite , gott @xcite and hiscock @xcite , and was investigated in detail already by marder @xcite . hence , if re - interpreted appropriately , the results of subsection [ subsec : mono ] apply to light rays in the plane perpendicular to a static string . for treating _ all _ light rays in a cylindrically symmetric and static spacetime one may introduce a modified lens equation , replacing the sphere @xmath24 with a cylinder .
|
lensing in a spherically symmetric and static spacetime is considered , based on the lightlike geodesic equation without approximations . after fixing two radius values @xmath0 and @xmath1 ,
lensing for an observation event somewhere at @xmath0 and static light sources distributed at @xmath1 is coded in a lens equation that is explicitly given in terms of integrals over the metric coefficients .
the lens equation relates two angle variables and can be easily plotted if the metric coefficients have been specified ; this allows to visualize in a convenient way all relevant lensing properties , giving image positions , apparent brightnesses , image distortions , etc .
two examples are treated : lensing by a barriola - vilenkin monopole and lensing by an ellis wormhole .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the basic paradigm of many body physics is to use analytical and numerical tools to investigate the low - lying states and in particular the ground state properties of a system governed by a given hamiltonian . at very low temperatures , when thermal fluctuations are dominated by quantum fluctuations , quantum phase transitions can occur due to the change of character of the ground state @xcite . what is exactly meant by `` character '' has been investigated in numerous works @xcite , @xcite , @xcite , especially in recent years after the discovery of exact measures @xcite of entanglement(purely quantum mechanical correlations ) . for example , it has been a matter of debate whether a quantum phase transition is always accompanied by a divergence of some property in entanglement of the ground state wave function . + unfortunately , except for a few exactly solvable examples , the task of finding the exact ground state of a given hamiltonian is notoriously difficult . as always in dealing with difficult problems , one way round the difficulty is to investigate the inverse problem , that is to start from states with pre - determined properties and investigate quantum phase transitions which occur by smoothly changing some continuous parameters of these states . the suitable formalism for following this path is the matrix product formalism @xcite , @xcite , @xcite which in recent years has been followed in constructing various models of interacting spins @xcite . in this paper we want to take one step in this direction and in particular , we want to construct as concrete models , ladders of spin one - half particles and see what happens to the entanglement between various spins when the system undergoes a phase transition . we construct general class of models having a number of discrete and continuous symmetries . these are symmetry under the exchange of legs of the ladder , symmetry under spin flip , and symmetry under parity ( left - right reflection of the ladder ) in addition to a continuous symmetry , namely rotation of spins around the @xmath1 axis or all three axes ( full rotational symmetry ) . + in these models we calculate the entanglement of one rung with the rest of the ladder as measured by the von neumann entropy of the state of the rung and the entanglement of the two spins of a rung with each other as be measured by the concurrence of the same state . + we will see that in all these models quantum phase transitions occur at a critical point of the coupling constant , and this point is where a rung of the ladder becomes completely disentangled from the rest of the ladder and the two spins of the rung become fully entangled with each other . the derivatives of these two types of entanglement are also divergent . + we should stress here that we are using the term phase transition in a wider sense than usual @xcite , that is we call any discontinuity in an observable quantity ( i.e. a two point correlation function ) , a phase transition . + the structure of this paper is as follows . in section [ mps ] , we review the formalism of matrix product states ( mps ) with emphasis on the symmetry properties of such states . in section [ ladder ] , we specify the mps construction to ladders of spin @xmath0 particles and set the general ground for construction of concrete models . in this same section we construct multi - parameter families of models which have specific symmetries . in section [ stateproperties ] , we study in detail the properties of the constructed states and in particular calculate the exact correlation functions of spins on the rungs . we also investigate the connection between quantum phase transitions and divergence of entanglement properties in these models . finally we derive in the appendix , the hamiltonian which governs the interaction of these models for which the state we have constructed is the exact ground state . first let us review the basics of matrix product states @xcite . consider a homogeneous ring of @xmath2 sites , where each site describes a @xmath3level state . the hilbert space of each site is spanned by the basis vectors @xmath4 . a state @xmath5 is called a matrix product state if there exists @xmath6 dimensional complex matrices @xmath7 such that @xmath8 where @xmath9 is a normalization constant given by @xmath10 and @xmath11 here we are restricting ourselves to translationally invariant states , by taking the matrices to be site - independent . for open boundary conditions , any state can be mps - represented , provided that we allow site - dependent matrices @xmath12 , where @xmath13 denotes the position of the site @xcite . the mps representation ( [ mat ] ) is not unique and a transformation such as @xmath14 leaves the state invariant . in view of this we can find the conditions on the matrices which impose discrete symmetries on the state . the state @xmath15 is reflection symmetric if there exists a matrix @xmath16 such that @xmath17 where @xmath18 is the transpose of @xmath19 and time - reversal invariant if there exist a matrix @xmath20 such that @xmath21 . + let @xmath22 be any local operator acting on a single site . then we can obtain the one - point function on site @xmath13 of the chain @xmath23 as follows : @xmath24 where @xmath25 in the thermodynamic limit @xmath26 , equation ( [ 1point ] ) gives @xmath27 where we have used the translation invariance of the model and @xmath28 is the eigenvalue of @xmath29 with the largest absolute value and @xmath30 and @xmath31 are the right and left eigenvectors corresponding to this eigenvalue , normalized such that @xmath32 . here we are assuming that the largest eigenvalue of @xmath29 is non - degenerate . + the n - point functions can be obtained in a similar way . for example , the two - point function @xmath33 can be obtained as @xmath34 where @xmath35 . note that this is a formal notation which allows us to write the n - point functions in a uniform way , it does not require that @xmath29 is an invertible matrix . in the thermodynamic limit the two point function turns out to be @xmath36 for large distances @xmath37 , this formula reduces to @xmath38 where @xmath39 is the second largest eigenvalue of @xmath29 for which the matrix element @xmath40 is non - vanishing and we have assumed that the eigenvectors of @xmath29 have been normalized , i.e. @xmath41 . thus the correlation length is given by @xmath42 any level crossing in the largest eigenvalue of the matrix @xmath29 signals a possible quantum phase transition . also , due to ( [ corr ] ) , any level crossing in the second largest eigenvalue of @xmath29 implies the correlation length of the system has undergone a discontinuous change . here we are using a broader definition of quantum phase transition , that is we call any non - analytical behavior of a macroscopic property , a quantum phase transition @xcite . of course in the models that we construct we observe a more direct change of observable physical properties , namely in one regime we have correlations between spin operators on different sites and in the other we have no such correlation . consider now a local symmetry operator @xmath43 acting on a site as @xmath44 where summation convention is being used . @xmath43 is a @xmath45 dimensional unitary representation of the symmetry . a global symmetry operator @xmath46 will then change this state to another matrix product state @xmath47 where @xmath48 a sufficient but not necessary condition for the state @xmath15 to be invariant under this symmetry is that there exist an operator @xmath49 such that @xmath50 thus @xmath43 and @xmath49 are two unitary representations of the symmetry , respectively of dimensions @xmath45 and @xmath6 . in case that @xmath43 is a continuous symmetry with generators @xmath51 , equation ( [ symm ] ) , leads to @xmath52,\ ] ] where @xmath51 and @xmath53 are the @xmath3 and @xmath54dimensional representations of the lie algebra of the symmetry . equations ( [ symm ] ) and ( [ symmalg ] ) will be our guiding lines in defining states with prescribed symmetries . given a matrix product state , the reduced density matrix of @xmath13 consecutive sites is given by @xmath55 the null space of this reduced density matrix includes the solutions of the following system of equations @xmath56 given that the matrices @xmath57 are of size @xmath58 , there are @xmath59 equations with @xmath60 unknowns . since there can be at most @xmath59 independent equations , there are at least @xmath61 solutions for this system of equations . thus for the density matrix of @xmath13 sites to have a null space it is sufficient that the following inequality holds @xmath62 let the null space of the reduced density matrix be spanned by the orthogonal vectors @xmath63 . then we can construct the local hamiltonian acting on @xmath13 consecutive sites as @xmath64 where @xmath65 s are positive constants . these parameters together with the parameters of the vectors @xmath66 inherited from those of the original matrices @xmath57 , determine the total number of coupling constants of the hamiltonian . if we call the embedding of this local hamiltonian into the sites @xmath67 to @xmath68 by @xmath69 then the full hamiltonian on the chain is written as @xmath70 the state @xmath15 is then a ground state of this hamiltonian with vanishing energy . the reason is as follows : @xmath71 where @xmath72 is the reduced density matrix of sites @xmath67 to @xmath68 and in the last line we have used the fact that @xmath73 is constructed from the null eigenvectors of @xmath74 for @xmath13 consecutive sites . given that @xmath75 is a positive operator , this proves the assertion . we now specify the above generalities to a ladder of spin @xmath0 particles . the ladder consists of @xmath2 rungs and obeys periodic boundary conditions . to each rung of the ladder we associate four matrices @xmath76 and @xmath77 respectively pertaining to the local states @xmath78 and @xmath79 . here we are using the qubit notation which corresponds to the spin notation as @xmath80 and @xmath81 the first and the second indices refer respectively to the states of sites on the legs 1 and 2 as shown in figure ( 1 ) . the rungs are numbered from 1 to n with a periodic boundary condition . the operator @xmath82 refers to the pauli operator @xmath83 on leg number @xmath84 and the rung number @xmath13 . the total spin operator on a rung at site @xmath13 is denoted by @xmath85 . let us label the legs of the ladder by @xmath86 and @xmath87 as in figure ( 1 ) . in each @xmath88 the first and the second indices refer respectively to legs number 1 and 2 . refers to the pauli operator @xmath83 on rung @xmath13 and leg number @xmath89 . , width=302,height=94 ] [ ladder ] any operator corresponding to leg number @xmath84 is designated with a superscript @xmath89 . with these conventions , one can easily use ( [ mpsop ] ) and write down the mps operator corresponding to an observable . for example , for the magnetization in the @xmath90 and @xmath1 directions in rung 1 , we have respectively @xmath91 and @xmath92 for the total magnetization in the @xmath1 direction in a rung we have @xmath93 other mps operators can be obtained in a similar way . in this paper we restrict the dimensions of our matrices to @xmath94 , and demand that our models have time reversal symmetry so that the matrices @xmath88 are chosen to be real . we will be looking for models which have rotational symmetry in the @xmath95 plane of spin space . thus we require that there be a matrix @xmath96 such that @xmath97=a_{00 } , \ \ \ \ [ { \cal t}_z , a_{11}]=-a_{11 } , \ \ \ \ [ { \cal t}_z , a_{01}]=[{\cal t}_z , a_{10}]=0.\ ] ] it is an easy exercise to show that the solution of these equations is @xmath98 where @xmath99 is found to be @xmath100 and we have excluded the solutions with @xmath101 or @xmath102 , which lead to trivial uncorrelated states . hereafter we use the freedom in re - scaling the matrices ( without changing the matrix product state ) to set the parameter @xmath103 . let us now consider extra discrete symmetries in addition to the above continuous symmetry . these are as follows : + * a : spin flip symmetry * represented by a matrix @xmath104 , such that @xmath105 where @xmath106 and @xmath107 means @xmath108 . this symmetry imposes the following condition on the solution ( [ tzsolution ] ) @xmath109 where @xmath104 is found to be @xmath110 . * b : symmetry under the exchange of legs of the ladder * represented by a matrix @xmath111 such that @xmath112 where @xmath113 . this symmetry imposes the following condition on ( [ tzsolution ] ) @xmath114 where @xmath111 is found to be @xmath115 . * c : parity * or a left - right symmetry , represented by a matrix @xmath116 , such that @xmath117 with @xmath118 . + this symmetry imposes the condition @xmath119 on ( [ tzsolution ] ) where @xmath116 is found to be @xmath120 . + thus when we impose any of these discrete symmetries we are left with two three - parameter families of models , each family being distinguished by the discrete parameter @xmath121 , @xmath122 or @xmath123 . + it is now readily seen that imposing any two of these symmetries makes the model symmetric under the third one too . a model which has all three symmetries is defined by the following set of matrices : @xmath124 such matrices satisfy ( [ y ] ) with @xmath125 . thus equation [ z23solution ] defines four two - parameter families of models on spin ladders which have @xmath126 symmetry in addition to three types of @xmath127 symmetries . the families are distinguished by the pair of discrete parameters @xmath128 . + * d : full rotational symmetry * let us now see if we can construct models which have full @xmath129 symmetry , symmetry under rotations in the spin space . to this order we note that to have full rotational symmetry , the matrices defined by @xmath130 and @xmath131 should respectively transform like the spin @xmath86 and spin @xmath132 representations of the @xmath133 algebra , that is , we should have @xmath134=0 , \h a = x,\ y,\ z,\ ] ] and @xmath135=m \ b_{l , m } , \h [ { \cal t}_{\pm } , b_{l , m}]= \sqrt{2-m(m\pm 1)}b_{l , m\pm 1 } , \h l=1 , \ m=-1,0,1,\ ] ] where @xmath136 form the two dimensional representation of the @xmath133 algebra . it is well known that the matrices @xmath137 transform like a vector under the adjoint action of @xmath129 . the matrix @xmath138 should also be a multiple of identity . thus we should set @xmath139 , and satisfy the following equations @xmath140 this puts the constraints @xmath141 with the unique solution @xmath142 where @xmath143 is an arbitrary real parameter . this will give us a one - parameter family of models with full rotational symmetry and spin - flip symmetry @xmath144 * remark : * note that comparison of these parameters with the constraints ( [ xsolution ] , [ ysolution ] ) and ( [ pisolution ] ) shows that full rotational symmetry is compatible with spin - flip symmetry for arbitrary values of the parameter @xmath143 and compatible with parity or leg - exchange symmetries only for @xmath145 . + the model ( [ rotsolution ] ) has already been studied in @xcite . to see the correspondence with that work , we can collect the above matrices in a vector - valued matrix @xmath146 defined as @xmath147 . in view of our notation for spins @xmath148 and the notation of @xcite in which the single and the triplet states are denoted respectively by @xmath149 and @xmath150 ) becomes @xmath151 which modulo an overall constant is identical to the matrix given in @xcite . we now study the properties of the states constructed above . for simplicity we consider in detail only two general classes . the first class is defined by ( [ z23solution ] ) and has @xmath126 symmetry in addition to three @xmath127 symmetries , and the second class is defined by ( [ rotsolution ] ) which has full rotational symmetry in addition to one @xmath127 symmetry , the spin flip symmetry . as mentioned in the previous section , full rotational symmetry is compatible only with spin flip symmetry for generic values of the parameter @xmath143 and is compatible with the other two symmetries only when @xmath145 . in order to derive results which can be specialized to the two classes mentioned above we study in detail the properties of the state , defined by the equation ( [ xsolution ] ) . the matrices are now given by @xmath152 the matrix @xmath29 for this class has the following form @xmath153 with eigenvalues @xmath154 for @xmath155 , the largest eigenvalue is @xmath156 and for @xmath157 it is @xmath158 . hence the point @xmath101 is a point of phase transition . the right and left eigenvectors of @xmath29 are simply obtained and one can determine all the relevant quantities of the ground state in closed form , in straightforward way after some rather lengthy calculations . the reduced one - rung density matrix is obtained from ( [ rho ] ) : @xmath159 where @xmath160 . this matrix can be rewritten as @xmath161,\ ] ] or @xmath162,\ ] ] where we have used the notation introduced in equation ( [ polisha ] ) . + from this density matrix one can obtain a lot of information about the observables pertaining to a single rung . for example it is readily seen that the average magnetization at each single site and hence each single rung is zero , i.e. @xmath163 where @xmath164 is the spin of a rung , implying an anti - ferromagnetic state in which every single site is in a completely mixed states . it is also seen that @xmath165 where @xmath166 is any unit vector in the @xmath95 plane . + defining the total spin of a single rung as @xmath167 , we find from the above result that @xmath168 thus for @xmath169 , each rung will be in a mixture of spin one states , but for @xmath101 , the spin one and spin zero multiplets can mix , depending on the value of @xmath170 and @xmath171 and @xmath121 . the entanglement of this rung with the rest of the lattice is measured by the von - neumann entropy of this state , defined as @xmath172 . from the eigenvalues of the one - rung density matrix , @xmath173 we readily find @xmath174.\ ] ] as @xmath169 @xmath175 . therefore in this limit each rung is still entangled with the rest of the ladder . this means that a rung is not in a pure state and the state of the ladder is not a product of single rung states . + the entanglement of the two spins of a single rung with each other is measured by the concurrence @xcite of this state which for the density matrix ( [ rhosolution ] ) is given by @xmath176 where @xmath177 is the largest eigenvalue of the matrix @xmath74 . a careful analysis of the eigenvalues shows that @xmath178 as @xmath169 , @xmath179 which means that although each single rung is not a pure state , it is a separable state . in fact these can also be verified directly by looking at the one - rung density matrix in this limit . from ( [ rhosolutionketbra ] ) we have @xmath180 finally we calculate the correlation functions of different components of spins of the rung as a function of the distance between the rungs . we find from ( [ 2pointthermodynamiclimit ] ) that @xmath181 with a longitudinal correlation length @xmath182 and @xmath183 with a transverse correlation length @xmath184 it is seen that the longitudinal correlation length depends on a single parameter @xmath185 as @xmath186 and the transverse correlation length depends on @xmath90 and another parameter @xmath187 as @xmath188 . figure ( 2 ) shows the behavior of these correlation functions for different values of the parameters @xmath90 and @xmath189 . [ correlationlengths ] , as a function of the parameter @xmath90 for different values of the parameter @xmath190 . at @xmath191 both the transverse and longitudinal correlation lengths diverge at @xmath192.,title="fig:",width=302,height=226 ] an interesting point about transverse correlation functions is that depending on the sign of @xmath121 , it becomes identically zero , on one side of the @xmath193 axis and different from zero on the other side . thus if one insists on a definition of an order parameter to be zero in one phase and non - zero in the other , then we can safely say that in these models , the transverse correlation function is an order parameter which signals a quantum phase transition . + let us see study some limiting cases in these general models . at the point of phase transition @xmath101 , as seen from ( [ rhosolutionketbra ] ) , the state of a rung becomes a mixture of spin - zero states , and the two spins of a rung become fully anti - correlated in the @xmath1 direction , a fact which is reflected in ( [ correlationsolution ] ) . near this point the correlation length @xmath194 becomes very large as seen from ( [ longitudinallength ] ) , although the amplitude becomes small since it is proportional to @xmath195 . thus at @xmath101 there is no long - range order in the model . also from ( [ concurrence ] ) and ( [ eigenvalues of rho ] ) it is readily seen that at @xmath101 , the concurrence ( or entanglement ) of the two spins of a rung becomes maximum and equal to @xmath196 on the other hand from ( [ entropysolution ] ) we see that as we approach the point @xmath101 from both sides , the von - neumann entropy decreases . in addition , the derivative of both types of entanglement become singular at @xmath101 . these facts show that the point of phase transition in these systems , is a point where the spins of a rung become highly entangled with each other and each rung becomes only slightly entangled to the rest of the lattice . + we can also obtain the explicit form of the state in this limit . from ( [ properties ] ) , and ( [ mat ] ) we see that in this limit , no two spins in a rung can be in a @xmath197 state . a little reflection shows that they can not be in the state @xmath198 either , since in the absence of @xmath199 , the only string of matrices with non - zero trace is a string of matrices @xmath200 and @xmath201 in arbitrary order . since these matrices commute with each other , the resulting state has a simple description . define two local states of a rung as @xmath202 where for convenience we have re - introduced the spin notations @xmath203 and @xmath204 instead of qubit notation @xmath132 and @xmath86 . define a global un - normalized state @xmath205 to be the equally weighted linear combination of all states with @xmath13 local @xmath143 states and @xmath206 local @xmath45 state . then from ( [ properties ] ) and ( [ mat ] ) we find that the ground state of the chain in the limit @xmath101 is given by @xmath207|u^k , d^{n - k}\ra,\ ] ] where @xmath9 is the normalization constant given by @xmath208 ^ 2 = 2\left[(a^2+b^2)^n+(2ab)^n\right].\ ] ] at the other extreme when @xmath209 , the state of each rung becomes a mixture of fully aligned spins either in the positive or negative @xmath1 direction . this is also reflected in ( [ correlationsolution ] ) . in this limit a rung becomes entangled with the rest of the lattice , sine @xmath210 and the two spins of a rung become disentangled from each other since @xmath211 . + the explicit form of the state can also be obtained in this limit . in this case the only strings of matrices with non - vanishing traces are strings of @xmath199 and @xmath77 in alternating order . thus if we define two local states @xmath212 then the ground state in this limit will be a ghz state @xmath213 in the next subsection we specialize these results to the two classes we discussed in the beginning of this subsection . this is the class which has the so(2 ) symmetry ( rotation around the @xmath1 axis in spin space ) and three @xmath127 symmetries ( spin flip , parity , leg exchange ) . for this class we have from ( [ pisolution ] ) that @xmath214 . the parameter @xmath90 defined after equation ( [ transverselength ] ) becomes equal to @xmath215 insertion of @xmath214 in various quantities of the previous subsection shows that all the quantities can be expressed as a function of @xmath90 , namely we find @xmath216 @xmath217 and @xmath218 we also find @xmath219 and @xmath220 figure ( 2 ) show the entropy and the concurrence of the state of a single rung for this model . [ finalladdercurvesabnew ] , which is taken to be dimensionless.,title="fig:",width=302,height=226 ] this is the class which has full so(3 ) symmetry ( rotation in spin space ) and one discrete symmetry ( spin flip ) for generic values of @xmath143 and all the three @xmath127 symmetries for @xmath145 . for this class we have from ( [ rotsolution ] ) @xmath221 , @xmath222 and @xmath223 . inserting these values in the equations of the previous section we find the following : @xmath224 and @xmath225 where @xmath166 is any direction in the spin space . the eigenvalues of the one - rung density matrix ( [ eigenvalues of rho ] ) in this case will be @xmath226 thus we find @xmath227 and @xmath228 figure ( 3 ) show the entropy and the concurrence of the state of a single rung for this model . [ ucurves ] , which is taken to be dimensionless.,title="fig:",width=302,height=226 ] although ladders of spin 3/2 models has already been studied in the context of vertex state models in @xcite , in which the ground state is constructed by a suitable concatenation of vertices assigned to single sites , the method developed in this work , in which each single rung is considered as a single site in a hyper - chain and the ground state is constructed as a matrix product state seems to us as more powerful and very easy to generalize to other spin models . in this work we have applied the matrix product formalism for construction of models on spin @xmath0 ladders . these models have been constructed to have special discrete and or continuous symmetries and to display a quantum phase transition in a broad sense , that is displaying non - analytical behavior in their correlation functions . naturally these non - analytical behavior can also be observed in the entanglement properties of pairs of spins in these models . this route can be followed to develop other models , e.g models with higher spins or alternating spins on the rungs , with alternating coupling constants , or next - nearest neighbor interactions . by exploiting higher dimensional matrices and having more free parameters at our disposal , we may be able to construct continuous families of models on spin ladders which have full rotational symmetry . in this article we have restricted ourselves to frustration - free hamiltonians and mps states with fixed - size . relaxing this latter condition allows one to represent any state @xmath229 as a matrix product state @xcite and then one may be able to study more diverse kinds of phase transitions on spin ladders . these matters will be taken up in separate publications . we would like to thank a. langari for very valuable discussions and the members of the quantum information group of sharif university , specially s. alipour and l. memarzadeh for instructive comments . s. sachdev , _ quantum phase transitions _ ( cambridge university press , cambridge , 1999 ) . a. osterloh , l. amico , g. falci , and r. fazio , nature 416 , 608 ( 2002 ) . t. osborne , m. nielsen , phys . rev . a , * 66 * , 032110 ( 2002 ) . m. c. arnesen , s. bose , and v. vedral , phys . lett . * 87 * , 277901 ( 2001 ) ; d. gunlycke et al , phys . rev . a * 64 * , 042302 ( 2001 ) . s. hill and w. k. wootters , phys . lett . * 78 * , 5022 ( 1997 ) ; w. k. wootters , phys . rev . letts . * 80 * , 2245 ( 1998 ) . a. klumper , a. schadschneider and j. zittartz , j. phys . a ( 1991 ) l293 ; z. phys . b , * 87 * ( 1992 ) 281 ; europhys . * 24 * ( 1993 ) 293 . m. fannes , b. nachtergaele and r. f. werner , commun . phys . * 144 * , 443 ( 1992 ) . m. m. wolf , g. ortiz , f. verstraete and i. cirac , phys . 97 , 110403 ( 2006 ) . d. peres garcia , et al , quant - ph/0608197 . m. a. ahrens , a. schadschneider , and j. zittartz , europhys . lett . * 59 * 6 , 889 ( 2002 ) . e. bartel , a. schadschneider and j. zittartz , eur . phys . jour . b , * 31 * , 2 , 209 - 216 ( 2003 ) . h. niggemann , and j. zittartz , j. phys . a : math . gen . 31 , 9819 - 9828 ( 1998 ) . f. verstraete , d. porras , j. i. cirac , phys . 93 * , 227205 ( 2004 ) . f. verstraete , j. j. garcia - ripoll , and j. i. cirac , phys . lett . * 93 * , 207204 ( 2004 ) . f. verstraete , j. i. cirac , phys . rev . b 73 , 094423 ( 2006 ) ; t. j. osborne , phys . 97 , 157202 ( 2006 ) ; m. b. hastings , phys . rev . b 73 , 085115 ( 2006 ) . g. vidal , phys . lett . * 91 * , 147902 ( 2003 ) . a. k. kolezhuk and h. j. mikeska , phys . * 80 * , 2709 ( 1998 ) ; int . b , vol.12 , 2325 - 2348 ( 1998 ) . in this appendix we briefly discuss the derivation of the explicit form of the hamiltonian in terms of local spin operators . for simplicity we consider models in class a. the hamiltonian for models in class b , can be constructed along similar lines . as explained in the main text , the starting point is to solve the system of equations @xmath230 this solution space is 12 dimensional since we have 4 equations for 16 unknowns . thus we should find a set of 12 orthogonal vectors which span this solution space and then form a non - negative linear combination of the corresponding one dimensional projectors . the hamiltonian constructed in this way does not necessarily have the symmetries imposed on the state , unless we choose new linear combinations of these vectors which transform suitably under the symmetry operators . these new basis vectors are found to be : @xmath233 \cr |2,0\ra & : = & \frac{1}{\sqrt{6}}\left[-{4a^2}(|0011\ra+|1100\ra ) + g\left(|0101\ra + \s\e |0110\ra+ \s\e|1001\ra + |1010\ra\right)\right]\cr & & \cr |1,1\ra & : = & \frac{1}{2}\left[\e|0001\ra + |0010\ra - \e|0100\ra - |1000\ra \right ] \cr |1,0\ra & : = & \frac{1}{\sqrt{2}}\left[|0110\ra-|1001\ra\right ] \cr & & \cr |1',1\ra & : = & \frac{1}{2}\left[-\s\e|0001\ra + |0010\ra + \e |0100\ra -\s |1000\ra\right ] \cr |1',0\ra & : = & \frac{1}{\sqrt{2}}\left[|0101\ra-|1010\ra\right ] \cr & & \cr |0,0\ra & : = & \frac{1}{2}\left[|0101\ra -\s\e |0110\ra - \s\e |1001\ra + |1010\ra\right ] , \end{aligned}\ ] ] with @xmath234 + here we have organized the states in multiplets , with a labeling reminiscent of the one used in labeling the states of @xmath235 representations . the reason is that in a certain limit ( @xmath236 ) these states actually comprise the irreducible representations of @xmath235 which emerge from the decomposition of the product of four spin one - half representations living on the sites of two rungs of the ladder . in fact for the product of these spin 1/2 representations we have @xmath237 however only the representations @xmath238 and @xmath132 span the solution space of ( [ equations ] ) in this limit . this labeling is useful because we can see under what conditions , the hamiltonian become fully rotational invariant . note also that we have used the labeling @xmath232 in accordance with the labeling of ( [ equations ] ) for @xmath231 ( see figure ( 5 ) . the reader can easily check that each of the above states ( or more precisely the corresponding one dimensional projector ) is invariant under the parity operation @xmath239 and leg - exchange of the ladder @xmath240 . they also transform to each other under the spin - flip operation @xmath241 . the local hamiltonian which has the three discrete symmetries mentioned in the text and the symmetry around rotation along the @xmath1 axis in spin space , is constructed as follows : @xmath242 where the coefficients @xmath243 are non - negative parameters and together with the parameters @xmath170 and @xmath244 form the 10 free parameters of the hamiltonian . of course the total number of coupling constants ( interaction strength ) is 8 , since we can always shift the ground state energy and also set the scale of energy by redefintion of these parameters . re - expressing the local operators in terms of pauli spin operators and rearranging terms , we find , after a rather lengthy calculation , the hamiltonian acting on the ladder where we have multiplied @xmath73 by a factor of @xmath245 for convenience , ( see figure ( 6 ) for labeling of sites of the ladder ) . note that we use @xmath246 as an abbreviation to denote the operator @xmath247 in the following . the total hamiltonian is then given by @xmath253 this is a complicated looking hamiltonian , with many types of interactions , but in view of the large number of parameters , it is possible to look at specific subsets of the parameter space , where some of the interactions are absent . indeed the parameter space consists of four disconnected parts , each of which corresponds to one choice of the pair @xmath254 . let us for example consider the subset on which the hamiltonian has full rotational symmetry . it is well known that any operator of the form @xmath255 is a scalar . thus in the limit @xmath256 if we set @xmath257 and @xmath258 , then we will have a hamiltonian which is fully rotational invariant . on can see that in this limit , all the coupling constant corresponding to non - scalar terms in the hamiltonian vanish and we are left with @xmath259
|
we investigate quantum phase transitions in ladders of spin @xmath0 particles by engineering suitable matrix product states for these ladders .
we take into account both discrete and continuous symmetries and provide general classes of such models .
we also study the behavior of entanglement between different neighboring sites near the transition point and show that quantum phase transitions in these systems are accompanied by divergences in derivatives of entanglement .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
recently andrei teleman considered moduli spaces of projectively anti - selfdual instantons in certain hermitian rank-2 bundles over a closed oriented 4-manifold with negative definite intersection form @xcite . these play a role in his classification program on class vii surfaces @xcite@xcite . however , in certain situations the instanton moduli spaces involved consist of projectively flat connections and therefore have very interesting topological implications . in this article we will study these ` casson - type ' moduli spaces . suppose @xmath8 is a hermitian rank-2 bundle with first chern - class a ( minimal ) characteristic vector @xmath9 of the intersection form . in other words , it is the sum of elements @xmath10 in @xmath11 which induce a basis of @xmath12 diagonalising the intersection form ( because of donaldson s theorem @xcite ) . then for one possible value of a strictly negative second chern class @xmath13 the moduli space is compact ( independently of the riemannian metric ) . in particular , if the manifold has second betti - number @xmath2 divisible by 4 and first betti - number @xmath14 the instanton moduli space consists of projectively flat connections and has expected dimension zero . this should be thought of as a ` casson - type ' moduli space because the holonomy yields a surjection onto the space of @xmath15 representations of @xmath16 with fixed stiefel - whitney class @xmath17 . non - emptiness of the casson - type moduli space implies that none of the elements @xmath18 can be poincar dual to an element representable by a sphere , i.e. to an element in the image of the hurewicz homomorphism . prasad and yeung @xcite constructed aspherical manifolds @xmath19 which are rational - cohomology complex projective planes , generalisations of mumford s fake projective plane @xcite . if @xmath20 denotes this manifold with the opposite orientation , a natural candidate of a manifold for which the moduli space might be non - empty is given by the connected sum @xmath21 of 4 copies of @xmath20 , and a candidate of a manifold for which the casson - invariant can be defined is given by a ` ring of 4 copies of @xmath20 ' ( the last summand in the 4-fold connected sum @xmath22 is taken a connected sum with the first ) . after recalling the gauge - theoretical situation considered in @xcite we show that if the casson - type moduli space is non - empty , then we can not have a connected sum decomposition @xmath4 unless both @xmath5 and @xmath6 are divisible by four . in particular the moduli space for the above mentioned @xmath21 - ring is empty . this result still leaves open the question of whether there is any @xmath0 with a non - empty casson - type moduli space . we give therefore in some detail a possible construction of suitable @xmath7-manifolds @xmath0 ( along with the correct representations of @xmath16 ) . we would like to point out that even though recent investigation leads us to believe that the casson - type invariant is vanishing @xcite , the casson - type moduli space may still be non - empty and is interesting from a topological perspective . our construction also suggests the possibility of considering casson - type moduli spaces for manifolds with boundary . a similar moduli space and invariant has been defined by ruberman and saveliev for @xmath23$]-homology hopf surfaces , going back to work of furuta and ohta @xcite , and for @xmath23$]-homology 4-tori @xcite . our situation is simpler than their first mentioned situation because of the absence of reducibles in the moduli space due to the condition on @xmath2 . the first author thanks simon donaldson for useful conversations . the second author is grateful to andrei teleman for turning his interest to low - energy instantons and for a stimulating conversation on them , and also wishes to express his gratitude to stefan bauer for helpful conversations . both authors thank kim fryshov profusely for invaluable advice and ideas . we are also grateful to the referee for the care taken in helping us substantially improve the article . after briefly recalling some general instanton gauge theory @xcite , and introducing our notations , we shall quickly turn to the special situation of ` low - energy instantons ' over negative definite 4-manifolds mentioned in the introduction . we show that the gauge - theoretical situation is indeed relatively simple , indicate a definition of an invariant , and set up the correspondance of the moduli space to representation spaces of the fundamental group in @xmath15 . let @xmath0 be a smooth riemannian four - manifold and @xmath8 a hermitian rank-2 bundle on @xmath0 . let further @xmath24 be a fixed unitary connection in the associated determinant line bundle @xmath25 . we define @xmath26 to be the affine space of unitary connections on @xmath27 which induce the fixed connection @xmath24 in @xmath28 . this is an affine space over @xmath29 , the vector space of @xmath30-valued one - forms on @xmath0 . let us denote by @xmath31 the principal @xmath32 bundle of frames in @xmath27 , and let @xmath33 be the bundle that is associated to @xmath31 via the projection @xmath34 , @xmath35 . the space @xmath36 of connections in the @xmath37 principal bundle @xmath33 and the space @xmath26 are naturally isomorphic . if we interpret a connection @xmath38 as a @xmath37 connection via this isomorphism it is commonly called a projective connection . the adjoint representation @xmath39 descends to a group isomorphim @xmath40 . the associated real rank-3 bundle @xmath41 is just the bundle @xmath30 of traceless skew - symmetric endomorphisms of @xmath27 . thus the space @xmath26 is also isomorphic to the space @xmath42 of linear connections in @xmath30 compatible with the metric . we shall write @xmath43 for connections in the @xmath37 principal bundle and denote the associated connection in @xmath30 by the same symbol . should we mean the unitary connection which induces the connection @xmath24 in @xmath44 we will write @xmath45 instead . let @xmath46 denote the group of automorphisms of @xmath27 of determinant @xmath47 . it is called the ` gauge group ' . this group equals the group of sections @xmath48 , where @xmath49 is given by conjugation . we shall write @xmath50 for the quotient space @xmath51 . a connection is called _ reducible _ if its stabiliser under the gauge group action equals the subgroup given by the centre @xmath52 which always operates trivially , otherwise _ irreducible_. equivalently , a connection @xmath45 is reducible if and only if there is a @xmath45 - parallel splitting of @xmath27 into two proper subbundles . let us point out that the characteristic classes of the bundle @xmath30 are given by @xmath53 for a connection @xmath43 we consider the anti - selfduality equation @xmath54 where @xmath55 denotes the curvature form of the connection @xmath56 , and @xmath57 its self - dual part with respect to the hodge - star operator defined by the riemannian metric on @xmath0 . the moduli space @xmath58 of antiself - dual connections , @xmath59 is the central object of study in instanton gauge theory . this space is in general non - compact and there is a canonical `` uhlenbeck - compactification '' of it . the anti - selfduality equations are elliptic , so fredholm theory provides finite dimensional local models for the moduli space . the often problematic aspect of donaldson theory is the need to deal with reducible connections and with a non - trivial compactification . we will consider special situations where these problems do not occur . we restrict now our attention to smooth riemannian four - manifolds @xmath0 with @xmath60 and @xmath61 . according to donaldson s theorem @xcite the intersection form of such a four - manifold is diagonal . let @xmath62 be a set of elements in @xmath11 which induce a basis of @xmath12 diagonalising the intersection form . [ no reductions ] @xcite suppose the hermitian rank-2 bundle @xmath8 has first chern class @xmath63 and its second chern class is strictly negative , @xmath64 . then @xmath8 does not admit any topological decomposition @xmath65 into the sum of two complex line bundles . _ proof : _ suppose @xmath66 . then @xmath67 and @xmath68 for some @xmath69 . therefore , @xmath70 @xmath71 let @xmath8 be as in the previous lemma . then the moduli space @xmath72 does not admit reducibles . for a connection @xmath73 chern - weil theory gives the following formula : @xmath74 in particular , for anti - selfdual connections the left hand side of this equation is always non - negative , and we can draw the following observation from the formula : @xcite 1 . for @xmath75 the moduli space @xmath72 is always compact , independently of the chosen metric or any genericity argument . in fact , the lower strata in the uhlenbeck - compactification consist of anti - selfdual connections in bundles @xmath76 with @xmath77 and @xmath78 for @xmath79 . for @xmath63 we have @xmath80 . thus , if @xmath81 and @xmath82 the moduli space @xmath72 will consist of projectively flat connections only . we recall the expected dimension of the moduli space @xmath72 . it is given by the formula @xmath83 in particular it can happen that @xmath84 in the situation we consider , namely , @xmath60 , @xmath85 , and @xmath64 , the latter condition assuring that we are in the favorable situation of lemma [ no reductions ] . + interesting is the following special case of ` casson - type ' moduli spaces that we consider from now on : [ flat ] let @xmath0 be a negative definite riemannian four - manifold with strictly positive second betti - number @xmath2 divisible by four , and @xmath14 . let @xmath8 be a hermitian rank-2 bundle with @xmath63 and with @xmath86 . then the moduli space @xmath72 of projectively anti - selfdual connections in @xmath27 is compact and consists of irreducible projectively flat connections only , and is of expected dimension zero . after suitable perturbations a gauge theoretic invariant can be defined in this situation : it is an algebraic count of a perturbed moduli space which consists of a finite number of points , the sign of each point is obtained by a natural orientation determined by the determinant line bundle of a family of elliptic operators . this has been done in the meantime in @xcite , where it is shown that this invariant is actually zero . we would like to emphasise that the vanishing of this invariant does nt imply emptiness of the unperturbed moduli space that we shall investigate further here . suppose we are in the situation that our moduli space @xmath72 consists of flat connections in @xmath87 , as for instance in the last proposition . then we must have @xmath88 by chern - weil theory . the holonomy establishes a correspondance between flat connections in the oriented real rank-3 bundle @xmath89 and representations of the fundamental group @xmath16 in @xmath15 with a prescribed stiefel - whitney class . more precisely , let @xmath90 be a representation of the fundamental group . let @xmath91 be the universal covering of @xmath0 ; it is a @xmath16 principal bundle over @xmath0 . we can form the associated oriented rank-3-bundle @xmath92 it admits a flat connection as it is a bundle associated to a principal bundle with discrete structure group . therefore it has vanishing first pontryagin class , @xmath93 , by chern - weil theory . its only other characteristic class @xcite is its second stiefel - whitney class @xmath94 . therefore we will say that the representation @xmath95 has stiefel - whitney class @xmath96 if @xmath97 . on the other hand , let @xmath98 be an oriented real rank-3 bundle with a flat connection @xmath56 . then the holonomy of @xmath56 along a path only depends up to homotopy on the path , and therefore induces a representation @xmath99 . in particular , the holonomy defines a reduction of the structure group to @xmath16 , and the bundle can therefore be reconstructed as @xmath100 . in particular the representation @xmath101 has stiefel - whitney class @xmath102 . the moduli space @xmath72 has been obtained by quotienting the space of antiself - dual connections in @xmath103 by the gauge group @xmath46 . from the perspective of the @xmath37 connections in @xmath30 this gauge group is not the most natural one . instead , the group @xmath104 is the natural group of automorphisms of connections in @xmath30 . not every element @xmath105 admits a lift to @xmath46 ; instead , there is a natural exact sequence @xmath106 quotienting by @xmath107 has the advantage of a simpler discussion of reducibles , as discussed above . let us denote by @xmath108 the moduli space of anti - self dual connections in @xmath30 modulo the full gauge group @xmath109 . then there is a branched covering @xmath110 with ` covering group ' @xmath111 . let us denote by @xmath112 the space of representations of @xmath16 in @xmath15 up to conjugation and of stiefel - whitney class @xmath96 . the above discussion implies that there is a homeomorphism @xmath113 where @xmath114 . in particular , @xmath72 surjects onto @xmath112 . we will use the above derived relation of the ` casson - type moduli space ' @xmath72 to the representation space @xmath112 to obtain the vanishing result which is mentioned in the introduction . the above construction of the bundle @xmath115 associated to a representation @xmath116 is functorial in the following sense : [ naturality ] suppose we have a map @xmath117 between topological spaces , and @xmath90 a representation of the fundamental group of @xmath0 . then there is a natural isomorphism @xmath118 between the pull - back of the bundle @xmath115 via @xmath119 and the bundle @xmath120 , where @xmath121 is the map induced by @xmath119 on the fundamental groups . _ proof : _ we have a commutative diagram @xmath122{\widetilde{x } } \arrow{s } \\ \node{w } \arrow{e , t}{f } \node[1]{x , } \end{diagram } \end{split}\ ] ] where the vertical maps are the universal coverings , and where @xmath123 is the unique map turning the diagram commutative ( we work in the category of pointed topological spaces here ) . it is elementary to check that the map @xmath123 is equivariant with respect to the action of @xmath124 , where this group acts on @xmath91 via @xmath125 and the deck transformation group of @xmath91 . the claimed isomorphism follows then from naturality of the associated bundle construction . @xmath126 + [ restrictions ] suppose the four - manifold @xmath0 splits along the connected 3-manifold @xmath127 as @xmath128 into two four - manifolds @xmath129 and @xmath130 . then any representation @xmath131 induces representations @xmath132 via @xmath133 where the map @xmath134 is the inclusion . for these representations we have @xmath135 conversely , given representations @xmath132 such that @xmath136 , where @xmath137 denotes the inclusion , there is a representation @xmath116 inducing @xmath138 and @xmath139 via the respective restrictions . _ proof : _ this follows from the theorem of seifert and van kampen and the lemma above or , equivalently , by gluing connections . @xmath126 + [ hurewicz ] let @xmath0 be a four - manifold with @xmath60 , and let @xmath96 be @xmath140 . suppose there is a representation @xmath90 with fixed second stiefel - whitney class @xmath9 . then none of the poincar dual of the basis elements @xmath18 is in the image of the hurewicz - homomorphism @xmath141 . _ proof @xcite : _ suppose we have a map @xmath142 such that @xmath143 $ ] , where @xmath144 \in h_2(s^2;{\mathbb{z}})$ ] denotes the fundamental cycle of @xmath145 , and @xmath146 denotes the poincar dual of @xmath18 . then we have @xmath147 \rangle \equiv \langle \sum e_j , pd(e_i ) \rangle = e_i^2 = -1 \ ( mod \ 2 ) .\ ] ] on the other hand , by naturality of the cohomology - homology pairing , we get @xmath148 \rangle & = \langle w_2(v_\rho ) , f_*[s^2 ] \rangle = \langle f^ * w_2(v_\rho ) , [ s^2 ] \rangle \ . \end{split}\ ] ] but the above lemma [ naturality ] implies that @xmath149 . as @xmath145 has trivial fundamental group the bundle @xmath150 is clearly the trivial bundle , so the left hand side of equation ( [ 0 ] ) must be zero modulo 2 , a contradiction to equation ( [ -1 ] ) . @xmath126 + by hopf s theorem on the cokernel of the hurewicz - homomorphism , expressed in the exact sequence @xmath151 the fundamental group has to have non - trivial second homology in order to obtain a non - empty casson - type moduli space . this proposition gives a topological significance of the zero - energy instantons : if the moduli space is non - empty then the elements @xmath146 are not representable by spheres ! one might wonder whether there exists any four - manifold where the elements @xmath146 are not representable by spheres . certainly this can not be a simply connected four - manifold because of the hurewicz - isomorphism theorem . interestingly , the answer is affirmative . generalising mumford s fake projective plane @xcite , prasad and yeung have constructed manifolds with the rational cohomology of the complex projective space @xmath152 whose universal cover is the unit ball in @xmath153 @xcite . such a manifold @xmath19 is therefore an eilenberg - maclane space @xmath154 . now let @xmath155 be the four - manifold that we obtain from the connected sum of four @xmath20 , where we do again a connected sum of the last summand with the first . the so obtained `` 4-@xmath20-ring '' is diffeomorphic to @xmath156 this manifold has negative definite intersection form and has betti - numbers @xmath157 and @xmath158 . in addition , no element of @xmath159 is representable by a 2-sphere , so we get no obstruction to non - emptiness from proposition [ hurewicz ] . thus the four - manifold @xmath155 is a prototype of a four - manifold on which to consider the moduli space of @xmath37 instantons associated to the bundle @xmath160 with @xmath63 and @xmath161 ( and therefore of representations of @xmath162 with fixed stiefel - whitney class @xmath163 ) . however , as we will see , there are no such instantons . [ doldwhitneyapplication ] let @xmath0 be a smooth closed negative definite four - manifold . if there is a representation @xmath164 with stiefel - whitney class @xmath165 , then the second betti - number @xmath2 must be divisible by four . _ proof : _ the bundle @xmath115 has @xmath166 and vanishing first pontryagin - class @xmath93 because this bundle admits a flat connection . now the dold - whitney theorem @xcite states that the second stiefel - whitney class @xmath167 and the first pontryagin class @xmath168 of any oriented real rank-3 bundle satisfy the equation @xmath169 here @xmath170 denotes the pontryagin square , a lift of the cup - product squaring @xmath171 to the coefficient group @xmath172 . if the class @xmath173 is the mod-2 reduction of an integral class @xmath174 then the pontryagin square is simply the mod-4 reduction of the square of @xmath175 , i.e. @xmath176 in our case the dold - whitney theorem thus implies that @xmath177 @xmath126 + hence we obtain the following let @xmath0 be a four - manifold with negative definite intersection form and suppose it admits a connected sum decomposition @xmath178 . suppose @xmath116 is a representation of the fundamental group of @xmath0 with fixed stiefel - whitney class @xmath179 . then both @xmath5 and @xmath6 must be divisible by four . _ proof : _ note first that the intersection form of both @xmath129 and @xmath130 must be diagonal . this follows from eichler s theorem on unique decomposition of symmetric definite forms over @xmath180 , see @xcite . therefore the basis vectors @xmath62 of @xmath11 are simply given by the union of basis vectors @xmath181 of @xmath182 , diagonalising the intersection form of @xmath129 , and basis vectors @xmath183 of @xmath184 , diagonalising the intersection form of @xmath130 . note that @xmath185 . the above proposition [ restrictions ] now applies yielding representations @xmath186 . its second stiefel - whitney class computes , using the above equation ( [ restriction ] ) , @xmath187 and likewise for @xmath188 . the above theorem therefore concludes the proof . @xmath126 + this implies that the above considered manifold @xmath189 does not admit a representation @xmath116 with stiefel - whitney class being the mod-2 reduction of the sum of basis elements diagonalising the intersection form . as a ` converse ' to the above vanishing theorem , suppose we are given a connected sum @xmath190 and representations @xmath132 with the desired stiefel - whitney classes on @xmath191 . according to proposition [ restrictions ] , we obtain the representation @xmath192 which has the desired stiefel - whitney class . this is in contrast to well - known vanishing theorems for connected sums of manifolds with @xmath193 as in ( * ? ? ? * theorem 9.3.4 and , in particular , proposition 9.3.7 ) . there is much interest in the relationship between the fundamental group of a @xmath7-manifold and its intersection form . the casson - type invariant considered in this paper gives rise to the natural question of whether there exists _ any _ @xmath7-manifold @xmath0 with non - empty casson - type moduli space . in this section we describe a construction that we hope will provide the first examples of such manifolds , by indicating how to construct non - empty representation spaces @xmath194 . let @xmath196 be a smooth immersion of @xmath197 @xmath195-spheres such that any points of self - intersection of @xmath198 occur with negative sign and between two branches of the same component of @xmath198 . suppose there are @xmath199 self - intersections . blowing up @xmath199 times and taking the proper transform we obtain an @xmath197-component embedded link @xmath200 . each component of @xmath201 intersects each exceptional sphere of @xmath202 either at no points or at one point positively and at one point negatively ( this is because each intersection point of @xmath198 occurred within a single component and with negative sign ) . hence each component of @xmath201 is trivial homologically and so the embedding of @xmath201 extends to a @xmath203-neighbourhood . we do surgery on @xmath201 by removing @xmath204 and gluing in @xmath205 . call the resulting @xmath7-manifold @xmath0 . the construction of @xmath0 was suggested by kim fryshov . it turns out to be very suited to our purposes ; we have so we shall be done if we can find @xmath199 embedded tori in @xmath0 which are pairwise disjoint and which each have self - intersection @xmath216 . figure [ torusbasis ] shows how to find these tori . working inside @xmath215 , each exceptional sphere @xmath27 intersects @xmath201 transversely in two points . connect these two points by a path on @xmath201 . the @xmath203-neighbourhood of @xmath201 pulls back to a trivial @xmath203-bundle over the path . the fibres over the two endpoints can be identified with neighbourhoods of these two points in @xmath27 . removing these neighbourhoods from @xmath27 we get a sphere with two discs removed and we take the union of this with the @xmath217 boundaries of all the fibres of the @xmath203-bundle over the path . we have shown how to associate to a given immersed @xmath195-link @xmath218 with only negative self - intersections and disjoint components , a smooth @xmath7-manifold @xmath219 which is diagonal and negative definite , with basis elements of @xmath220 represented by embedded tori . [ [ so3-representations - of - pi_1-and - presentations - of-2-links ] ] @xmath15 representations of @xmath221 and presentations of @xmath195-links ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ using the same notation as in the previous subsection , we give a method to describe links @xmath198 that come with representations @xmath222 with the correct stiefel - whitney class @xmath223 . this method may not at first appear entirely general , but we show that if there is such a link @xmath198 then it must admit a description of this form . 1 . @xmath226-handles ( circle creation ) . 2 . simple crossing changes ( see figure [ crossing ] ) . ribbon - type reidemeister moves of type ii ( see figure [ ribbon ] ) . ribbon - type @xmath47-handle addition ( see figure [ ribbon ] ) . [ [ representations - of - the - fundamental - group - and - ribbon - presentations - of-2-links ] ] representations of the fundamental group and ribbon presentations of @xmath195-links ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ for notation , let @xmath228 be a height function corresponding to lemma [ pres_lem ] with exactly @xmath195 critical points that restricts to a morse function on @xmath198 such that all the @xmath229-handles of @xmath198 occur in @xmath230 and the self - intersections of @xmath198 occur in @xmath231 . then there is a morse function on the blow - up @xmath232 with one maximum and one minimum , and @xmath199 index @xmath195 critical points . these index 2 critical points all occur at @xmath233 , and h restricts to a morse function on the proper transform @xmath201 with the @xmath229-handles of @xmath201 occurring in @xmath234 . we can use the same movie of @xmath198 to describe the embedding of @xmath201 . recall that @xmath235 . we compute @xmath236 using the van kampen theorem . first note that @xmath237 is the boundary connect sum of @xmath199 copies of the complement of @xmath195 fibres in the @xmath203-bundle over @xmath145 of euler class @xmath216 , and @xmath238 copies of @xmath239 where the @xmath203 with @xmath240 is trivially embedded . here @xmath199 is the number of self - intersections of @xmath198 ( and hence the number of blow - ups required on the way to constructing @xmath219 ) and @xmath238 is the number of extra @xmath226-handles used in the movie presentation of @xmath198 satisfying lemma [ pres_lem ] . since by assumption @xmath219 has a non - empty casson - type moduli space and dim@xmath241 , we can write @xmath242 by theorem [ doldwhitneyapplication ] . it is easy to compute that @xmath243 is the free ( non - abelian ) group on @xmath244 generators . we fix representatives of a basis for this group as simple loops coming down from infinity , linking the relevant circle by small meridians and heading back up again . for each of the @xmath224 generators coming from the blowups we allow ourselves two representatives - one for each circle . note that our representatives live in the _ boundary _ of @xmath237 . to get the space @xmath245 we attach the complements of some @xmath47-handles to @xmath246 . what this means is that for every @xmath47-handle of @xmath201 , we glue a @xmath247 to @xmath237 , via a homeomorphism of @xmath248 with a subset of @xmath249 . ( all discs in this discussion are trivially embedded ) . since @xmath250 , @xmath251 , and the map on @xmath221 induced by inclusion is onto , the van kampen theorem tells us that adding the complement of a @xmath47-handle adds a single , possibly trivial , relation to @xmath221 . in other words , we obtain a presentation of @xmath252 with @xmath244 generators and as many relators as there are @xmath47-handles . since we obtain @xmath253 from @xmath245 by gluing on the complement of some trivially embedded @xmath203 s ( one for each @xmath195-handle of @xmath201 ) in @xmath254 , it follows that @xmath255 . hence we have a presentation of @xmath256 . now by assumption , @xmath219 has a non - empty casson - type moduli space , so we choose some representation @xmath257 that has the correct associated characteristic classes . each generator of the presentation is associated to some circle or hopf link in figure [ 0-handles ] . we decorate each circle or hopf link with the image of the associated generator under @xmath95 . we call these images @xmath258 . each @xmath47-handle complement that we attach appears in the movie of @xmath198 as a ribbon - type @xmath47-handle addition as illustrated in figure [ ribbon ] . once we have added each ribbon - type handle then by assumption we have an unlink . [ [ representations - of - the - fundamental - group - and - a - singular - link - diagram ] ] representations of the fundamental group and a singular link diagram ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 1 . each component of @xmath198 has genus @xmath226 . 2 . suppose two circles of figure [ 0-handles ] are joined by three paths of @xmath47-handle cores @xmath262 in the singular diagram @xmath261 . if @xmath262 meet the first circle in three points that go clockwise ( respectively anticlockwise ) around the circle , then @xmath262 must meet the second circle in three points that go anticlockwise ( respectively clockwise ) around that circle . 1 . self - intersections of @xmath198 only occur within a component and not between two components of the preimage of @xmath198 . the singular diagram @xmath261 describes an obvious singular link in @xmath263 . given a hopf link in figure [ 0-handles ] , we require that the two circles comprising it are part of the same component in this singular link . 1 . the representation + @xmath264 + determined by the labelling @xmath258 factors through + @xmath265 2 . each circle in figure [ 0-handles ] bounds an obvious oriented disc which has no double points when projected to the plane of the diagram . consider a core of a @xmath47-handle @xmath56 in the singular link diagram @xmath261 . suppose @xmath56 connects circles decorated by @xmath15 elements @xmath266 and @xmath267 , and that the arc , given the orientation from @xmath266 to @xmath267 , intersects discs bounded by circles which are decorated by elements @xmath268 . define the element @xmath269 , where the @xmath270 index is the sign of the intersection of the arc with the disc . we require + @xmath271 -handle , which will take place within the dotted circle . we have indicated @xmath7 generators of @xmath273 . by the van kampen theorem , adding the @xmath47-handle imposes the relation that the rightmost generator is a conjugate of the leftmost generator as in item @xmath7 of our checklist . thinking of the ribbon as a thickened arc , we note that this calculation does not depend on whether the arc is locally knotted , but only on the order in and parity with which it intersects the discs bounded by the @xmath226-handles . also , since a small loop encircling both strands of a ribbon clearly bounds a disc in @xmath245 , it is also immaterial how the arcs link each other.,title="fig:",width=384,height=240 ] 1 . the representation @xmath274 has the correct stiefel - whitney class @xmath223 . 2 . * for each hopf link of figure [ 0-handles ] , choose a path of cores of @xmath47-handles in the singular diagram @xmath261 which connects the circles of the hopf link . ( such a path exists by lemma [ selfint ] ) . + say it consists of cores @xmath275 . we order and orient these cores so that the start point of @xmath276 and the end point of @xmath277 are on different components of the hopf link and the end point of @xmath278 is on the same circle of figure [ 0-handles ] as the start point of @xmath279 for @xmath280 . write @xmath266 for the element decorating the hopf link . then we require that + @xmath281 * the elements @xmath282 are each conjugate to the element @xmath283 ( in other words each element @xmath284 is a rotation by @xmath285 radians ) . by naturality , this means that the map @xmath288 has to give the non - trivial flat bundle over @xmath289 . given a basis for @xmath290 this is equivalent to asking that @xmath291 sends each of the two basis elements to rotations by @xmath285 , but around orthogonal axes . for each basis element of @xmath292 , there is an associated hopf link in figure [ 0-handles ] . say the hopf link is decorated by @xmath293 and there is a path connecting the two components of the hopf link as in lemma [ selfint ] . consider the loop which we gave as a generator of @xmath294 corresponding to the hopf link , and a loop based at @xmath295 which goes down to the path and follows it around until returning to the hopf link and then returns back up to @xmath295 . this gives two basis elements for @xmath221 of a @xmath289 representing the basis element of @xmath296 . the former is sent to @xmath266 by @xmath291 and the latter is sent to @xmath297 . since necessarily @xmath297 commutes with @xmath266 , the requirement that @xmath298 , ensures that @xmath297 is a rotation by @xmath285 around an axis orthogonal to that of @xmath266 . if we can find a presentation of some @xmath198 with decoration by some @xmath260 satisfying the conditions of lemmas [ genus0 ] , [ selfint ] , [ relatorsgoto0 ] , and [ swclass ] , then we have seen that we can construct a negative definite @xmath7-manifold @xmath219 with non - empty casson - type moduli space . in particular we have exhibited a particular representation giving such a presentation of @xmath198 is equivalent to giving first the singular link diagram @xmath261 and then giving a framing to the cores of each @xmath47-handle . therefore we have the following : [ constructionresult ] suppose we give a singular link diagram @xmath261 in the sense of definition [ g ] , starting with figure [ 0-handles ] and then adding arcs which begin and end at points of figure [ 0-handles ] . further suppose that there is a decoration of @xmath261 by @xmath260 that satisfies the conditions on the singular link diagrams given as the latter statements of lemmas [ genus0 ] , [ selfint ] , [ relatorsgoto0 ] , and [ swclass ] . then , if there exists a framing of the arcs of @xmath261 such that the corresponding @xmath47-handle additions to figure [ 0-handles ] gives a diagram of a trivial link , there exists a @xmath7-manifold with non - empty casson - type moduli space . @xmath126 + the symmetry group on @xmath7 elements @xmath300 can be embedded in @xmath15 as the rotational symmetry group of a cube . under this embedding , all elements of @xmath301 are taken to rotations by @xmath285 around some axis . in figure [ example ] we have given an example of a diagram ( of labelled hopf links , simple circles , and arcs ) satisfying all the conditions of theorem [ constructionresult ] . the group element decorations of the simple circles and the hopf links are given in the cycle notation for @xmath302 . if we can find a way to add more arcs , each satisfying the conditions of theorem [ constructionresult ] ( the steifel - whitney condition of lemma [ swclass ] has already been satisfied in the diagram ) such that when we replace each arc by a ribbon we get the unlink , then we will have described an immersion @xmath303 such that @xmath219 has non - empty casson - type moduli space . [ onepoint ] if figure [ example ] is an intermediary diagram of a movie presentation of an immersed @xmath198 satisfying theorem [ constructionresult ] , then @xmath219 has exactly @xmath47 point in the representation space @xmath112 . recent discoveries @xcite have indicated that the invariant defined as the signed count of the casson moduli space may always be @xmath226 . results such as proposition [ onepoint ] are still valuable as they may be useful in showing that links are not slice ( for more in this direction see @xcite ) . suppose that we have some new decoration satisfying theorem [ constructionresult ] . call the decorating elements of @xmath15 @xmath305 where the initials stand for _ _ t__op , _ _ b__ottom , _ _ l__eft , _ _ r__ight . each of the four decorations is a rotation by @xmath285 around some axis , so each element is equivalent to a choice of axis , and we use the same labels for these axes . by lemma [ swclass ] , we must have @xmath306 perpendicular to @xmath307 and @xmath308 perpendicular to @xmath309 . there is an arc connecting @xmath306 to @xmath307 . by condition lemma [ relatorsgoto0 ] we can interpret this as meaning that the unique axis perpendicular to both @xmath308 and @xmath309 lies in the same plane as @xmath306 and @xmath307 and is at an angle of @xmath310 to both of them . similarly , there is an arc connecting @xmath308 and @xmath309 , which implies that the axis perpendicular to @xmath306 and @xmath307 is in the same plane as @xmath308 and @xmath309 and at an angle of @xmath311 to both of them . it is a simple matter to convince oneself that any two ordered pairs of ordered pairs of perpendicular axes satisfying the conditions of the previous paragraph must be equivalent via the action of an element of @xmath15 . hence , up to conjugation , there is exactly @xmath47 representation @xmath304 of the correct characteristic class . @xmath126 + 99999 a. dold , h. whitney , _ classification of oriented sphere bundles over a 4-complex _ , the annals of mathematics vol . 3 , ( 1959 ) , 667 - 677 . s. donaldson , _ an application of gauge theory to four - dimensional topology _ , j. differential geom . 18 , no . 2 ( 1983 ) , 279 - 315 . s. donaldson , _ the orientation of yang - mills moduli spaces and 4-manifold topology _ , j. differential geom . 26 , no . 3 ( 1987 ) , 397 - 426 s. donaldson , p.b . kronheimer , _ the geometry of four - manifolds _ , oxford mathematical monographs , ( 1990 ) . m. furuta , h. ohta , _ differentiable structures on punctured 4-manifolds _ , topology and its applications , no . 51 ( 1993 ) , 291 - 301 . d. husemoller , j. milnor , _ symmetric bilinear forms _ , ergebnisse der mathematik und ihrer grenzgebiete , band 73 ( 1973 ) . p. b. kronheimer , _ four - manifold invariants from higher - rank bundles _ , j. differential geom . 70 , no . 1 ( 2005 ) , 59 - 112 . a. lobb , _ the dold - whitney theorem and slicing 2-component links _ , preprint d. mumford , _ an algebraic surface with @xmath312 ample , @xmath313 , @xmath314 . _ , amer . j. math . 101 ( 1979 ) , 233 - 244 . g. prasad ; s .- k . yeung , _ fake projective planes _ , invent . 168 , no . 2 ( 2007 ) , 321 - 370 . d. ruberman , n. saveliev , _ casson - type invariants in dimension four _ , geometry and topology of manifolds , fields inst . commun . 47 , ams ( 2005 ) , 281 - 306 . a. teleman , _ harmonic sections in sphere bundles , normal neighborhoods of reduction loci , and instanton moduli spaces on definite 4-manifolds _ , geometry & topology 11 ( 2007 ) , 1681 - 1730 . a. teleman , _ donaldson theory on non - khlerian surfaces and class vii surfaces with @xmath315 _ , invent . 162 ( 2006 ) , 493 - 521 . a. teleman , _ gauge theoretical methods in the classification of non - khlerian surfaces _ , preprint 2007 , to appear in postnikov memorial volume , banach center publications . r. zentner , _ a vanishing result for a casson - type instanton invariant _ , in preparation ( 2009 ) .
|
recently andrei teleman considered instanton moduli spaces over negative definite four - manifolds @xmath0 with @xmath1 .
if @xmath2 is divisible by four and @xmath3 a gauge - theoretic invariant can be defined ; it is a count of flat connections modulo the gauge group .
our first result shows that if such a moduli space is non - empty and the manifold admits a connected sum decomposition @xmath4 then both @xmath5 and @xmath6 are divisible by four ; this rules out a previously naturally appearing source of @xmath7-manifolds with non - empty moduli space .
we give in some detail a construction of negative definite @xmath7-manifolds which we expect will eventually provide examples of manifolds with non - empty moduli space .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
transport properties of small conducting structures are strongly influenced by size effects . oscillation of magnetoresistance in thin metallic films , and quantization of conductance in narrow wires and point contacts are examples of such effects . size effects in superconducting tunneling have attracted attention since early experiments by tomasch @xcite . in these experiments , oscillations of the tunnel conductance as a function of applied voltage were found for tunneling from a superconductor to a thin superconducting film of an ns proximity bilayer . the geometric resonance nature of the effect was clearly indicated by the dependence of the period of oscillations on the thickness of the superconducting film . similar conductance oscillations for tunneling into a normal metal film of ns bilayers were reported by rowell and mcmillan @xcite . later on an even more pronounced effect steps on the current - voltage characteristics of sins junctions at applied subgap voltages , @xmath5 was observed by rowell@xcite ( for a review see ref . ) . in addition to the dependence on the thickness of the n - film , the period of the current steps also shows temperature dependence which scales with the temperature dependence of the superconducting gap @xmath3 . the current steps occur at applied subgap voltages , @xmath5 , and they are understood as resonant features due to quasiparticle tunneling through superconducting bound states existing in ins wells at energies lying within the superconducting gap , @xmath6 , de gennes saint - james levels@xcite . recently , properties of superconducting bound states have attracted new attention in connection with observations of conductance anomalies in mesoscopic ns structures . resonant oscillations of the subgap conductance in mesoscopic quasi - ballistic ns junctions have been reported by morpurgo et al@xcite . the zero bias conductance peak , a related resonant phenomenon in diffusive ns junctions , was discovered even earlier@xcite and then intensively studied both theoretically and experimentally ( for a review see ref . ) . in these recent studies of mesoscopic junctions , attention was switched from the single - particle current through superconducting bound states to the two - particle ( andreev ) current . the traditional view of subgap current transport in proximity nins and sins structures considers single - particle tunneling into bound states in the normal region of the ins well@xcite , implicitly assuming that the normal region of the ins well plays the role of equilibrium reservoir . such a model is appropriate for low - transmission tunnel junctions with low tunneling rate compared to the inelastic relaxation rate . however , transparent mesoscopic structures are in a different transport regime where the bound levels are well decoupled from the superconducting reservoirs , and where injected quasiparticles escape from the ins well via andreev reflection.@xcite resonant two - particle current in quantum nins junctions has been theoretically studied in refs .. in sns junctions , the quasiparticles may undergo multiple andreev reflections ( mar ) before they escape into the reservoirs.@xcite in a number of recent experiments with ballistic sns devices fabricated with high mobility 2d electron gas ( 2deg ) @xcite the highly coherent mar transport regime has been realized . the purpose of this paper is to develop a theory of coherent multiple andreev reflections which will be applicable to 2deg - based sns junctions . in 2deg devices the separation of the superconducting electrodes @xmath7 is typically larger than 200 nm , which is of the same order of magnitude as the superconducting coherence length , @xmath8 ( @xmath9 is the fermi velocity of the 2d electrons ) , and superconducting bound states are formed well inside the energy gap . the presence of bound states in the junctions of finite length gives rise to resonances in the mar transport , which dramatically affects the subgap current . furthermore , it is possible in 2deg devices to use electrostatic gates to reach the quantum transport regime with a small number of electron modes and variable transmissivity . a theory of coherent mar has earlier been developed for short superconducting junctions,@xcite @xmath10 , where superconducting bound states do not play any significant role.@xcite such a theory is consistent with the physical situation in atomic - size superconducting point contacts , @xcite where quantization of conduction modes has turned out to be very helpful for detailed comparison between theory and experiment . the purpose of our study is to investigate the interplay between superconducting bound state resonances and coherent mar in long sns junctions , @xmath11 , in the quantum transport regime . in a number of publications , the coherent mar approach has been applied to long sns junctions@xcite . however , these studies were restricted to fully transparent junctions where the bound states are strongly washed out and the resonances are not pronounced ( in fact , as we will show , at zero temperature the current in such junctions does not show any structures ) . we will study junctions with arbitrary transmissivity , @xmath12 and pay special attention to the low - transparency limit , @xmath13 , where the resonance effects are most pronounced . the paper is organized as follows . in sec . [ sninsjunc ] we derive a 1d model for a gated ballistic 2deg device with one transport mode . in sec . [ details ] we construct a scheme for calculating mar amplitudes in terms of wave propagation in energy space . in sec . [ lowtransp ] , single current resonances are studied , and sec . [ interres ] is devoted to a discussion of the interplay between resonances in multi - particle currents . the properties of the total subgap current is discussed in sec . [ discussion ] . we consider an sns junction similar to the one discussed by takayanagi et al.@xcite schematically shown in fig . [ s2degs - dev ] . the junction consists of a normal conducting channel fabricated with a high mobility 2deg , which is confined between superconducting electrodes . the distance between the electrodes is comparable to the superconducting coherence length and small compared to the elastic and inelastic mean free paths and to the normal electron dephasing length . the superconductor-2deg interfaces are highly transmissive , the transmission coefficient typically exceeding a value 0.75 , and the number of conducting modes in the 2deg channel is controlled by a split gate . under these conditions , electrons ballistically move from one electrode to the other while occasionally being scattered by rare impurities or junction interfaces . under a voltage bias applied to the junction , the transport regime corresponds to fully coherent multiple andreev reflections ( mar ) . to calculate the dc current we will apply the scattering theory approach @xcite generalized for superconducting junctions ; see refs . and references therein . the normal electron propagation through the junction is generally described by the @xmath14-channel scattering matrix . by assuming the split gate to select only one transport mode , we will characterize the transport through this mode by energy - dependent transmission amplitude , @xmath15 , and reflection amplitudes , @xmath16 ( the energy @xmath17 is counted from the fermi energy ) . the scattering amplitudes satisfying the unitarity relations , @xmath18 , @xmath19 . the energy dispersion of the scattering amplitudes will introduce the normal electron ( breit - wigner ) and superconducting ( andreev ) resonances in the scattering problem . the effect of narrow breit - wigner resonances on coherent mar was earlier studied by johansson et al.@xcite and levi yeyati et al.@xcite here we will focus on the effect of andreev resonances and only consider breit - wigner resonances which are wide on the scale of the energy gap . this will allow us to neglect the energy dispersion of the junction transparency , @xmath20const . however , the scattering phases may depend on the energy , which yields the andreev resonances . in the simplest case , this dependence is a linear function within the energy interval @xmath21 , and we will write it on the form @xmath22 where @xmath23 are constant . in this case , the scattering properties of the normal channel are similar to those of a 1d nin junction . indeed , the corresponding 1d transfer matrix , @xmath24 can be decomposed into a product of three transfer matrices , @xmath25 where @xmath26 is a pauli matrix . the first and the last matrices describe ballistic propagation of an electron , with wave vector @xmath27 , through the right and left n - regions of an effective junction with lengths @xmath28 and @xmath29 respectively ( from right to left ) , and the matrix @xmath30 describes an effective barrier ( i ) . quasiparticle propagation through the effective 1d snins junction is described by means of the time - dependent bogoliubov - de gennes ( bdg ) equation , @xcite @xmath31 where @xmath32 is the normal electron hamiltonian , @xmath33 is the impurity potential and @xmath34 is the applied voltage . the superconducting order parameter @xmath35 is constant within the superconducting electrodes and zero within the normal region @xmath36(see fig . [ potentialdiag ] ) . in the further calculations , the impurity potential is described by the transfer matrix @xmath37 in eq . ( [ t ] ) . the spatial distribution of the applied potential along the channel is modeled with a step - like function @xmath38 . in fact , the actual spatial distribution of the potential does not play any role in this system : it can be included in the transfer matrix in eq . ( [ t ] ) , leading to an additional energy - independent shift in the scattering phases in the matrix @xmath37 . as we will see later [ comment after eq . ( [ resphase ] ) ] , the energy - independent phases in the @xmath37-matrix do not affect the current , and can therefore be excluded . the phase difference between the two superconductors follows from the josephson relation ( @xmath39 ) and introduces time dependence into the problem . the superconducting electrodes are considered to be equilibrium reservoirs where the quasiparticle wave function is a superposition of electron- and hole - like plane waves , @xmath40 in this equation , the @xmath41 signs in the time - dependent factors refer to the left / right electrode , @xmath42 is the wave vector of electron / hole - like quasiparticles , and @xmath43 , @xmath44 are the bogoliubov amplitudes . the ratio of the bogoliubov amplitudes equals the amplitude of andreev reflection for particles incoming from the neighboring normal region , @xmath45 since the time dependencies of the wave functions in the two reservoirs are different , the quasiparticle scattering by the junction is inelastic , and one has to consider a superposition of plane waves with different energies in order to construct scattering states . we will now proceed with the construction of recurrences for the scattering amplitudes following the method suggested by johansson et al . @xcite . to this end we introduce the wave functions in the left / right normal region ( @xmath46 ) of the junction with respect to the position of the impurity . a particular scattering state , labeled with the energy @xmath17 of the incoming quasiparticle , will consist of a superposition of plane waves with energies @xmath47 , where @xmath48 is an integer , @xmath49 , @xmath50\ ] ] @xmath51.\ ] ] the normal electron / hole wave vector @xmath52 is here defined as @xmath53 , @xmath54 . the meaning of the labels for the scattering ( mar ) amplitudes @xmath55 will be explained below . continuity of the scattering state wave function across the left and right ns interfaces determines the relation between the electron and hole amplitudes in the vicinity of each interface , @xmath56 which describes elastic andreev reflection ( l / r indices are omitted ) . it is convenient to consider scattering amplitudes near the impurity ( at @xmath57 ) rather than at the ns interfaces , and to rewrite eq . ( [ andrreflection ] ) for such amplitudes , combining the amplitudes of the ballistic propagation through the normal regions with the andreev reflection amplitude . then , in vector notation , @xmath58 the modified relation ( [ andrreflection ] ) takes the form @xmath59 where @xmath60 the phase @xmath61 , characterizing @xmath62 , is real inside the energy gap , @xmath63 , where it describes the total energy dependent phase shift due to ballistic propagation and andreev reflection . outside the gap , @xmath64 , the phase @xmath65 has an imaginary part which describes leakage into the superconducting reservoirs due to incomplete andreev reflections . by matching harmonics with the same time dependence in eq . ( [ ansatz ] ) , we derive a relation between scattering amplitudes at the left and the right side of the barrier : @xmath66 where the effective barrier transfer matrix @xmath67 is defined in eqs . ( [ te1 ] ) , ( [ t ] ) . the recursion relations in eqs . ( [ urelation ] ) and ( [ trelation ] ) couple the scattering amplitudes @xmath68 into an infinitely large equation system . this equation system describing coherent mar is illustrated by the mar diagram in fig . [ coeffpic ] . the electron part of the quasiparticle injected at the left ns interface propagates upwards along the energy axis , the amplitudes for this propagation being labeled with @xmath69 . at the injection energy @xmath70 ( amplitude @xmath71 ) , the quasiparticle is accelerated across the barrier ( i ) , where the potential drops . thus , it enters the right normal part of the junction with energy @xmath72 ( @xmath73 ) , undergoes andreev reflection and goes back as a hole ( @xmath74 ) , entering the left normal part of the junction having been accelerated to energy @xmath75 ( @xmath76 ) , and is then again converted into an electron ( @xmath77 ) . the @xmath41 indices label the amplitudes after ( @xmath78 ) and before ( @xmath79 ) the andreev reflection for propagation upwards along the energy axis . there is a similar trajectory of injected holes , which descends in energy , with the mar amplitudes labeled with @xmath80 . due to electron back scattering at the barrier , the upward and downward propagating waves are mixed , e.g. @xmath71 being not only forward scattered into @xmath73 , but also back - scattered into @xmath81 , which opens up the possibility of interference . injection from the left reservoir , shown in fig . [ coeffpic ] , generates a mar path which only connects even side bands at the left side of the junction with odd side bands at the right side . injection from the right reservoir will generate a different mar path , with even side bands at the right side of the junction , i.e. the diagram in fig . [ coeffpic ] will effectively be mirrored around the barrier ( i ) . thus , there are two independent equation systems for the mar amplitudes : injection from the left and from right . the @xmath82 labels in the mar amplitudes can then be omitted since they are uniquely defined by the source term and the side band index . the transport along the energy axis generated by mar , from energy @xmath17 to @xmath83 , is conveniently described by the effective transfer matrix @xmath84 , @xmath85^{-1}\hat { c}_{0- } , \ \ n<0\ ] ] @xmath86 where @xmath87 and @xmath88 for the injection from the left ( for the injection from the right , the even and odd side band indices are interchanged ) . for paths within the superconducting gap , @xmath89 , the matrix @xmath84 satisfies the standard transfer matrix equation , @xmath90 , which provides conservation of probability current along the energy axis , @xmath91 an important consequence of the coherence of mar is the possibility of transmission resonances in energy space . from the form of the @xmath92-matrix , @xmath93 it is evident that when @xmath94 , the two matrices @xmath95 and @xmath67 will cancel each other , and the probability of transmission through this part will be unity , which leads to resonant enhancement of mar . the solutions @xmath96 of the resonance equation , @xmath97 coincide with the spectrum of the de gennes saint - james levels localized in ins quantum wells @xcite . the corresponding bound states are located either on the left or the right side of the junction . without loss of generality , the calculations can be performed with a real matrix @xmath67 . the transformation to such a real matrix is given by @xmath98 , with diagonal unitary matrices @xmath99 whose elements are constructed with the scattering phases , which are energy - independent . it is clear from eq . ( [ mresstruc ] ) that since these energy - independent matrices commute with the matrices @xmath100 , they cancel each other , and the matrix @xmath84 undergoes a similar transformation . this will lead to an overall phase shift of the scattering state which does not affect the current . it is interesting to consider the special case of fully transparent junctions , @xmath101 , which has been studied in the literature @xcite . in this case , all matrices @xmath102 in eq . ( [ mdef ] ) are equal to the unity matrix , and the @xmath92-matrix takes the simple form @xmath103 . the length of the junction then enters only through the phase of the mar amplitudes , which drops out of the side band current . thus the dc current of fully transparent sns junctions is independent of length and equal to the current in quantum constrictions @xcite . in particular , at zero temperature this current does not show any structures in the subgap current . it is also worth mentioning that in this particular case of fully transparent sns junctions , the @xmath92-matrix is diagonal and therefore a closed set of recursive relations can be derived for the mar probabilities ( not just for the mar amplitudes , as in the general case ) , equivalent to the equations for distribution functions derived in the original paper by klapwijk , blonder and tinkham @xcite . equation ( [ mar ] ) describes `` source - free '' propagation along the mar ladder . to complete the set of equations for the mar amplitudes we need to take into account quasi - particle injection , which introduces a source term in eq . ( [ mar ] ) . to this end , let us consider a quasiparticle incoming from the left superconducting electrode with energy @xmath17 , having a wave function of the form @xmath104 .\ ] ] the two terms in this equation refer to electron- ( @xmath105 ) and hole - like ( @xmath106 ) injected quasiparticles . we now include this wave function into the continuity condition at the ns interface at energy @xmath17 , which gives us the following relation between the mar amplitudes @xmath107 and @xmath108 , @xmath109 @xmath110 for quasiparticles injected from the right , a similar equation holds with the substitutions @xmath111 , and @xmath112 . equations ( [ mar ] ) and ( [ source ] ) give a complete set of equations for the mar amplitudes with the boundary conditions @xmath113 at infinity . a formal solution of eqs . ( [ mar ] ) and ( [ source ] ) , which is useful both for numerical calculations and analytical investigations , can be constructed by reducing this infinite set of recursion relations to a finite set by representing the mar process above @xmath114 and below @xmath115 by boundary conditions involving reflection amplitudes @xmath116 and @xmath117 , defined as @xmath118 and @xmath119 . this gives the following representation for the vectors in eq . ( [ coeffdef ] ) , @xmath120 the reflection amplitudes @xmath116 and @xmath117 are independent of the injection , in contrast to the coefficients @xmath121 . furthermore , they are determined by the boundary conditions @xmath113 and can be expressed in terms of the matrix elements of @xmath122 and @xmath123 , where @xmath124 , @xmath125^{-1}\left ( \begin{array}{c } r_{0- } \\ 1 \end{array } \right)=0.\ ] ] in other words , the vectors in eq . ( [ lim ] ) are equal to the asymptotical values of the eigenvectors of @xmath126-matrices corresponding to the eigenvalues which decrease when @xmath14 goes to infinity . the advantage of introducing the reflection amplitudes @xmath116 and @xmath117 is that although they have to be calculated numerically , the recurrences which they obey do not contain resonances and converge rather quickly . this is in contrast to the matrix @xmath84 , which does possess resonances , but which can be calculated analytically in a straight forward way for any given @xmath48 . the solutions of the recursion equations eqs . ( [ urelation ] ) and ( [ trelation ] ) can now be explicitly written down . for any given energy @xmath17 we get four different sets of solutions for four scattering states including electron / hole injection from the left and the right . using the formal expression in eq . ( [ nullvectors ] ) and the matrix elements of @xmath127 , the solutions for injection from the left ( @xmath128 ) have the form , @xmath129 } { m_{22}+m_{21}r_{0-}e^{2i\varphi_{0 } } -m_{12}r_{n+}e^{2i\varphi_{n } } -m_{11}r_{0-}r_{n+}e^{2i\varphi_{0}+2i\varphi_{n } } } \left(\begin{array}{c } 1\\ r_{n+ } \end{array}\right).\ ] ] the solutions for injection from the right can be found through interchanging @xmath130 and calculating all quantities with respect to injection from the right . the solutions for @xmath131 are calculated in a similar manner . now turning our attention to the current , we calculate it in the normal region next to the barrier , using the wave function in this region , @xmath132 , and assuming quasiparticle equilibrium within the electrodes . the current then takes the form , @xmath133 where @xmath134 is the superconducting density of states , and the sum is over the four scattering states at a given energy @xmath17 associated with the electron- and hole - like quasiparticles ( e / h ) injected from the left and right ( l / r ) . the current can be divided into parts with different time dependence and expressed as a sum over harmonics : @xmath135 focusing on the dc ( @xmath136 ) component , and calculating the contribution of each scattering state at the injection side of the junction , we express the current spectral density @xmath137 through the probability currents of electrons and holes at energies @xmath138 ( fig . [ coeffpic ] ) , @xmath139 @xmath140 these currents coinside with the probability currents @xmath141 , eq.([jp ] ) , flowing along the energy axis . it is convenient to introduce a leakage current @xmath142 , defined as the difference of the probability currents before and after andreev reflection , @xmath143 @xmath142 represents the amount of probability current from all the scattering states injected at energy @xmath17 and leaking out of the junction at energy @xmath114 ( fig . [ coeffpic ] ) . the leakage current is zero inside the energy gap due to complete andreev reflection , @xmath144 , @xmath145 [ cf . ( [ jp ] ] . the explicit expression for the leakage current for @xmath146 follows from eq . ( [ leakage ] ) after insertion of eqs . ( [ coeffsol ] ) and ( [ urelation ] ) , @xmath147 it follows from eq . ( [ ipn ] ) that the leakage currents are positive for all @xmath146 , @xmath148 . one can also show that they satisfy the inequality @xmath149 , which is a consequence of the conservation of probability current : the leakage current of all side bands except of the side band @xmath150 does not exceed the probability current injected into four scattering states . furthermore , the leakage current satisfies the important detailed balance equation @xcite , @xmath151 i.e. the leakage at energy @xmath152 due to the injection at energy @xmath17 is the same as the leakage at energy @xmath17 due to injection at energy @xmath152 . using the continuity of current across the barrier , @xmath153 , guaranteed by the transfer matrix @xmath67 , we can express the probability currents in eq . ( [ dccurrent ] ) through the leakage current , eq . ( [ leakage ] ) , @xmath154 @xmath155 by adding and subtracting consecutive terms in the sum . the spectral density of the dc charge current eq . ( [ dccurrent ] ) can then be written on the form @xmath156 since @xmath142 appears in @xmath48 probability currents . this formula has a clear physical meaning : the contribution to the charge current of the @xmath48-th side band is proportional to the leakage current of the side band times the effective transferred charge @xmath157 . the detailed balance of the leakage currents , eq . ( [ cancel ] ) , allows us explicitly to prove that at zero temperature the scattering processes between ( occupied ) states with negative energies , @xmath158 do not contribute to the current , in agreement with the pauli exclusion principle . indeed , by separating the contributions from side bands with @xmath131 and remembering that the leakage current is zero within the gap , we get for zero temperature , @xmath159 where the first and the third terms cancel each other by virtue of eq . ( [ cancel ] ) . at finite temperature , these two terms produce current of thermal excitations while the second term gives the current of real excitations created by the voltage source . keeping only this term which dominates at low temperature , we finally get , @xmath160 we end this section by noting a technically useful symmetry in the current density , namely @xmath161 , seen from the explicit form of the @xmath84-matrix . this allows us to reduce the integration interval in eq . ( [ npartcurrent ] ) to @xmath162 . the approach formulated above provides necessary foundations for numerical calculation of the current for arbitrary transparency and length . however , to get a full understanding of the rich subgap structure in the current - voltage characteristics , which may seem quite random , especially for intermediate transparencies and lengths ( see figs . [ itotd1plots]-[itotl4plots ] ) , we will conduct a detailed analytical study of the limit of low transparency @xmath13 . the separation of currents into @xmath48-particle currents , eq . ( [ npartcurrent ] ) , is our basis for analysis , and we will study each current @xmath163 separately . as explained in the previous section , the de gennes saint - james levels , eqs . ( [ mresstruc ] ) and ( [ resphase ] ) , are important for the current transport through the junction , leading to resonant enhancement of the current . our main attention in this and the next section is on the calculation of the position , height and width of the main current peaks and oscillations which have the magnitude of order @xmath2 . to simplify notations , left / right injection indices are omitted in most cases . the single - particle current , which dominates at large applied voltages , has , according to eq . ( [ npartcurrent ] ) , an onset at @xmath164 . the full numerical solution for the single - particle current is plotted in fig . [ i1d1plots ] . the current shows pronounced oscillations , and the magnitude of the slope at the current onset strongly depends on the junction length . to understand this behavior , we analyze eq . ( [ ipn ] ) in the limit of small transparency @xmath0 , i.e. in the tunnel limit . first we note ( see appendix [ refexp ] ) that the reflection amplitudes @xmath116 and @xmath117 may be expanded as @xmath165 @xmath166 after inserting the explicit form of @xmath167 together with the expansion ( [ reflectionexp ] ) into eq . ( [ ipn ] ) and putting @xmath168 , we can write the single particle current on the form @xmath169 \tanh(|e|/2k_bt),\ ] ] where @xmath170 in analogy with the tunnel formula for the current , @xcite @xmath171 is identified as the tunneling density of states ( dos ) on the left / right side of the junction . in fig . [ densplot ] the energy dependence of the dos is presented . the deviation of this dos from the normal metal density of states is a manifestation of the proximity effect . the expression ( [ densityofstates ] ) for the dos has earlier been derived for proximity ns sandwiches @xcite . note that the dos in our case is constant throughout the n - regions . in junctions with arbitrary length , the dos usually approaches zero at the gap edge @xmath172 ( ref . ) . exceptions are junctions with lengths @xmath173 where a bound state splits off from the gap edge . in this case , the dos diverges at the gap edge . the quantum well structure of the sns junctions also give rise to quasi - bound states in the continuum spectrum , @xmath174 , seen as oscillations in the dos . the single particle current in eq . ( [ i1smalld ] ) is written as the integral over the product of the dos at the entrance energy @xmath17 and the exit energy @xmath175 . the latter depends on the applied voltage , as well as the integration interval , and therefore the dos oscillations produce oscillations of the current @xmath176 as a function of voltage ( rowell - mcmillan oscillations @xcite ) . the oscillations become more pronounced when the junction is sufficiently long , and the differential conductance may even become negative . it is also clear that the dos oscillates as a function of the length of the junction , which give rise to oscillations also in @xmath176 . in short junctions , @xmath177 , the current onset at @xmath164 is very steep , see fig . [ i1d1plots ] . in junctions with finite length , the current onset is smeared and replaced with a smooth oscillating behavior . this can be directly related to the smearing of the singularity in the dos at the gap edge . the length where the crossover between these two behaviors occurs can be taken as a measure of when finite length effects become important . to estimate this length , we write equation ( [ ipn ] ) for small lengths @xmath178 , near the threshold , @xmath179 , @xmath180 , keeping the first order terms in @xmath2 in the denominator . for a symmetric junction , @xmath181 , we get @xmath182 from this formula it is clear that for short junctions ( @xmath183 ) , the current onset has the width @xmath184 . if @xmath7 is of the order of @xmath185 , the size of the onset has substantially diminished , and there is no visible onset at @xmath164 when @xmath186 . this crossover between steep onset and smooth behavior , which happens already for quite short lengths if @xmath2 is small , can be interpreted in terms of a bound state which is situated exactly at the gap edge in short junctions ( @xmath183 ) , and which moves down into the gap when @xmath187 , the effect becoming fully pronounced when the distance from the gap edge , @xmath188 , exceeds the dispersion of the andreev state , @xmath189 , in symmetric junctions@xcite . when @xmath190 approaches @xmath191 , the lowest quasi - bound state in the continuum spectrum approaches the gap edge . this leads to an accumulation of the spectral weight at the gap edge and reappearance of the singularity in the dos , which results in the reappearance of a sharp current onset at @xmath164 , but with smaller magnitude ; see fig . [ i1d1plots ] ( @xmath192 ) . it is of interest to note that in our calculations , based on the scattering theory approach , the bound states are not directly involved in the single - particle transport , which therefore is non - resonant and shows no subgap resonance peaks . within the tunnel model approach the situation is qualitatively different : the dos in eq . ( [ i1smalld ] ) usually includes the contribution of the broadened bound states , and therefore the single - particle current exists and has pronounced resonant features at subgap voltages @xmath5 . this difference results from the fact that , within the tunnel model approach , the superconducting bound states are implicitly assumed to be connected to the reservoirs ( broadening due to inelastic interaction ) , which allows a stationary current to flow through the bound states . in contrast , within the scattering approach , the bound states are disconnected from the reservoirs and have zero intrinsic width . in this case the bound states obtain their width only due to higher order tunneling processes involving andreev reflections , which are manifested by the resonant multi - particle currents . in practice , the relevance of the multi - particle versus single - particle mechanism of the subgap current transport is determined by physics and depends on the ratio of the corresponding dwelling and relaxation times@xcite . in this paper , the inelastic relaxation time @xmath193 which determines the width of the single - particle resonances is assumed to be much larger than the dwelling time of the most important two - particle current , @xmath194 . the two - particle current @xmath195 in quantum point contacts , ( @xmath178 ) is of order @xmath196 when @xmath5 and of order @xmath197 when @xmath198 ( ref . ) . for finite length junctions , the situation is different . for the mar paths where the energy of the andreev reflection coincides with a bound state , the current spectral density @xmath199 is of order unity , due to resonant transmission through this state . for low transparency @xmath0 , this gives a sharp concentration of the current density around the resonant energies . in this limit , the two - particle current is well described by the sum of contributions from these resonances , and to evaluate them we examine the energy dependence of @xmath200 close to the resonant energies , @xmath201 . let us consider the contribution to the leakage current @xmath202^{l}$ ] from quasi - particles injected from the left . as shown in appendix [ twopartres ] , in this case eq . ( [ ipn ] ) reduces to the standard breit - wigner resonance form @xmath203^{l}=\frac{\gamma_{0}^{(m ) } \gamma_{2}^{(m)}}{\displaystyle \left(\frac{\delta e-\delta e^{(m)}}{\delta}\right)^{2}+ \left(\frac{\gamma_{0}^{(m)}+\gamma_{2}^{(m)}}{2 } \right)^{2 } } , \ ] ] where the tunneling rates @xmath204 are given by @xmath205 , @xmath206 , and @xmath207 and the position of the resonance is shifted by @xmath208 an analogous result is valid for quasiparticles injected from the right . after integrating over energy , the two - particle current in the resonance approximation may be written on the form @xmath209 where the summation is over the positive bound level energies , @xmath210 , and the dos @xmath211 should be calculated at the injection side of the junction and @xmath212 $ ] . according to eq . ( [ i2approx ] ) , the two - particle current @xmath213 increases in a step - like manner in the voltage region @xmath214 . the steps occur at every voltage where a new resonant channel through a bound state opens up , at @xmath215 . we note that the step positions depend on temperature and approximately scale with @xmath3 . each current step has the height of order @xmath2 . as seen from eq . ( [ i2approx ] ) , the contribution to the current of a particular bound state @xmath96 is modulated , as a function of voltage , by the oscillations of the density of states at the entrance and exit energies , @xmath216 . in other words , the pronounced oscillations of the two - particle current seen in fig . [ i2d1plots ] reflect how close the entrance and exit energies @xmath217 are to a quasi - bound state in the continuum . for @xmath198 , the two - particle current @xmath195 oscillates around a constant value with an amplitude of oscillation decreasing as @xmath218 for large voltages . it is interesting to compare the resonant structures of the two - particle current with the resonant structures in nins junctions.@xcite in nins junctions , the resonant current steps occur at @xmath219 , and they do not have any modulation because the dos on the normal side of the junction is constant . the distance between the resonances and the resonance widths are proportional to the bound level spacing , and they decrease in long junctions . for sufficiently long junctions , the two - particle current may thus give the appearance to include a series of peaks , as shown on fig . [ i2comp ] . in symmetric junctions , the bound state energies at both sides of the barrier will coincide , reducing the number of steps with a factor of two , and giving current steps of double height . we will conclude this subsection by noting that the difference between the full numerical calculation of the two - particle current and the resonant approximation given in eq . ( [ i2approx ] ) is rather small already when @xmath220 , as can be seen in fig . [ i2comp ] . excess current in sns junctions , i.e. the difference between the current in superconducting junction and in the normal junction at large voltage , @xmath221 is commonly considered as a measure of the intensity of andreev reflection . in tunnel sis junctions and low - transmissive point contacts the excess current is small , @xmath222 , @xmath13 , while in fully transparent contacts the excess current is large , @xmath223 , @xmath101.@xcite accordingly , one would expect large excess current in long sns junctions due to the resonant enhancement of the two - particle current . however , the excess current is small because of a large deficiency , of order @xmath2 , of the single - particle current caused by the broadening of the current onset at the threshold . as we will show , the single - particle and two - particle currents undergo a fine cancellation , yielding small net excess current of order @xmath196 when @xmath13 . the excess current has contributions only from the single- and two - particle currents , since all higher order currents include at least one andreev reflection outside the gap whose probability is of order @xmath224 . in the limit of large voltage , @xmath225 , the relevant part of the current in eq . ( [ npartcurrent ] ) then takes the form @xmath226 @xmath227 these equations are written for symmetric junctions , @xmath228 , and for zero temperature ; small andreev reflection amplitudes @xmath229 have been neglected in eq . ( [ ipn ] ) . the behavior of the current in eq . ( [ ihighvoltage ] ) as a function of voltage is presented in fig . [ excessapproach ] for different lengths . it is clearly seen that the limiting value of the excess current is approached much faster in finite length sns junctions compared to point contacts ( @xmath183 ) . in fig . [ excessplot ] the excess current behavior with respect to the junction length is presented for different transparencies . to analytically examine the excess current in the limit of small transparency , @xmath13 , it is convenient to start with equations ( [ i1smalld ] ) and ( [ i2approx ] ) . to first order of @xmath2 the excess current assumes the form ( @xmath230 ) , @xmath231 @xmath232de,\ ] ] @xmath233 let us consider the contributions to the single - particle current from the left electrode , @xmath234^{l}=-\frac{2ed\delta}{h } + \frac{2ed}{h}\int_{\delta}^{\infty}\left [ n^{l}(e)-1\right]de.\ ] ] inserting @xmath235 from eq . ( [ densityofstates ] ) , this equation can be transformed to the form , @xmath236^{l}= \frac{2ed}{h}\int_{\delta}^{\infty}\left(\frac{e\xi } { \xi^{2}+\delta^{2}\sin^{2}(2el_{l}/\delta\xi_{0})}- \frac{e}{\xi}\right)de= -\frac{ed}{h}\int_{-\infty}^{\infty}d\xi \frac{\sin^{2}(2el_{l}/\delta\xi_{0})}{\xi^{2}+\sin^{2 } ( 2el_{l}/\delta\xi_{0})},\ ] ] where @xmath237 . it is now possible to analytically continue the integral in the upper half plane which will reduce the integral to a sum over the residues of the poles given by the equation @xmath238 . comparing this equation with eq . ( [ resphase ] ) we find that the poles coincide with the energies of the bound states in the gap . the excess current contribution from the left - injected single particle current is thus @xmath236^{l } = -\frac{2d\pi e\delta}{h}\sum_{m\geq0 } \frac{1}{\eta^{(m)}}=-\left[i^{exc}_{2}\right]^{l},\ ] ] where @xmath239^{l}$ ] is the contribution to the two - particle current from the bound state resonances at the left electrode . a similar relation is derived for current from the right electrode . thus , there is exact cancellation of the excess single - particle and two - particle currents to first order in @xmath2 . it is interesting to note that the cancellation effect is related to the conservation of the number of states in a proximity normal metal compared to the conventional normal metal . it follows from eq . ( [ i1excl ] ) that @xmath240 is equal to the difference between the number of continuum states in the proximity metal and the total number of states in a conventional metal , while , on the other hand , the number of the bound states , is equal to @xmath241 according to eq . ( [ iexcsmalld ] ) . for processes with several andreev reflections ( @xmath242 ) , the possibilities for resonances increase . every andreev reflection energy may coincide with a bound state energy and thus be resonant . for some specific voltages , more than one resonance is important , creating a situation of overlapping resonances , which can enhance the current giving peaks in the current - voltage characteristics at these voltages . the three - particle current @xmath243 has a non - resonant value of order @xmath244 . however , @xmath243 is enhanced to order @xmath196 when the energy of one of the two andreev reflections coincides with a bound state energy . for the applied voltage equal to the difference between two bound state energies , @xmath245 , two resonances occur simultaneously , i.e. form a resonance consisting of two overlapping single resonances ; see the inset in fig . [ i3d1plots ] . this will enhance the current to order @xmath2 close to this voltage , giving a peak in the ivc . the number of peaks is equal to the number of bound state pairs . the peaks are located in the voltage interval @xmath246 ; we note that the peak positions are weakly dependent on temperature . to evaluate the height and the width of these peaks , we study the contribution from overlapping resonances at @xmath247 and at @xmath248 . close to these energies , the phases @xmath249 and @xmath250 , defined in eq . ( [ resphase ] ) , are close to zero , and we find the current spectral density for injection of a quasiparticle from the left ( see appendix [ threepartres ] ) , @xmath251^{(km),l}=\frac { d^{3 } n^{l}(e)n^{r}(e_{3 } ) } { \left|d-4\varphi_{1}^{(k)}\varphi_{2}^{(m ) } + id\left(\varphi_{1}^{(k)}n^{r}(e_{3})+\varphi_{2}^{(m)}n^{l}(e ) \right)\right|^{2 } } .\ ] ] we now expand @xmath252 , @xmath253 in the deviation from perfect overlap in energy , @xmath254 , and in voltage , @xmath255 , and find , using @xmath0 , from eq . ( [ ip3res ] ) @xmath256^{(km),l } = \frac{d \gamma_{0}^{(k ) } \gamma_{3}^{(m ) } } { \left(\delta e_{+}\delta e_{-}/\delta^{2 } - d/4\eta^{(k ) } \eta^{(m ) } \right)^{2}+\lambda^{2}},\ ] ] where @xmath257 , @xmath258 . the energy dependence of the current in eq . ( [ doublebreit ] ) has the form of two resonant peaks with width @xmath259 split by the energy interval @xmath260 at @xmath261 , the peak splitting increasing with increasing @xmath262 . after integration over energy , the overlapping resonances give a current contribution in the form of a current peak ( @xmath263 ) , @xmath264 in this equation , the densities of states @xmath171 are taken at the entrance and exit energies , @xmath265 and @xmath266 , and the temperature is taken to be zero . a factor of @xmath267 has been included in eq . ( [ i3res ] ) to take into account the similar resonant process for injection from the right , where @xmath268 and @xmath269 . the curve for the three - particle current versus voltage thus consists of peaks with heights of order @xmath2 and half - width @xmath270 on top of a background of order @xmath196 . the background current increases with voltage in the interval @xmath271 as more single resonances come into the integration region . in the interval @xmath272 , the background current decreases due to broadening of the resonances because of leakage associated with incomplete andreev reflection outside the gap . in long symmetric junctions the current peaks form an interesting triangular pattern . to see this , we first note that if the bound state spectrum were perfectly linear , several of the peaks described by eq . ( [ ip3res])-([i3res ] ) will be situated at the same voltage since @xmath273 ( see also the inset in fig . [ i3long ] ) , and thus the total number of peaks will be reduced while their respective height will be increased . since the bound state spectrum is not linear , the peaks show splitting . however , the deviation from linearity is small and in practice the peaks form clusters , giving combined peaks with height roughly equal to the number of clustering peaks . in the interval @xmath271 , the number of peaks in a cluster increases in steps of three from @xmath274 to @xmath275 , etc . , up to the number of bound states . in the interval @xmath214 the number of peaks in a cluster decreases in steps of one . this gives an appearance of a `` peak triangle '' for very long junctions , shown in fig . [ i3long ] . this `` peak triangle '' is further enhanced by the background current , which has a similar triangular form , as explained above . the four - particle current has a non - resonant value of order @xmath276 , which is enhanced to order @xmath244 when the energy of one of the three andreev reflections coincides with a bound state energy . similar to the three - particle current , overlapping resonances can enhance the magnitude of the current @xmath277 to the order @xmath2 for those voltages where both the first and the third andreev reflections coincide with the bound states , as shown in the upper inset in fig . [ i4d1plots ] . indeed , it is clear from the explicit form of @xmath278 that when @xmath279 and @xmath280 , then @xmath281 , i.e. the transparency of the mar trajectory is enhanced to unity . other combinations of the resonances , e.g. when the first and the second andreev reflection occur at bound state energies , will produce peaks of order @xmath196 or smaller , as described in appendix [ fourpartres ] . focusing on the double resonances which produce large ( @xmath282 ) current peaks , we find that in short junctions with just one pair of bound states , @xmath283 , the double resonance will occur at voltage @xmath284 , provided the energy of the bound state is within the interval @xmath285 . the spectral density of the current has a form similar to the one in eq . ( [ doublebreit ] ) , the major difference being the small peak splitting@xcite , @xmath286 . the height of the resulting current peak ( @xmath263 ) is @xmath287_{max}=\frac{\pi de\delta}{h } \frac{1-\left[a(2e^{(0)})\right]^{4 } } { 1+\left[a(2e^{(0)})\right]^{4 } } , \ ] ] where @xmath288 is the andreev reflection amplitude at energy @xmath289 . for longer junctions , there are many possibilities to have overlapping resonances . two bound states at one side of the junction with energies @xmath290 and @xmath291 can give a peak in @xmath277 if @xmath292 . although the height of all peaks is roughly proportional to @xmath2 , numerically the heights ( and widths ) of the peaks may vary considerably depending on the position of the second andreev reflection . if the second andreev reflection does not occur at the energy of a bound state , the situation is similar to the one described above ; see lower inset in fig . [ i4d1plots ] . however , if a bound state is close to the energy of the second andreev reflection , then the current spectral density @xmath293 consists of the three full - transmission peaks with widths @xmath286 which are split within the interval @xmath294 ( triple resonance ) . the triple resonance has larger spectral weight compared to the double resonance , which results in the larger height and width of the current peak . rigorously speaking , a triple resonance can only occur in asymmetric junctions because it requires equal distance between neighboring resonances , while the bound state spectrum in symmetric junctions is not equidistant . however , in long junctions , the deviation from the equidistant spectrum is small , and quasi - triple resonances may therefore occur also in long symmetric junctions . this effect can be observed in fig . [ i4d1plots ] , where the four - particle current for a symmetric junction with length @xmath295 consists of three peaks with different heights : the central peak corresponding to the quasi - triple resonance while the two side peaks corresponding to the double resonances with the heights given by eq . ( [ i4max ] ) . finally , it is worth noting that , similar to the situation for the three - particle current , the peaks will form clusters , giving a smaller number of current peaks than the number of pairs of bound states in long junctions . the studied properties of multiple resonances in three- and four - particle currents allow us to make some general conclusions about resonant behavior of the high order multi - particle currents which determine the total current at small voltage . the non - resonant magnitude of an @xmath48-particle current is of order @xmath296 at the threshold voltage , @xmath297 , and therefore the total non - resonant current exponentially decreases with the applied voltage ( in transparent junctions , @xmath298 , the current is exponentially small at@xcite @xmath299 ) . however , multiple resonances may enhance the magnitude of the current by several orders of @xmath2 . the major question of interest here concerns the maximum value of the resonant current , in particular whether it can be of order @xmath2 at arbitrary small voltage . to obtain such large current at small voltage , it is necessary to achieve a transmission probability through a high order mar path equal to unity , which implies that the energy of at least every other andreev reflection must coincide with a bound state ( cf . the discussion in the previous subsection ) . for @xmath300 , this means that three or more bound states must be approximately equidistant in energy . since the bound state spectrum is non - equidistant , eq . ( [ resphase ] ) , this is generally not possible if the resonances are narrow ; therefore , in junctions with arbitrary geometry and small transmissivity there are no large current peaks below the voltage @xmath301 . however , the possibility of a large resonant current exists for junctions with sufficiently large transparency . to find the relevant transparency , let us consider a very long symmetric junction and assume for the moment that the bound state spectrum is equidistant , @xmath302 . then , from mapping of the @xmath48-th order mar process on a 1d multi - barrier structure ( see fig.[quasi - resonance ] ) , it is clear that if the applied voltage is commensurate with the level spacing , e.g. , @xmath303 , the multi - barrier structure is periodic , and full transmission is achieved leading to a current peak . this conclusion is valid also for a non - equidistant spectrum if the variation of the interlevel distance does not exceed the width of the full - transmission band . the deviation of the bound state spectrum from the best linear fit does not exceed the value 0.33@xmath304 , fig.[quasi - resonance ] . on the other hand , the width of the full - transmission energy band is @xmath305 for equidistant spectrum and for @xmath306 . thus one should expect large current structures in long symmetric junctions with transparency @xmath307 to occur at voltages @xmath308 . in junctions with smaller transparency , large current structures may appear only at @xmath309 , as explained before ; see fig . [ itotl4plots ] . it is also easy to see that in asymmetric junctions , where the width of the full transmission band for an equidistant spectrum is @xmath286 ( since the relevant resonances at one side of the junction are weakly coupled to each other through the mar process ) , large resonant current at small voltage may exist if @xmath310 . our numerical investigations confirm that in symmetric junctions when @xmath2 is of the order @xmath311 , the multiple resonances are completely blocked and current peaks are exponentially suppressed at @xmath312 . adding up the contributions to the current calculated in this paper , we arrive at a rather complex form of current - voltage characteristics ( ivc ) at subgap voltages , as shown in figs . [ itotd1plots]-[itotl4plots ] . nevertheless , the analysis of the tunnel limit allows us to classify various subgap current structures . here we will summarize the results of this classification . as a reference system we will take a short ( @xmath183 ) junction where the form of the ivc is well studied @xcite . the current structures in short junctions can be interpreted as resonant features due to quasi - bound states situated at the edges of the energy gap @xcite , the resonant conditions selecting voltages equal to the gap subharmonics , @xmath313 . this subharmonic gap structure of the short junction gradually changes with increasing junction length as bound states move down into the gap , giving rise to ivc structures with steps , oscillations and peaks . major points are : \(i ) the current in the subgap region is considerably enhanced , compared to the short junction case . this effect is present as soon as the effective length @xmath314 is comparable to , or larger than , the square root of transparency of the junction , @xmath315 . \(ii ) the main onset of the current in short junctions at @xmath164 shifts downwards in voltage to the value @xmath316 where @xmath317 is the energy of the bound state . this shift is caused by the resonant two - particle current giving a contribution to the total current of the order of the single - particle current . \(iii ) for longer junctions , the current onset transforms into a staircase within the voltage interval @xmath214 with the number of steps corresponding to the number of bound states , the step positions being given by @xmath318 . this is due to the resonances in the two - particle current transported through bound states . resonant channels open up , one by one , as the voltage increases and bound states enter the `` energy window '' available for two - particle processes . the current plateaus are not flat but modulated because of oscillations of the density of continuum states . the period of the modulation is roughly equal to the interlevel distance , and it decreases with the junction length . the amplitude of the modulation , on the other hand , increases with the junction length . thus , in long junctions , the current structures take the form of a series of peaks ( see fig . [ itotd1plots ] ) within the voltage interval @xmath214 . the position of the peaks has pronounced temperature dependence , scaling with the temperature dependence of the order parameter , while the distance between peaks has a weak temperature dependence . \(iv ) there is another series of the current peaks whose positions only weakly depend on temperature and are entirely determined by the bound state spectrum : @xmath319 and @xmath320 . these peaks are caused by the overlap of two resonances in the three- and four - particle currents and they exist in the intervals of applied voltage @xmath246 and @xmath321 respectively . the heights of these peaks are comparable with the heights of the two - particle current structures ( @xmath282 ) . \(v ) at voltage smaller than @xmath301 the resonant current structures generally become smaller in magnitude ( at least by one order in @xmath2 ) if the junction transparency is sufficiently small ( @xmath322 ) , and the current decays exponentially when @xmath323 approaches zero ( although for some particular junction lengths there could be huge ( @xmath282 ) current peaks caused by multiple resonances ) . this qualitative difference of the ivc below and above @xmath301 allows one to expect a cross over from power to exponential dependence of ivc in multichannel junctions . \(vi ) in transparent junctions , all current structures will persist but become smooth ; appreciable current will appear below @xmath301 as soon as @xmath324 . the current structures completely disappear in fully transparent junctions , @xmath101 , where the ivc does not depend on the junction length ; see fig . [ itotl4plots ] . \(vii ) at voltage larger than @xmath325 , the current undergoes oscillations , similar to rowell - mcmillan oscillations @xcite , and the excess current is approached much faster than in short junctions . in low transparency junctions the excess current is small , @xmath326 , despite strong andreev reflection and large pair current @xmath327 . we thank e. n. bratus , j. lantz and t. lfwander for discussions . support from nfr and nutek ( sweden ) and from nedo ( japan ) is gratefully acknowledged . in this appendix , the expansion is derived for the reflection amplitudes in eq . ( [ reflectionexp ] ) for a quasiparticle injected from the left . from the definition of @xmath117 and @xmath328 , eq . ( [ nullvectors ] ) , we know @xmath329 @xmath330 they are related as @xmath331 where @xmath332 and @xmath333 . from this relation , we find @xmath117 in terms of @xmath328 as @xmath334 where @xmath335 . when @xmath336 , we can make an expansion in this parameter to get to the form @xmath337 similarly we also get @xmath338 in this appendix , we derive the resonant form of the two - particle current , eq . ( [ ip2res ] ) , for a quasiparticle injected from the left . the definition of @xmath339 is @xmath340 , which using the pseudo - unitarity of the transfer matrices @xmath341 can be written on the form @xmath342 it simplifies in the limit @xmath0 , @xmath343 to @xmath344 inserting the simplified expansion of @xmath339 and the expansion of @xmath345 and @xmath117 from eq . ( [ reflectionexp ] ) into eq . ( [ ipn ] ) , as well as putting @xmath168 , the leakage current density takes the form @xmath346^{l}={\frac{\displaystyle \frac{1- { \left|1-e^{2i\varphi_{0 } } \right|^{2 } } \frac{1-|a_{2}|^{4}}{\left|1-e^{2i\varphi_{2 } } \right|^{2 } } d^{2 } } { \left|2i\varphi_{1}^{(m)}+\displaystyle\frac{d}{2}\left ( \displaystyle\frac{1+e^{2i\varphi_{0}}}{1-e^{2i\varphi_{0}}}+ \displaystyle\frac{1+e^{2i\varphi_{2}}}{1- e^{2i\varphi_{2}}}\right ) \right|^{2}}.}\ ] ] we make an expansion of the phase @xmath347 , where @xmath348 the two - particle current density now takes a breit - wigner form @xmath346^{l}=\frac{\gamma_{0}^{(m)}\gamma_{2}^{(m ) } } { \displaystyle \left(\frac{\delta e-\delta e^{(m)}}{\delta}\right)^{2}+ \left(\frac{\gamma_{0}^{(m)}+\gamma_{2}^{(m)}}{2 } \right)^{2}}\ ] ] where the tunneling rates are given by @xmath349 where @xmath350 are equal to the dos , eq . ( [ densityofstates ] ) at energy @xmath351 . the resonance is slightly shifted from @xmath96 with @xmath352 in this appendix , the resonant form of the three - particle current , eq . ( [ ip3res ] ) , is derived . the @xmath353-matrix , which by definition is @xmath354 can be transformed using eq . ( [ m20 ] ) to @xmath355 which can be written in the form @xmath356 it simplifies in the limit of @xmath0 , @xmath357 and @xmath358 to @xmath359.\ ] ] inserting this form of the @xmath353-matrix and the expansion ( [ reflectionexp ] ) for @xmath117 and @xmath360 into eq . ( [ ipn ] ) , as well as putting @xmath168 , the probability current density for injection of a quasiparticle from the left takes the form @xmath361^{l}=\frac{(1-|a_{0}|^{4})(1- { \left|1-e^{2i\varphi_{0 } } \right|^{2 } \left|1-e^{2i\varphi_{3 } } \right|^{2}\left|q\right|^{2}}\ ] ] @xmath362 where @xmath0 is once again used . since @xmath363 and @xmath364 and the dos at energies @xmath365 , eq . ( [ densityofstates ] ) , are equal to @xmath366 @xmath367 we arrive at the form @xmath361^{l}=\frac{n^{l}(e)n^{r}(e_{3})d^{3 } } { \left|d-4\varphi_{1}^{(k)}\varphi_{2}^{(m ) } + id\left(\varphi_{1}^{(k)}n^{r}(e_{3})+\varphi_{2}^{(m)}n^{l}(e ) \right)\right|^{2 } } .\ ] ] in this appendix , we discuss the structure of the resonance in the four - particle current . the matrix @xmath368 can be written as @xmath369 @xmath370+\ ] ] @xmath371 @xmath372.\ ] ] from eq . ( [ mmatris ] ) it is clear that in general @xmath373 . when both @xmath374 and @xmath375 are close to a multiple of @xmath376 , @xmath377 , while close to other double resonances , e.g. when @xmath374 and @xmath378 are close to a multiple of @xmath376 , @xmath379 .
|
we study coherent multiple andreev reflections in quantum sns junctions of finite length and arbitrary transparency .
the presence of superconducting bound states in these junctions gives rise to great enhancement of the subgap current .
the effect is most pronounced in low - transparency junctions , @xmath0 , and in the interval of applied voltage @xmath1 , where the amplitude of the current structures is proportional to the first power of the junction transparency @xmath2 .
the resonant current structures consist of steps and oscillations of the two - particle current and also of multiparticle resonance peaks .
the positions of the two - particle current structures have pronounced temperature dependence which scales with @xmath3 , while the positions of the multiparticle resonances have weak temperature dependence , being mostly determined by the junction geometry . despite the large resonant two - particle current ,
the excess current at large voltage is small and proportional to @xmath4 .
+ pacs : 74.50.+r , 74.80.fp , 74.20.fg , 73.23.ad
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
nonequilibrium processes , their stationary states and their phase transitions have been of considerable interest in natural science as well as in medicine and sociology for many years . here we are interested in processes that can be modelled by growth and decay of populations with spatially local interaction rules . the transition between survival and extinction of a population is a nonequilibrium continuous phase transition phenomenon and is characterized by universal scaling laws . it is well known that also for systems far from equilibrium the concept of universality classes with respect to their critical properties in the vicinity of a continuous phase transition is applicable . for the description of transitions in systems that show active and absorbing inactive states , percolation models play an outstanding role . some years ago it was conjectured @xcite that markovian growth models with one - component order parameters displaying a transition into an absorbing state in the absence of any special conservation law generically belong to the universality class of directed percolation ( dp ) . besides dp @xcite this universality class includes e.g. reggeon field theory ( rft ) @xcite , the contact process @xcite , certain cellular automata @xcite and some catalysis models @xcite ( for a recent review of dp processes see @xcite ) . despite the fact that a large variety of different models belong to the dp universality class , _ there is still no experiment where the critical behavior of dp was seen _ @xcite . in a recent paper @xcite , hinrichsen compares suggested experiments and discusses possible reasons why the observation of dp critical exponents is obscured or even impossible . one of these reasons might be that the basic feature of the dp class , the existence of an absorbing state , is quite difficult to realize in nature . small fluctuations will always affect this state and may be strong enough to soften the transition like a small particle source , which works as an external field @xcite . another reason might be the influence of spatial quenched disorder which is abundant in reality . we have shown @xcite that in contrast to equilibrium systems , the critical scaling properties of dp processes are not only altered by frozen randomness , but fully destroyed . for the analytic description of universal behavior near a critical nonequilibrium transition , it is often useful to model the universality class by mesoscopic stochastic processes involving the order parameter and other relevant fields . in case of the dp class a representation by the langevin equation for the time - development of the particle density , the gribov process ( the stochastic version of the so - called schlgl model @xcite ) , is appropriate . the name gribov process was coined by grassberger who showed that rft is a markov process in disguise rather than a quantum theory @xcite . on the level of a formulation of stochastic processes by means of path integrals , there is superficially no difference between rft and the gribov process . however rft uses creation and annihilation operators for particles as the principal fields in contrast to the particle density and its conjugate response field in the gribov process . microscopically the rft - description of dp starts with special reactions between diffusing individuals on a given @xmath1-dimensional lattice such as birth : @xmath2 , competition : @xmath3 , and death : @xmath4 . these reactions are represented by a master equation that is mapped onto a second - quantized bosonic operator representation , which is in turn mapped onto a bosonic field theory using the continuum limit @xcite . at the critical point the rates of birth and death have to balance to yield a vanishing overall production of individuals . but this condition leads to strong local correlations and the microstates of the system consist typically of clusters of individuals embedded in the vacuum @xcite . thus , a fluctuating density description for the gribov process is appropriate on a mesoscopic level . note that _ the replacement of the spontaneous single particle death reaction by a two particle reaction @xmath5 leads to strong anticorrelations because now the birth rate itself has to vanish at the critical point_. in this case , the microstates consist of separated lonely wanderers that only sporadically interact and no clustering of individuals sets in . _ for such problems of branching and annihilating random walks one is forced to use the creation - annihilation operator formulation _ @xcite and a naive stochastic density description for the fundamental field would lead to wrong results . as a general rule : one can show that branching processes lead to positive correlations and annihilation processes to anticorrelations with the exception of spontaneous decay , which does not generate any correlations . nonvanishing branching and spontaneous decay are needed to yield positive correlations together with the possibility of a vanishing overall production of particles at the critical point . in such cases a mesoscopic density description is correct . _ instead of considering only one species of particles as is usually done for processes belonging to the dp class , it is of interest to introduce processes with several interacting species as for instance in mathematical biology @xcite , which are also of relevance for a special model of surface growth @xcite . it is the purpose of this paper to describe the detailed field - theoretic investigation based on renormalization group methods of such _ colored and flavored directed percolation processes _ ( mdp ) realized by the gribov process for several species . we group different colored species in the same flavor class , if they have equal transport properties . in the next chapter we introduce the model and define its renormalization and the one - loop calculation in the third chapter . in chapter iv , we present the general renormalization group analysis and find the asymptotic scaling behavior to one - loop order . in chapter v , we include the two - loop results of the appendices and show the crossover to unidirectionality of the couplings between the different species . in chapter vi , we present our considerations on symmetries and the general fixed point properties of the model . we show that the permutation symmetry of a multicolored process is spontaneously broken . a brief account of this work has been presented in @xcite . in addition , in chapter vii , we will show that the universal multicritical features of a recently introduced model of unidirectional coupled directed percolation processes ( ucdp ) by tuber et al . @xcite , which contains an additional linear coupling between the species , is completely described by the mdp class . in the last chapter we discuss the results and give an application to biomathematics . three appendices present technical details , e.g. the @xmath6-expansion of the dp - exponents , known for a long time , but as yet unpublished . the mesoscopic description of the dynamics of physical systems is based on a correct choice of the complete set of fundamental slowly - developing fields . in general these are the order parameter densities and the densities of conserved quantities . the multispecies processes under consideration are completely described by the particle densities @xmath7 of the percolating colored and flavored individuals . we assume that there does not exist any conservation law . next one has to find out the general form of the stochastic equations of motions of the fundamental fields as timelocal ( markovian ) langevin equations . these langevin equations have to respect symmetries and general principles characterizing the universality class under consideration . the mdp - class is characterized by the following four principles : 1 . errorfree self - reproduction ( `` birth '' ) and spontaneous annihilation ( `` death '' ) of individuals . the rates for birth and death may be different for each color . interaction between the individuals ( `` competition '' , `` saturation '' ) with color - dependent couplings . 3 . diffusion ( `` motion '' , `` spreading '' ) of the individuals in a @xmath1-dimensional space with flavor - depending transport coefficients . 4 . the states with at least one extinct color are absorbing . in the language of chemistry , the mdp may be realized microscopically by an autocatalytic reaction scheme of the form @xmath8 , @xmath9 , @xmath10 , where the last reaction subsumes the interactions of individuals with colors @xmath11 , @xmath12 and @xmath13 may be the integers @xmath14 . a description in terms of particle densities typically arises from a coarse - graining procedure where a large number of microscopic degrees of freedom are averaged out . the influence of these is simply modelled by gaussian noise - terms in the langevin equation which however have to respect the absorbing state condition . the stochastic reaction - diffusion equations for the particle densities in accordance with the four principles given above are of the form @xmath15 where the first term on the right hand side models the ( diffusive ) motion , and the @xmath16 are the overall reproduction rates of the particles with color @xmath11 . these deterministic terms are constructed proportional to @xmath17 in order to ensure the existence of an absorbing state for each species . near the absorbing transition the particle densities @xmath18 are small quantities . expanding the rates @xmath16 in powers of @xmath18 results in @xmath19 the gaussian noises @xmath20 must also respect the absorbing state condition , whence @xmath21 subleading terms in the expansions ( [ 2],[3 ] ) as well as additional terms with derivatives of the spatial @xmath22-function in the first line of eq . ( [ 3 ] ) are not displayed . it can be shown that they are irrelevant in the renormalization group sense as long as the stability condition @xmath23 for all @xmath24 is fulfilled . the breakdown of this condition signals the occurrence of a discontinuous transition to compact growth and the appearance of tricritical phenomena at the border between first and second order transitions . such a behavior is expected in microscopic models based on more complicated particle reactions @xcite . the `` temperature '' variables @xmath25 measure the difference of the rates of death and birth of the color @xmath11 . thus the temperatures may be positive ore negative . we are interested in the case where all @xmath26 ( up to fluctuation corrections ) which defines the multicritical region . under these conditions all the species live on the border of extinction . the next step is a mean field investigation of homogeneous steady state solutions of the equations of motion . neglecting all fluctuations in eq.([1 ] ) and using the expansion ( [ 2 ] ) up to first order in @xmath27 we find easily that all @xmath28 as long as all @xmath29 , meaning the vacuum is absorbing for each color . as soon as some of the temperatures become negative , stationary solutions with @xmath30 emerge , satisfying the corresponding equations @xmath31 . thus , in general a multitude of hypersurfaces of first and second order transitions exist in the phase space spanned by the relevant temperature variables @xmath32 which separate the phases where a specific color becomes extinct . whenever @xmath33 changes from a negative to a positive value , the order parameter @xmath34 undergoes a continuous or a discontinuous phase transition from an inactive absorbed state with @xmath35 to an active state with @xmath30 . all hypersurfaces of phase transitions meet in the multicritical point where all temperatures are zero . all homogeneous states are globally stable because the evolution of the total particle density of homogeneous states in time is given by @xmath36 . thus all solutions of the mean field equations of motion are bounded to a finite region in the space of positive @xmath34 as long as the stability condition mentioned in the foregoing paragraph holds . in the following we focus on the effect of fluctuations on the scaling behavior of correlation and response functions in the vicinity of the multicritical point where the strongly relevant parameters \{@xmath37 } are small . in order to apply field - theoretic methods and the renormalization group equation in conjunction with an @xmath38-expansion about the upper critical dimension @xcite , it is convenient to use the path - integral representation of the underlying stochastic processes @xmath39 @xcite . with the imaginary - valued response fields denoted by @xmath40 , the generating functional of the connected response and correlation functions , the greens functions , takes the form @xmath41=\ln \int { \cal d}\bigl[\widetilde{n},n \bigr]\exp \biggl(-{\cal j}\bigl[\widetilde{n},n\bigr]+\int d^{d}x\int dt \bigl(hn+\widetilde{h}\widetilde{n}\bigr)\biggr)\ . \label{pathint}\ ] ] the response fields @xmath42 correspond to the conjugated auxiliary variables of the operator formulation of statistical dynamics by kawasaki @xcite and martin , siggia , rose @xcite ) . the dynamic functional @xmath43 $ ] and the functional measure @xmath44 $ ] , which in symbolic notation is proportional to @xmath45 , is understood to be defined using a prepoint ( ito ) discretization with respect to time . the prepoint discretization leads to the causality rule @xmath46 in response functions which forbids response propagator loops in the diagrammatical perturbation expansion @xcite . the langevin equations ( [ 1]-[3 ] ) are recast as a dynamic functional @xmath47 in the second line we have neglected subleading terms . correlation and response functions can now be expressed as functional averages of monomials of the @xmath17 and @xmath48 with weight @xmath49 . a glance at eq . ( [ pathint ] ) shows that the responses are defined with respect to additional local particle sources @xmath50 in the langevin equations ( [ 1 ] ) . a rescaling of the fields @xmath51 , @xmath52 leaves the functional @xmath53 forminvariant but transforms the coupling constants @xmath54 and @xmath55 . thus invariant coupling constants are given by @xmath56 . a suitable rescaling is defined by @xmath57 . if we choose this normalization , we denote the rescaled fields by @xmath58 and @xmath59 . the scaling by a suitable mesoscopic length and time scale , @xmath60 and @xmath61 ( with @xmath62 ) respectively , leads to @xmath63 , @xmath64 where @xmath65 . hence @xmath66 is the upper critical dimension . it is now easy to show that all neglected possible subleading terms in the expansion of @xmath67 and the noise correlation , as well as higher gradient terms in the langevin equation ( [ 1 ] ) and non - gaussian and non - markovian noise correlations , have coupling constants with negative @xmath68-dimensions near the upper critical dimension . therefore , under the renormalization group flow , they are renormalized to zero and we can safely neglect them because we are interested in the leading universal critical behavior . these couplings can be reintroduced if one is interested in corrections to scaling . in a renormalized field theory context @xcite , the @xmath68-dimensions of the coupling constants are equal to the so called naive or engineering dimensions . all coupling constants with positive naive dimensions ( the relevant couplings with respect to the gaussian fixed point as the starting point of the perturbation expansion ) need a renormalization because the corresponding vertex functions are the only ones which develop primitive divergencies in perturbation theory . thus the field theory based on the functional @xmath69 ( eq . ( [ 4 ] ) ) is renormalizable and the calculated scaling properties are universal for the full class of mdp - processes . now we are in a position to develop the perturbation theory in the inactive phase with all the @xmath29 . to begin with we separate the anharmonic `` interaction '' terms from the dynamic functional @xmath69 , eq . ( [ 4 ] ) , and retain only the harmonic ones : @xmath70 here we have included the external sources @xmath71 and @xmath72 . the gaussian path integral @xmath73\exp ( -{\cal j}_{0})=\exp ( ( h , g\widetilde{h}))$ ] involves only non - negative fields @xmath74 as long as @xmath75 . it yields the propagators for the fourier transformed fields @xmath76 etc . , as @xmath77 where the heaviside theta - function is defined with @xmath78 following from the ito - discretization of the path - integral and ensuring causality . besides the propagators , the anharmonic coupling terms in @xmath53 , eq.([4 ] ) , define the elements of the graphical perturbation expansion . they are depicted in fig . 1 where an arrow marks a @xmath79-leg and we draw diagrams with the arrows always directed to the left ( ascending time ordering from right to left ) . 1 . elements of the graphical perturbation expansion fig . 1 shows that the color of a @xmath79-leg on the left side of a vertex is not annihilated : at least one @xmath80-leg on the right side displays the same color . thus , going from left to right , i.e. backward in time , through a diagram , colors have only sources and no sinks . this property is a consequence of the existence of absorbing states in the model . the perturbation expansion of a translationally invariant field theory can be analyzed by the calculation of the vertex functions @xmath81 corresponding to the one - particle irreducible amputated diagrams . here the sets @xmath82 and @xmath83 denote the colors of the amputated outer @xmath79-legs and @xmath80-legs respectively . going backward in time we conclude from the conservation property of the @xmath79-colors that the colors of the set @xmath84 must appear as a subset of @xmath83 . therefore the only nonzero two and three point vertex functions are @xmath85 , @xmath86 , and @xmath87 . moreover , another property follows directly from color conservation : the vertex function @xmath88 does not depend on parameters @xmath89 , @xmath90 , and @xmath91 with colors other than the ones of the vertex functions itself . thus , @xmath85 , @xmath92 , and @xmath93 are only functions of the parameters @xmath94 , @xmath25 , and @xmath57 and are in particular independent from the interspecies couplings @xmath95 with @xmath96 . therefore these vertex functions are the same as the corresponding functions of the well analyzed one - species gribov process . to calculate them one can set the interspecies couplings @xmath97 . in this case the model shows rapidity reversal invariance @xmath98 from which follows the equality @xmath99 . now we are ready to consider the renormalization of the model . it is known that the perturbation expansion of a field theory with a momentum cutoff @xmath100 develops divergencies if @xmath101 @xcite . if the model is renormalizable one can absorb all these `` primitive divergencies '' order by order in a loop expansion in a suitable reparametrization of the parameters . absorbing the primitive divergencies regularizes the full model . the primitively divergent vertex functions have nonnegative naive @xmath68 -dimensions . here , they are @xmath102 and @xmath103 @xmath104 , logarithmic at the upper critical dimension , ( fig . 2 ) . fig . 2 . primitively divergent vertex functions taking into account the general properties of the vertex functions found in the foregoing paragraph , we see that the following renormalization scheme renders the theory finite @xmath105 \tau _ { \alpha } & \rightarrow & \mathaccent"7017{\tau } _ { \alpha } = \frac { z_{\tau } ^{(\alpha ) } } { z_{\lambda } ^{(\alpha ) } } { \tau } _ { \alpha } \ , \quad f_{\alpha \beta } \rightarrow \mathaccent"7017{f}_{\alpha \beta } = g_{\varepsilon } ^{\,-1}\mu ^{\varepsilon } \,\frac{z_{u}^{(\alpha \beta ) } } { z_{s}^{(\beta ) } z_{\lambda } ^{(\alpha ) } z_{\lambda } ^{(\beta ) } } \,u_{\alpha \beta } \ . \label{7}\end{aligned}\ ] ] here @xmath106 is a convenient constant . instead of a momentum - cutoff regularization , we use dimensional regularization and minimal renormalization in the following . from the discussion above we learn that the renormalization factors @xmath107 with @xmath108 and @xmath109 , which are determined from @xmath85 and @xmath110 , are already known from the one - species gribov process and depend only on @xmath111 . thus @xmath112 . the new renormalization factors @xmath113 with @xmath114 stem from the interspecies couplings . they depend only on the couplings @xmath115 , @xmath116 , @xmath117 , @xmath118 , and the ratio @xmath119 . we will now explicitly calculate the renormalizations to one - loop order . the primitively divergent one - loop diagrams are shown in ( fig . 3 ) . primitively divergent one - loop diagrams using dimensional regularization , we express the contribution of the self - energy diagram , fig . 3(a ) , as a function of external momentum and frequency , @xmath120 and @xmath121 : @xmath122 here , we have retained only terms linear in @xmath121 and @xmath123 . these are the terms that display poles in @xmath124 . to determine the primitive divergencies of the vertex functions we can set external momenta and frequencies to zero and use equal temperatures @xmath125 as infrared regularisators . the contributions of the three diagrams fig . 3(c , d , e ) add to @xmath126 from the zero - loop contributions and the results of our short calculation , eqs . ( [ 8],[9 ] ) , we obtain the ( as yet unrenormalized ) one - loop vertex functions to the desired order in @xmath121 and @xmath123 : @xmath127 and @xmath128 an explicit calculation of diagram ( b ) of fig . 3 demonstrates that indeed @xmath129 if @xmath130 . to absorb the @xmath6-poles in the renormalization @xmath131-factors we note that the vertex functions are renormalized by the scheme eq . ( [ 7 ] ) as @xmath132 using again the renormalization scheme eq . ( [ 7 ] ) , we find the renormalized vertex functions from eqs . ( [ 10],[11 ] ) as @xmath133 and @xmath134 therefore the vertex functions become finite by choosing @xmath135 up to higher orders in the coupling constants @xmath136 . as anticipated , for @xmath137 we have found the well known renormalization factors of the reggeon field theory . next we will explore the scaling properties of multicolored directed percolation . scaling properties describe how physical quantities will transform under a change of length scales . at the end of chapter ii we have introduced the arbitrary mesoscopic length scale @xmath68 . the freedom in the choice of @xmath68 , keeping the unrenormalized bare parameters @xmath138 , and in cutoff regularization the momentum cutoff @xmath139 fixed , can be used to derive in a routine fashion the renormalization group ( rg ) equation for the connected correlation and response functions , the green functions @xmath140 we denote @xmath68-derivatives at fixed bare parameters by @xmath141 . from @xmath142 and the renormalization scheme eq . ( [ 7 ] ) , which leads to @xmath143 , we then find the rg equations @xmath144g^{\{n,\widetilde{n}\}}=0 \label{17}\ ] ] with the renormalization group differential operator @xmath145 here we have introduced the gell - mann low functions @xmath146 with @xmath147 , and the wilson functions @xmath148 the rg equations ( [ 17 ] ) can be solved in terms of a single flow parameter @xmath149 using characteristics . following this method , flowing parameters are defined by the characteristic equations @xmath150 and the rg equations ( [ 17 ] ) of the green functions become @xmath151g^{\{n,\widetilde{n}\}}(\{{\bf x } , t\},\{\bar{\tau}(l)\},\{\bar{u}(l)\},\{\bar{\lambda}(l)\},l\mu ) = 0\ . \label{21b}\ ] ] here , the functions @xmath152 , @xmath153 , and @xmath154 are independent of the color @xmath155 and the interspecies couplings . the flow equations ( [ 21a ] ) describe how the parameters transform if we change the momentum scale @xmath68 according to @xmath156 . being interested in the infrared ( ir ) behavior of the theory , we must study the limit @xmath157 . in general we expect that in this ir limit the coupling constants @xmath158 flow to a stable fixed point @xmath159 according to the first set of eq . ( [ 21a ] ) . in particular , the intraspecies couplings @xmath160 then flow to a color independent fixed point @xmath161 because @xmath162 , and this gell - mann low function @xmath163 is equal to the corresponding function known from the one - species gribov process . thus , the fixed point value @xmath164 is independent from any coupling to other species . the solutions of the second and third set of the flow equations ( [ 21b ] ) are readily found in terms of the functions @xmath165 . in the ir limit @xmath166 they have the scaling form @xmath167 where the @xmath168 are nonuniversal amplitude factors . the scaling exponents @xmath169 are already known from directed percolation . from eq . ( [ 21b ] ) we find the solution in the ir limit as @xmath170 with @xmath171 , @xmath172 , and the dp anomalous field scaling exponent @xmath173 omitting the nonuniversal amplitude factors @xmath168 and taking into account dimensional analysis in space and time @xmath174 where @xmath175 , we get by combination of eqs . ( [ 24],[26 ] ) the asymptotic scaling form of the green functions @xmath176 this has the important consequence that all scaling properties of the dp processes remain unaffected by the introduction of many colors . moreover , all intraspecies green functions are completely independent of the coupling between the species , which follows from the absorbing state conditions for each color . however , the interspecies coupling constants @xmath177 with @xmath178 determine the properties of the interspecies scaling functions . therefore we will now consider the consequences of the flow equations for these parameters . for this purpose we need the gell - mann low functions @xmath179 explicitly . from the last of eqs . ( [ 19 ] ) we know that each of these functions begins with the zero - loop term @xmath180 , and the higher order terms are determined by the wilson functions . these functions , the logarithmic derivatives of the @xmath131 -factors , are given by @xmath181 . in minimal renormalization the @xmath131-factors have a pure laurent expansion with respect to @xmath6 : @xmath182 . thus recursively in the loop expansion the wilson functions also have a pure laurent expansion and , because they are finite for @xmath183 , this expansion reduces to the constant term , i.e. all @xmath6-poles have to cancel in the logarithmic derivation . hence , we obtain the wilson functions simply from the formula @xmath184 . now it is easy to get these functions from the one loop results of the @xmath131-factors eqs . ( [ 15 ] ) . we find to this order @xmath185 from which one finds the gell - mann low functions eq . ( [ 19 ] ) as @xmath186 the gell - mann low functions of the intraspecies couplings , the `` diagonal '' @xmath163-functions , follow as @xmath187 giving the stable fixed point values @xmath188 . with the help of eqs.([19],[23],[25 ] ) , this stable fixed point leads to the well known one loop order dp exponents @xmath189 , @xmath190 , and @xmath191 . using the fixed point values @xmath192 to obtain the gell - mann low functions of the interspecies couplings of a pair of colors @xmath114 , we get @xmath193 in addition to the unstable decoupled fixed point values @xmath194 , the equations @xmath195 are solved by a fixed point line @xmath196 this equation is the key result of this section . clearly , however , the one - loop calculation can not give us any information whether the degeneracy of all the points on this line is fundamental to our model or is lifted by higher loop corrections . thus , we must proceed to two - loop order , presented in the following chapter . we note , however , that there are two special points of unidirectional coupling on the fixed line : @xmath197 in the next chapter we will show that for colors with the same flavor , meaning @xmath198 , these two points are indeed the only stable points on the line . because @xmath199 or @xmath200 are fixed point values at any loop order , we conjecture that unidirectionality is generic for the asymptotic behavior of coupled dp processes , irrespective of whether colors belong to different flavors or not . from the analysis in the foregoing chapters we know that the interaction of two colors does not depend on the existence of other ones . thus in this chapter we consider a coupled model of two colors @xmath201 and @xmath202 with the same flavor , i.e. equal kinetic coefficients @xmath203 and intraspecies couplings @xmath204 . in contrast , the temperatures @xmath25 may be different . thus , the ( unrenormalized ) dynamic functional is now given by @xmath205 note that in the case @xmath206 , the dynamics of species @xmath207 completely decouples from species @xmath202 and vice versa . it follows that @xmath208 , if @xmath209 and @xmath114 . the detailed calculation of the two - loop contributions is presented in appendices a to c. adding eq . ( [ a9a ] ) to the zero- and one - loop parts of the selfenergy eq . ( [ 10 ] ) , we get after renormalization , using eq . ( [ 12 ] ) and the scheme eq . ( [ 7 ] ) , @xmath210 where the constants @xmath211 are given in eq . ( [ a9b ] ) , and @xmath212 . note that in comparison with eq . ( [ 13 ] ) now the one - loop terms acquire renormalizations to @xmath213 in order to be consistent in @xmath214 of the perturbation expansion . these renormalized one - loop terms are needed to compensate non primitive singular terms proportional to @xmath215 arising now by the @xmath38-expansion of @xmath216 . of course only primitive uv - divergencies , which means here @xmath217-independent ones , have to be regularized by renormalization to make the theory finite . eliminating the @xmath218-poles from eq . ( [ 35 ] ) by the @xmath131-factors , we find @xmath219 in the same way as for the selfenergy we find the other vertex functions . renormalizing eq . ( [ 11 ] ) and adding the two - loop contribution eq . ( [ a22 ] ) we get @xmath220 where @xmath221 . here the @xmath6-expansion also shows that the non primitive divergencies @xmath222 cancel . we eventually find @xmath223 & & + \frac{1}{16\varepsilon } \biggl(\biggl(\frac{36}{\varepsilon } -\frac{19}{2 } \biggr)u^{2}+\biggl(\frac{8}{\varepsilon } -\frac{5}{4}\biggr)uu_{\beta \alpha } + \biggl(\frac{2}{\varepsilon } -1\biggr)u_{\alpha \beta } u_{\beta \alpha } \nonumber \\ & & + \biggl(\frac{8}{\varepsilon } -\frac{9}{4}+3\ln \frac{4}{3}\biggr ) uu_{\alpha \beta } + \biggl(\frac{1}{\varepsilon } -\frac{3}{2}\ln \frac{4}{3 } \biggr)\bigl(u_{\alpha \beta } ^{2}+u_{\beta \alpha } ^{2}\bigr)\biggr ) + o(u^{3})\ , \label{38}\end{aligned}\ ] ] and in particular @xmath224 we are now in a position to calculate the renormalization group functions eq . ( [ 19 ] ) from eqs . ( [ 36],[38],[39 ] ) with the help of eq . ( [ 20 ] ) . they are given by @xmath225 and @xmath226 setting @xmath227 , we find the nontrivial stable fixed point value @xmath228 from which the scaling exponents eqs . ( [ 23],[25 ] ) of the green functions eq . ( [ 27 ] ) follow to second order of the @xmath6 -expansion as @xmath229 the first two expansions have been known for a long time from reggeon field theory @xcite where a different definition of the exponents is used . the expansion of the exponent @xmath230 was presented by the author in @xcite . the order parameter exponent @xmath163 , which enters the scaling law for the mean particle number in the active state @xmath231 we now turn to the interspecies coupling constants @xmath177 with @xmath232 . the fixed point values as the solutions of the equation @xmath233 , where @xmath179 is the gell - mann low function eq . ( [ 40a ] ) , are of three different types : 1 . the decoupled fixed point @xmath234 , totally instable for @xmath235 ; 2 . the two unidirectional coupled fixed points @xmath236 , @xmath237 and @xmath238 , @xmath239 ; 3 . the symmetric fixed point @xmath240 . to discuss the stability and the crossover between the last two types we try an ansatz of the form @xmath241 with @xmath242 . we already know from the one - loop result that @xmath177 is driven by the renormalization flow to the fixed line @xmath243 with a crossover exponent @xmath244 . setting therefore @xmath245 in @xmath179 we get @xmath246 for @xmath247 this equation shows that the symmetric fixed point @xmath248 is instable in contrast to the stable unidirectional coupled fixed points @xmath249 . the solution of the flow equation @xmath250 leads to the crossover @xmath251 with the very small crossover exponent @xmath252 a qualitative picture of the flow of the interspecies couplings in the plane @xmath247 under renormalization is shown in fig . 4 . fig . flow of the interspecies couplings under renormalization in addition to the four fixed points d ( decoupled ) , s ( symmetric ) , u ( unidirectional ) , the topology of the flow is determined by the symmetry line @xmath253 which acts in the first quadrant as a separatrix between the regions of attraction of the two unidirectional fixed points . there exists another separatrix at the border of these regions , given to first order in @xmath6 by @xmath254 , where the flow is driven to the hatched line @xmath255 . on the left of this line we have the region of instability i , in which the condition @xmath256 for all positive @xmath257 is violated . therefore we conjecture that interspecies couplings with @xmath258 ultimately lead to first order transitions . in fig . 4 the fixed point line of the one - loop calculation is also shown . this line is the support of the slow crossover to the unidirectional fixed points which therefore describe the ultimate critical behavior of the mdp universality class . we conjecture that in the case of colors with different flavors , @xmath259 , all properties are smooth functions of the ratio @xmath119 as long as this ratio is sufficiently close to one . thus generalizing to different flavors , the topology of the renormalization flow displayed in fig . 4 should only be smoothly deformed but not destroyed . in particular , the unidirectional coupled fixed points u should be stable also for @xmath260 . having found the detailed fixed point structure of the model eq.([33 ] ) of two colors with the same flavor in the two - loop approximation , we will now investigate which results are valid to all orders of perturbation theory . according to the considerations of the third chapter , one demonstrates easily that for @xmath114 the vertex function @xmath261 if @xmath262 . thus the lines of unidirectionally coupled models in fig . 4 , namely @xmath263 or @xmath264 , respectively , are invariant under the renormalization flow . trivially , the decoupled fixed point d is the intersection point of these lines . for @xmath265 the dynamic functional @xmath53 eq . ( [ 33 ] ) possesses the symmetry @xmath266 , @xmath267 , and @xmath268 from which we find that @xmath269 . thus , these two vertex functions need the same @xmath131-factor : @xmath270 . it follows that the symmetry line in fig . 4 , @xmath253 , is invariant under renormalization . is there a condition that determines the crossover line ? the answer is yes . we change to variables corresponding to the total and relative particle numbers , respectively , @xmath271 such linear transformations do not alter the measure of the functional integrals , and the dynamic functional changes , in the special case @xmath272 , to @xmath273 we see that in the case @xmath274 the dynamics of the total particle number decouples from the dynamics of the relative one in the sense that all vertex functions containing @xmath79- , but no @xmath275 -legs are zero if @xmath276-legs are attached . in particular @xmath277 and @xmath278 , which leads to the renormalized interspecies couplings with @xmath279 and determines the crossover line to all loop - orders . note that on the symmetry line @xmath280 holds , and the functional @xmath53 is then invariant against @xmath281 , @xmath282 . the different fixed point values of the interspecies couplings are now fully determined as the intersection points of the several invariant lines ( see fig . 4 ) : the unidirectional lines @xmath263 , @xmath264 , the symmetry line @xmath253 , and the crossover line @xmath279 . thus , we find to all orders for the symmetric fixed point s : @xmath283 and for the stable unidirectional fixed points u : @xmath284 @xmath285 and @xmath286 @xmath287 , respectively . these are of course the results that we have found explicitly in the two - loop approximation . the unidirectionally coupled model , which describes the generic scaling properties of the mdp processes , exhibits another symmetry . using @xmath206 and @xmath288 , we write the dynamic functional of this model in the form @xmath289 here we have introduced a further harmonic unidirectional coupling @xmath290 , which corresponds to an additional linear term : @xmath291 in the langevin equation for species @xmath202 . this term was first considered by tuber et al . @xcite in their study of the nonequilibrium critical behavior in unidirectionally coupled dp processes . such a term does not alter the general renormalizations as long as the corresponding composed field @xmath292 is treated as a soft insertion . of course , @xmath293 is a relevant parameter like the temperatures @xmath25 and needs its own renormalization factor @xmath294 determined by @xmath295 in such a way that the renormalized vertex function with an insertion @xmath296 is finite . the settings @xmath297 define the multicritical point . from reggeon field theory one knows a transformation called rapidity reversal : @xmath298 , which is broken by the dp transition to an active state . here we generalize it to read @xmath299 under this transformation the functional @xmath300 changes to @xmath301 we learn from this relation that in the case @xmath302 , @xmath303 gains a higher symmetry at the multicritical point , which is not destroyed by renormalization . it follows in this case that @xmath293 renormalizes in the same way as the @xmath25 , which implies @xmath304 , and that the renormalized couplings are related by @xmath305 at the fixed point , as we already know from above . from these considerations follows that the crossover exponent , which is defined by the scaling invariants @xmath306 , is given by @xmath307 . recently tuber et al . @xcite ( in the following abbreviated by thhg ) introduced a general unidirectionally coupled dp process with species - independent diffusion coefficients , that was motivated by a study of alon et al . @xcite on a nonequilibrium growth model for adsorption and desorption of particles which displays a roughening transition . the thhg - model reads in our dynamic functional language as @xmath308 in contrast to thhg , we have introduced an additional cross diffusion term @xmath309 here that is indispensable for a complete renormalization of the general model . in physical terms , coarse graining will always produce cross diffusion in this coupled model . but coarse graining does more : it also produces a term proportional to the time derivative of the density of the first species in the langevin equation of the second one besides further irrelevant couplings . accordingly , in the functional @xmath310 a term proportional to @xmath311 arises . however , such a term can be eliminated by a suitable redefinition of the fields @xmath312 so that the harmonic parts with the time derivatives in the dynamic functional remain diagonal . note that in the special case @xmath313 , @xmath314 the functional @xmath310 eq . ( [ 53 ] ) is identical to @xmath303 , eq . ( [ 49 ] ) , from which we know that it is fully renormalizable , and , in particular , in the case @xmath315 by the @xmath131-factors eqs . ( [ 36],[39 ] ) . in the following we will demonstrate that the asymptotic properties of the thhg - model indeed belong to the universality class described by @xmath300 . it is well known @xcite that the infinitesimal generators of a continuous transformation of the fundamental fields , which leads to a forminvariance of the describing statistical functional , define redundant operators ( composite fields ) . in the case that these redundant operators are relevant or marginal , they unnecessarily contaminate the renormalization group . therefore they should be avoided from the outset . here we introduce a linear , homogeneous transformation between the fields , which does not change the form of the functional @xmath310 eq.([53 ] ) . let us call it the @xmath11-transformation : @xmath316 note the difference to the @xmath163-transformation eq . ( [ 54 ] ) , which is exploited to eliminate a @xmath317-term from @xmath318 . this @xmath11-transformation leads to new coupling constants that we denote with a bar : @xmath319 now we renormalize the thhg - model by the scheme @xmath320 where @xmath321 , @xmath322 , and the @xmath323 with @xmath324 are given by eqs . ( [ 36],[39 ] ) . besides @xmath136 , dimensionless coupling constants are defined as @xmath325 . the scheme eq.([57 ] ) is chosen in such a way that the renormalized dynamic functional reads @xmath326 note that the counter term @xmath327 serves to cancel primitive divergencies arising in the vertex function @xmath328 . in dimensional regularization and minimal renormalization the counterterms @xmath329 and @xmath330 are given by series in @xmath331 beginning with simple poles @xmath332 and @xmath333 , respectively . we have formerly shown that only the residua of these poles determine the renormalization group functions . it is now appropriate to define @xmath11-transformation invariant dimensionless coupling constants as @xmath334 the somewhat lengthy but simple calculation of all the one - loop renormalizations leads to the gell - mann low functions of the renormalization group equation . in part , in the case @xmath335 , they can be derived from the results of thhg @xcite . if we set @xmath136 to its fixed point value @xmath336 and define as usual @xmath337 , where @xmath338 is any of the coupling constants , we obtain for the @xmath163 -functions of the invariant couplings @xmath339 the function @xmath340 results from the additive renormalization @xmath341 and mixes the renormalized fields in a correlation or response function under application of the renormalization group as @xmath342\bigl\{\tilde{s}_{1},s_{1 } , \tilde{s}_{2},s_{2}\bigr\}=\bigl\{-a\tilde{s}_{2},0,0,-as_{1}\bigr\}\;. \label{64}\ ] ] here @xmath343 is the renormalization group differential operator now given by @xmath344 with @xmath345 @xmath346 , and eq . ( [ 64 ] ) acts on green functions . the @xmath347 result from the additive renormalizations @xmath330 . the function @xmath348 is found to be @xmath349 the new renormalizations yield @xmath350 in order to determine the fixed - point solutions of eqs . ( [ 63 ] ) , @xmath351 , we impose the condition @xmath352 . this yields @xmath353 , with @xmath354 ( unstable ) or @xmath355 ( stable ) . it can easily be checked that these solutions are consistent with the full set of eqs . ( [ 63 ] ) and ( [ 66 ] ) . these are the solutions found by thhg @xcite . note that on the fixed point lines generated from the stable fixed point by the @xmath356-transformation a minimally coupled fixed point with @xmath357 is found . now one has to prove stability of the fixed points of the full equations ( [ 63 ] ) without using the constraints @xmath352 . a linearization about @xmath358 and either @xmath359 or @xmath360 shows that the flow of @xmath361 is unstable for @xmath359 , whereas it shows full stability of the fixed point for @xmath362 . this vindicates the neglect of @xmath363 and the corresponding counterterms @xmath341 and @xmath364 in @xcite but only at the stable fixed point line , which is generated from the fixed point by the @xmath11-transformation . however without further knowledge this statement is only correct in the one - loop calculation and could be violated in higher loop orders . we will show that the stable fixed point is given by @xmath365 , @xmath355 to all orders of the loop expansion . as a consequence , the fixed point values @xmath366 are zero , and @xmath367 . this leads to a crossover exponent @xmath368 where @xmath369 determines the scaling of @xmath370 . indeed , we see from eq . ( [ 59 ] ) that the stable one - loop order fixed point belongs , up to an @xmath11-transformation , to the dynamic functional @xmath318 , eq . ( [ 58 ] ) , with coupling constants @xmath371 and @xmath372 , i.e. model @xmath373 , eq . ( [ 53 ] ) , with the additional constraint @xmath315 . above it was shown that this equality leads to rapidity reversal eq.([51 ] ) as a higher symmetry . this higher symmetry is preserved under renormalization and , because @xmath300 is fully renormalizable , we have the result @xmath374 , @xmath375 and @xmath307 to all loop orders . computations based on the dynamic functional @xmath300 , eq . ( [ 53 ] ) , are much easier to perform than calculations using the complete model @xmath376 , eq . ( [ 58 ] ) . thus it may be possible to find the equation of state for @xmath377 to second order and check the assumptions made in @xcite on the reexponentiation of logarithms to yield the new order parameter exponent @xmath378 of that paper ( for a calculation of the equation of state for @xmath379 to two - loop order see @xcite ) . the model @xmath300 describes the coupled dp processes near the multicritical point @xmath380 . what is needed for a thorough calculation of @xmath378 is a theory that comprises the limit @xmath381 . therefore our considerations here do not solve the problem addressed in @xcite , namely the determination of the scaling exponent @xmath378 that controls the scaling @xmath382 where species @xmath207 is in its active phase . thhg calculate @xmath383 by reexponentiation of logarithms . ( we have done a recalculation and find a slightly different value @xmath384 . the difference arises from a subleading term resulting from the ominous peculiar diagram fig . 9(c ) in @xcite . ) however the approach of thhg relies on the assumption that simple reexponentiation is possible . to derive such scaling properties faithfully , one indeed has to solve the crossover problem @xmath385 which ( possibly ! ) induces a new scaling at infinity for the correlations of species @xmath202 . some features of this crossover remind us of the crossover from special to ordinary behavior in the theory of surface transitions @xcite , with @xmath293 corresponding to the surface enhancement @xmath276 and the species @xmath207 and @xmath202 corresponding to the bulk and surface respectively . the crossover problem of interest here is thus as yet unsolved . in this paper , we have studied multicolored directed percolation processes ( mdp ) . in particular , we have shown that the scaling behavior of these coupled dp processes near their absorbing state transition is determined by the same critical exponents as known from simple one - species dp and therefore is independent from the number of colors . the characteristic asymptotic feature of mdp shows up in the asymptotic unidirectional coupling of each pair of colors . a special result of our analysis is the very slow crossover to this asymptotic unidirectionality which may be seen in computer simulations . the unidirectional behavior of the couplings of an interacting population is summarized in the following graphical picture . consider a graph where each node represents one color . the stable fixed points are then represented by the so - called tournaments , that are the complete graphs with directed edges @xcite . the directed edge from color @xmath11 to color @xmath163 stands for the influence of @xmath386 on @xmath163 in the respective equations of motion . in particular , for a population of three species there exist two different tournaments ( up to permutation of the colors ) which we call `` cyclic '' and `` hierarchic '' , fig . 5 . fig cyclic and hierarchic tournament of three species one gets a graphical picture for the not fully stable fixed points either by deleting the directionality of the edges between the nodes for symmetric couplings of the corresponding colors , or by completely deleting the edges for the uncoupled pairs ( uncomplete tournaments ) . to get a simple qualitative impression of the behavior of the dynamic system of a population corresponding to a tournament of colors with the same flavor , we consider , in the active region with all@xmath387 , a renormalized mean - field theory of the equations ( [ 1],[2 ] ) for spatially homogeneous densities @xmath388 and set @xmath389 with @xmath390 . we redefine the time scale @xmath391 and get @xmath392 then the decision , which of all the @xmath393 equal @xmath394 , defines the tournament . the edge between a pair of species is directed from @xmath163 to @xmath11 if @xmath395 and vice versa . the directions of the edges therefore represent the unidirectional `` pressure '' on the reproduction rate resulting from one color to another . despite the simplicity of the eqs . ( [ 68 ] ) they can generate a complex dynamic behavior ( see e.g. @xcite ) . first , let us consider the stationary states of eqs . ( [ 68 ] ) for a population consisting of three species . in the ternary phase diagram spanned by the positive rates @xmath396 in the subspace @xmath397 , one finds different regions with one , two ore all three species alive , fig . 6 . fig . 6 . phase diagrams of the cyclic and hierarchic tournament these regions are bounded by critical lines with absorbing state transitions where some colors become extinct . the dynamic behavior of the hierarchic tournament fig . 5(b ) is relatively simple : from each nonequilibrium initial state @xmath398 , the system relaxes to the stationary state which is a stable node . in the case of the cyclic tournament fig . 5(a ) , for rates @xmath399 such that we have a three species stationary state , one also finds regions for which the ultimate relaxation behavior is characterized by attracting nodes . but for rates which lead to stationary states belonging to the crosshatched area in the ternary diagram fig . 7 spanned by the @xmath400 , we find stable spirals ( damped cyclic relaxation ) as the ultimate relaxation behavior ( qualitatively pictured by the trajectory in fig . 7 ) . fig . 7 . dynamic behavior of the cyclic tournament for stationary points near the middle of fig . 7 the damping is very small and cyclic behavior dominates the dynamics . especially in the middle of the diagram , i.e. for equal rates @xmath401 , the motion is not damped anymore . in this case , the dynamic system eq . ( [ 68 ] ) is known as a special form of the may - leonard model @xcite that has been extensively studied in mathematical biology @xcite . finally , after a relaxation in the plane @xmath402 , there exists another constant of motion @xmath403 and the dynamic behavior is characterized by limit cycles around the neutral stationary point @xmath404 . summarily we have shown that also in stochastic multispecies models of populations that evolve near the extinction threshold of all colors and therefore have many absorbing states , the critical properties at the multicritical point and at all continuous transitions are governed by the well known gribov process ( reggeon field theory ) exponents ( for a previous simulational result on a two - species system which seems to agree with our findings see @xcite ) . in other regions of the phase diagram of course more complicated critical behavior may arise such as multicritical points with different scaling exponents @xcite . the models considered here have many absorbing states ( each combination of colors may go extinct irrespective of the other ones ) , and therefore are different from models which were considered by grinstein et al . @xcite where it was shown that multispecies systems with one absorbing state belong to the gribov universality class . in addition we have shown that the universal properties of interspecies correlations and the phase diagram are determined by totally asymmetric fixed point values of the renormalized interspecies coupling constants . this eventually leads to a system working cooperatively . it is interesting that the asymmetry between the species seems to be the condition for this cooperation near extinction . the model considered here is a simple but universal model of such a cooperative society and should therefore have many applications in all fields of natural and even social science . competition and extinction is of course a subject much considered in theoretical biology . the main differences between the present work and the topics covered e.g. in the monograph of hofbauer and sigmund @xcite are the more realistic local description of the interactions between the species , their diffusional motion , and the inclusion of local fluctuations . thus here the equations of motion are local stochastic partial differential equations . coarse graining and renormalization lead to an universal macroscopic picture of cooperativity near the critical states of extinction . we thank uwe tuber for many fruitful discussions , stephan theiss for a critical reading of the paper and beate schmittmann for numerous valuable remarks leading to the final version of the manuscript . this work has been supported in part by the sfb 237 ( `` unordnung und groe fluktuationen '' ) of the deutsche forschungsgemeinschaft . in the calculation we will encounter momentum integrals of the type @xmath405 where @xmath406 , @xmath407 , and @xmath408 . they can be derived from two `` mother '' integrals @xmath409 by taking derivatives with respect to the parameters @xmath410 . discarding nonsingular terms , we find in dimensional regularization @xmath411 these formulas yield the singular parts of the integrals ( [ a1 ] ) as @xmath412 fig . 8 . two - loop selfenergy diagrams in fig . 8 the two - loop selfenergy diagrams are drawn . diagram fig . ( 8a ) leads to @xmath413 where @xmath414 is the propagator ( eq . ( [ 6 ] ) ) and we always set @xmath415 as an ir regulator . noting that e@xmath416e@xmath417 e @xmath418e@xmath419 factorizes in the integral , the time integrations over the intervals @xmath420 , @xmath421 , @xmath422 are easily performed . the expansion in @xmath423 and @xmath424 to linear order eventually yields @xmath425 where the @xmath426 denote the integrals defined in eq . ( [ a1 ] ) . extracting the singular parts using eq . ( [ a4 ] ) , we obtain @xmath427 in the same way we calculate the second selfenergy diagram fig . 8(b ) : @xmath428 up to higher orders in @xmath423 and @xmath424 . extracting the singular parts again , we find @xmath429 summing up , we finally get from @xmath430 and @xmath431 the two - loop contribution of the selfenergy as @xmath432 where @xmath433 fig . 9 . two - loop vertex diagrams ; the numbers show the other time orderings fig . 9 presents the eleven two - loop vertex diagrams . they can be calculated by the same method as the selfenergy diagrams . here the external frequencies and momenta can be set to zero because the expansion in these variables does not lead to primitive divergencies . as an example we show the calculation of the diagram fig . 9(i ) explicitly . the symmetry factor of the diagrams is one , but one also has to add the two diagrams arising from the interchange of the color indices @xmath163 and @xmath434 . after the factorization of the propagators , the integration over the time intervals , indicated by the broken lines , is trivial und leads to the expression ( with the abbreviations @xmath435 , @xmath436 ) @xmath437 thus we obtain the contribution of the diagram fig . 9(i ) to the vertex function @xmath87 : @xmath438 m. doi , j. phys . a : math . * 9 * , 1479 ( 1976 ) ; p. grassberger and p. scheunert , fortschr . * 28 * , 547 ( 1980 ) ; l. peliti , j. phys . ( france ) * 46 * , 1469 ( 1984 ) ; b. p. lee , j. phys . a : math . gen . * 27 * , 2633 ( 1994 ) . u. c. tuber , m. j. howard , and h. hinrichsen , phys . lett . * 80 * , 2165 ( 1998 ) ; y. y. goldschmidt , phys . * 81 * , 2178 ( 1998 ) ; y. y. goldschmidt , m. j. howard , h. hinrichsen , and u. c. tuber , phys . e * 59 * , 6381 ( 1999 ) . k. kawasaki , in _ proc . varenna summer school on critical phenomena _ , edited by m. s. green , ( academic , new york , 1971 ) ; in _ phase transitions and critical phenomena _ , vol . * 5a * , edited by c. domb and m. s. green , ( academic , london , 1976 ) .
|
a model of directed percolation processes with colors and flavors that is equivalent to a population model with many species near their extinction thresholds is presented .
we use renormalized field theory and demonstrate that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the one - species gribov process ( reggeon field theory ) . in addition
this universal model shows an instability that generically leads to a total asymmetry between each pair of species of a cooperative society , and finally to unidirectionality of the interspecies couplings .
it is shown that in general the universal multicritical properties of unidirectionally coupled directed percolation processes with linear coupling can also be described by the model .
consequently the crossover exponent describing the scaling of the linear coupling parameters is given by @xmath0 to all orders of the perturbation expansion .
as an example of unidirectionally coupled directed percolation , we discuss the population dynamics of the tournaments of three colors .
key words : multicolored directed percolation , field - theoretic renormalization group , stochastic population dynamics ,
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the most intriguing features of qcd for massless quarks is its underlying conformal invariance , invariance under both scale ( dilatation ) and special conformal transformations @xcite . for example , in the case of perturbative qcd , the running coupling @xmath0 becomes constant in the limit of zero @xmath1-function and zero quark mass , and conformal symmetry becomes manifest . in fact , the renormalization scale uncertainty in pqcd predictions can be eliminated by using the principle of maximum conformality ( pmc ) @xcite . using the pmc / blm procedure @xcite , all non - conformal contributions in the perturbative expansion series are summed into the running coupling by shifting the renormalization scale in @xmath2 from its initial value , and one obtains unique , scale - fixed , scheme - independent predictions at any finite order . one can also introduce a generalization of conventional dimensional regularization which illuminates the renormalization scheme and scale ambiguities of pqcd predictions , exposes the general pattern of nonconformal terms , and allows one to systematically determine the argument of the running coupling order by order in pqcd in a form which can be readily automatized @xcite . the resulting pmc scales and finite - order pmc predictions are both to high accuracy independent of the choice of initial renormalization scale . for example , pmc scale - setting leads to a scheme - independent pqcd prediction @xcite for the top - quark forward - backward asymmetry which is within one @xmath3 of the tevatron measurements . the pmc procedure also provides scale - fixed , scheme - independent commensurate scale relations @xcite , relations between observables which are based on the underlying conformal behavior of qcd such as the generalized crewther relation @xcite . the pmc satisfies all of the principles of the renormalization group : reflectivity , symmetry , and transitivity , and it thus eliminates an unnecessary source of systematic error in pqcd predictions @xcite . anti - de sitter space in five dimensions ( ads@xmath4 ) provides a geometric representation of the conformal group . one can modify ads space by using a dilaton factor in the ads metric @xmath5 to introduce the qcd confinement scale . although such a mass scale explicitly breaks the dilatation invariance of the equations of motion , the action can still be conformally invariant , as was first shown by v. de alfaro , s. fubini and g. furlan ( daff ) @xcite in the context of one - dimensional quantum field theory ; i.e , the change in the mass scale of the potential can be compensated by the time scale in the action . however , this manifestation of conformal symmetry can only occur if the dilaton profile @xmath6 is constrained to have the specific power @xmath7 , a remarkable result which follows from the daff construction of conformally invariant quantum mechanics @xcite . the quadratic form @xmath8 leads to linear regge trajectories @xcite in the hadron mass squared . a remarkable holographic feature of dynamics in ads space in five dimensions is that it is dual to hamiltonian theory in physical space - time , quantized at fixed light - front ( lf ) time @xcite . for example , the equation of motion for mesons on the light - front has exactly the same single - variable form as the ads / qcd equation of motion ; one can then interpret the ads fifth dimension variable @xmath9 in terms of the physical variable @xmath10 , representing the invariant separation of the @xmath11 and @xmath12 at fixed light - front time . as discussed in the next sections , this light - front holographic principle provides a precise relation between the bound - state amplitudes in ads space and the boost - invariant lf wavefunctions describing the internal structure of hadrons in physical space - time . the resulting valence fock - state wavefunction eigensolutions of the lf qcd hamiltonian satisfy a single - variable relativistic equation of motion analogous to the nonrelativistic radial schrdinger equation . the quadratic dependence in the quark - antiquark potential @xmath13 in the light - front schrdinger equation ( lfse ) is determined uniquely from conformal invariance , whereas the constant term @xmath14 is fixed by the duality between ads and lf quantization , a correspondence which follows specifically from the separation of kinematics and dynamics on the light - front @xcite . the lf potential thus has a specific power dependence in effect , it is a light - front harmonic oscillator potential . it is confining and reproduces the observed linear regge behavior of the light - quark hadron spectrum in both the orbital angular momentum @xmath15 and the radial node number @xmath16 . the pion is predicted to be massless in the chiral limit @xcite - the positive contributions to @xmath17 from the lf potential and kinetic energy is cancelled by the constant term in @xmath18 for @xmath19 the derived running qcd coupling displays an infrared fixed point @xcite . the construction of daff retains conformal invariance of the action despite the presence of a fundamental mass scale . the ads approach , however , goes beyond the purely group - theoretical considerations of daff , since features such as the masslessness of the pion and the separate dependence on @xmath20 and @xmath15 are consequences of the potential derived from the holographic lf duality with ads for general @xmath20 and @xmath15 @xcite . the quantization of qcd at fixed light - front time @xcite ( dirac s front form ) provides a first - principles hamiltonian method for solving nonperturbative qcd . it is rigorous , has no fermion - doubling , is formulated in minkowski space , and it is frame - independent . given the boost - invariant light - front wavefunctions @xmath21 ( lfwfs ) , one can compute a large range of hadron observables , starting with structure functions , generalized parton distributions , and form factors . it is also possible to compute jet hadronization at the amplitude level from first principles from the lfwfs @xcite . a similar method has been used to predict the production of antihydrogen from the off - shell coalescence of relativistic antiprotons and positrons @xcite . the lfwfs of hadrons thus provide a direct connection between observables and the qcd lagrangian . solving nonperturbative qcd is thus equivalent to solving the light - front heisenberg matrix eigenvalue problem . angular momentum @xmath22 is conserved at every vertex . the lf vacuum is defined as the state of lowest invariant mass and is trivial up to zero modes . there are thus no quark or gluon vacuum condensates in the lf vacuum the corresponding physics is contained within the lfwfs themselves @xcite , thus eliminating a major contribution to the cosmological constant . the simplicity of the front form contrasts with the usual instant - form formalism . current matrix elements defined at ordinary time @xmath23 must include the coupling of photons and vector bosons fields to connected vacuum - induced currents ; otherwise , the result is not lorentz - invariant . thus the knowledge of the hadronic eigensolutions of the instant - form hamiltonian are insufficient for determining form factors or other observables . in addition , the boost of an instant form wavefunction from @xmath24 to @xmath25 changes particle number and is an extraordinarily complicated dynamical problem . it is remarkable fact that ads / qcd , which was originally motivated by the ads / cft correspondence between gravity on a higher - dimensional space and conformal field theories in physical space - time @xcite , has a direct holographic mapping to light - front hamiltonian theory @xcite . the ads mass parameter @xmath26 maps to the lf orbital angular momentum . the formulae for electromagnetic @xcite and gravitational @xcite form factors in ads space map to the exact drell - yan - west formulae in light - front qcd @xcite . thus the light - front holographic approach provides an analytic frame - independent first approximation to the color - confining dynamics , spectroscopy , and excitation spectra of the relativistic light - quark bound states of qcd . it is systematically improvable in full qcd using the basis light - front quantization ( blfq ) method @xcite and other methods . in the limit of zero quark masses the longitudinal modes decouple from the invariant lf hamiltonian equation @xmath27 with @xmath28 . the generators @xmath29 , @xmath30 , are constructed canonically from the qcd lagrangian by quantizing the system on the light - front at fixed lf time @xmath31 , @xmath32 @xcite . the lf hamiltonian @xmath33 generates the lf time evolution with respect to @xmath31 , whereas the lf longitudinal @xmath34 and transverse momentum @xmath35 are kinematical generators . it is advantageous to reduce the full multiparticle eigenvalue problem of the lf hamiltonian to an effective light - front schrdinger equation which acts on the valence sector lf wavefunction and determines each eigensolution separately @xcite . in contrast , diagonalizing the lf hamiltonian yields all eigensolutions simultaneously , a complex task . the central problem for deriving the lfse becomes the derivation of the effective interaction @xmath36 which acts only on the valence sector of the theory and has , by definition , the same eigenvalue spectrum as the initial hamiltonian problem . in order to carry out this program one must systematically express the higher fock components as functionals of the lower ones . this method has the advantage that the fock space is not truncated , and the symmetries of the lagrangian are preserved @xcite . the light - front hamiltonian for qcd can be derived directly from the qcd lagrangian @xcite . the result is relativistic and frame - independent . the @xmath37 lf fock state wavefunction for a meson can be written as @xmath38 thus factoring the angular dependence @xmath39 and the longitudinal , @xmath40 , and transverse mode @xmath41 . in the limit of zero quark masses the longitudinal mode decouples and the lf eigenvalue equation @xmath42 takes the form of a light - front wave equation for @xmath43 @xcite @xmath44 \phi_{j , l , n}(\zeta^2 ) = m^2 \phi_{j , l , n}(\zeta^2),\ ] ] a relativistic _ single - variable _ lf schrdinger equation . this equation describes the spectrum of mesons as a function of @xmath16 , the number of nodes in @xmath10 , the total angular momentum @xmath20 , which represent the maximum value of @xmath45 , @xmath46 , and the internal orbital angular momentum of the constituents @xmath47 . the variable @xmath9 of ads space is identified with the lf boost - invariant transverse - impact variable @xmath10 @xcite , thus giving the holographic variable a precise definition in lf qcd @xcite . for a two - parton bound state @xmath48 , where @xmath49 is the longitudinal momentum fraction and @xmath50 is the transverse - impact distance between the quark and antiquark . in the exact qcd theory @xmath36 is related to the two - particle irreducible @xmath37 green s function . the potential in the lfse is determined from the two - particle irreducible ( 2pi ) @xmath51 greens function . in particular , the higher fock states in intermediate states leads to an effective interaction @xmath52 for the valence @xmath53 fock state @xcite . a related approach for determining the valence light - front wavefunction and studying the effects of higher fock states without truncation has been given in ref . @xcite . unlike ordinary instant - time quantization , the light - front hamiltonian equations of motion are frame independent ; remarkably , they have a structure which matches exactly the eigenmode equations in ads space . this makes a direct connection of qcd with ads methods possible . in fact , one can derive the light - front holographic duality of ads by starting from the light - front hamiltonian equations of motion for a relativistic bound - state system in physical space - time @xcite . recently we have derived wave equations for hadrons with arbitrary spin starting from an effective action in ads space @xcite . an essential element is the mapping of the higher - dimensional equations to the lf hamiltonian equation found in ref . this procedure allows a clear distinction between the kinematical and dynamical aspects of the lf holographic approach to hadron physics . accordingly , the non - trivial geometry of pure ads space encodes the kinematics , and the additional deformations of ads encode the dynamics , including confinement @xcite . a spin-@xmath20 field in ads@xmath54 is represented by a rank @xmath20 tensor field @xmath55 , which is totally symmetric in all its indices . in presence of a dilaton background field @xmath56 the effective action is @xcite @xmath57 where the indices @xmath58 , @xmath59 and @xmath60 is the covariant derivative which includes parallel transport . the coordinates of ads are the minkowski coordinates @xmath61 and the holographic variable @xmath9 , @xmath62 . the effective mass @xmath63 , which encodes kinematical aspects of the problem , is an _ a priori _ unknown function , but the additional symmetry breaking due to its @xmath9-dependence allows a clear separation of kinematical and dynamical effects @xcite . the dilaton background field @xmath56 in ( [ seff ] ) introduces an energy scale in the five - dimensional ads action , thus breaking conformal invariance . it vanishes in the conformal ultraviolet limit @xmath64 . a physical hadron has plane - wave solutions and polarization indices along the 3 + 1 physical coordinates @xmath65 , with four - momentum @xmath66 and invariant hadronic mass @xmath67 . all other components vanish identically . the wave equations for hadronic modes follow from the euler - lagrange equation for tensors orthogonal to the holographic coordinate @xmath9 , @xmath68 . terms in the action which are linear in tensor fields , with one or more indices along the holographic direction , @xmath69 , give us the kinematical constraints required to eliminate the lower - spin states @xcite . upon variation with respect to @xmath70 , we find the equation of motion @xcite @xmath71 \phi_j = m^2 \phi_j,\ ] ] with @xmath72 , which is the result found in refs . @xcite by rescaling the wave equation for a scalar field . similar results were found in ref . @xcite . upon variation with respect to @xmath73 we find the kinematical constraints which eliminate lower spin states from the symmetric field tensor @xcite @xmath74 upon the substitution of the holographic variable @xmath9 by the lf invariant variable @xmath10 and replacing @xmath75 in ( [ phijm ] ) , we find for @xmath76 the lfse ( [ lfwe ] ) with effective potential @xcite @xmath77 provided that the ads mass @xmath78 in ( [ phijm ] ) is related to the internal orbital angular momentum @xmath79 and the total angular momentum @xmath80 according to @xmath81 . the critical value @xmath82 corresponds to the lowest possible stable solution , the ground state of the lf hamiltonian . for @xmath83 the five dimensional mass @xmath78 is related to the orbital momentum of the hadronic bound state by @xmath84 and thus @xmath85 . the quantum mechanical stability condition @xmath86 is thus equivalent to the breitenlohner - freedman stability bound in ads @xcite . a particularly interesting example is a dilaton profile @xmath87 of either sign , since it leads to linear regge trajectories @xcite and avoids the ambiguities in the choice of boundary conditions at the infrared wall . for the confining solution @xmath88 the effective potential is @xmath89 and eq . ( [ lfwe ] ) has eigenvalues @xmath90 , with a string regge form @xmath91 . a discussion of the light meson and baryon spectrum , as well as the elastic and transition form factors of the light hadrons using lf holographic methods , is given in ref . @xcite . as an example the spectral predictions for the @xmath92 light pseudoscalar and vector meson states are compared with experimental data in fig . [ pionspec ] for the positive sign dilaton model . parent and daughter regge trajectories for the @xmath93-meson family ( left ) with @xmath94 gev ; and the @xmath95-meson family ( right ) with @xmath96 gev.,title="fig:",width=232 ] parent and daughter regge trajectories for the @xmath93-meson family ( left ) with @xmath94 gev ; and the @xmath95-meson family ( right ) with @xmath96 gev.,title="fig:",width=232 ] the effective interaction @xmath97 is instantaneous in lf time and acts on the lowest state of the lf hamiltonian . this equation describes the spectrum of mesons as a function of @xmath16 , the number of nodes in @xmath98 , the internal orbital angular momentum @xmath99 , and the total angular momentum @xmath100 , with @xmath80 the sum of the orbital angular momentum of the constituents and their internal spins . [ the @xmath101 casimir @xmath102 corresponds to the group of rotations in the transverse lf plane . ] it is the relativistic frame - independent front - form analog of the non - relativistic radial schrdinger equation for muonium and other hydrogenic atoms in presence of an instantaneous coulomb potential . the ads / qcd harmonic oscillator potential could in fact emerge from the exact qcd formulation when one includes contributions from the lfse potential @xmath36 which are due to the exchange of two connected gluons ; _ i.e. _ , `` h '' diagrams @xcite . we notice that @xmath36 becomes complex for an excited state since a denominator can vanish ; this gives a complex eigenvalue and the decay width . the correspondence between the lf and ads equations thus determines the effective confining interaction @xmath36 in terms of the infrared behavior of ads space and gives the holographic variable @xmath9 a kinematical interpretation . the identification of the orbital angular momentum is also a key element of our description of the internal structure of hadrons using holographic principles . if one starts with a dilaton profile @xmath103 with @xmath104 the existence of a massless pion in the limit of massless quarks determines uniquely the value @xmath7 . to show this , one can use the stationarity of bound - state energies with respect to variation of parameters @xcite . the quadratic dilaton profile also follows from the algebraic construction of hamiltonian operators by ( daff ) @xcite . the action @xmath105 is invariant under conformal transformations in the variable @xmath23 , and there are in addition to the hamiltonian @xmath106 two more invariants of motion for this field theory , namely the dilation operator @xmath107 and @xmath108 , corresponding to the special conformal transformations in @xmath23 . specifically , if one introduces the the new variable @xmath109 defined through @xmath110 and the rescaled fields @xmath111 , it then follows that the the operator @xmath112 generates evolution in @xmath109 @xcite . the hamiltonian corresponding to the operator @xmath113 which introduces the mass scale is a linear combination of the old hamiltonian @xmath106 , @xmath107 , the generator of dilations , and @xmath108 , the generator of special conformal transformations . it contains the confining potential @xmath114 , that is the confining term in ( [ u ] ) for a quadratic dilaton profile and thus @xmath115 . the construction of new hamiltonians from the generators of the conformal group has been used by daff to construct algebraically the spectra and the eigenfunctions of these operators . the conformal group in one dimension is locally isomorphic to the group of pseudo - rotations @xmath116 . the compact operator @xmath117 generates rotations in the euclidean 1 - 2 plane , whereas the non - compact operators @xmath118 and @xmath119 generate pseudo - rotations ( boosts ) in the non - euclidean 2 - 3 and 1 - 3 plane respectively . as in the familiar case of angular momentum , one can introduce raising and lowering operators @xmath120 and construct the spectrum and the eigenfunctions analogously to the angular momentum operators . interestingly , this is just the method employed in ref . @xcite to obtain an integrable lf confining hamiltonian by following infeld s observation that integrability follows immediately if the equation of motion can be factorized as a product of linear operators @xcite . the method can be extended to describe baryons in ads while preserving the algebraic structure @xcite . our approach has elements in common with those of ref . @xcite , where the scale of the confinement potential arises from a boundary condition when solving gauss equation . the triple complementary connection of ( a ) ads space , ( b ) its lf holographic dual , and ( c ) the relation to the algebra of the conformal group in one dimension , is characterized by a quadratic confinement lf potential , and thus a dilaton profile with the power @xmath121 , with the unique power @xmath7 . in fact , for @xmath122 the mass of the @xmath123 pion is automatically zero in the chiral limit , and the separate dependence on @xmath20 and @xmath15 leads to a mass ratio of the @xmath95 and the @xmath124 mesons which coincides with the result of the weinberg sum rules @xcite . one predicts linear regge trajectories with the same slope in the relative orbital angular momentum @xmath15 and the principal quantum humber @xmath16 . the ads approach , however , goes beyond the purely group theoretical considerations of daff , since features such as the masslessness of the pion and the separate dependence on @xmath20 and @xmath15 are a consequence of the potential ( [ u ] ) derived from the duality with ads for general @xmath20 and @xmath15 . the constant term in the potential , which is not determined by the group theoretical arguments , is fixed by the holographic duality to lf quantized qcd @xcite . the resulting lagrangian , constrained by the conformal invariance of the action , has the same form as the ads lagrangian with a quadratic dilaton profile . in their discussion of the evolution operator @xmath125 as a model for confinement , daff mention a critical point , namely that `` the time evolution is quite different from a stationary one '' . by this statement they refer to the fact that the variable @xmath109 is related to the variable @xmath23 for the case @xmath126 by @xmath127 _ i.e. _ , @xmath109 has only a limited range . the finite range of invariant lf time @xmath128 can be interpreted as a feature of the internal frame - independent lf time difference between the confined constituents in a bound state . for example , in the collision of two mesons , it would allow us to compute the lf time difference between the two possible quark - quark collisions . the treatment of the chiral limit in the lf holographic approach to strongly coupled qcd is substantially different from the standard approach based on chiral perturbation theory . in the conventional approach @xcite , spontaneous symmetry breaking by a non - vanishing chiral quark condensate @xmath129 plays the crucial role . in qcd sum rules @xcite @xmath130 brings in non - perturbative elements into the perturbatively calculated spectral sum rules . it should be noted , however , that the definition of the condensate , even in lattice qcd necessitates a renormalization procedure for the operator product , and it is not a directly observable quantity . in bethe - salpeter @xcite and light - front analyses @xcite , the gell mann - oakes - renner relation @xcite for @xmath131 involves the decay matrix element @xmath132 instead of @xmath133 . in the color - confining ads / qcd light - front model discussed here , the vanishing of the pion mass in the chiral limit , a phenomenon usually ascribed to spontaneous symmetry breaking of the chiral symmetry , is obtained specifically from the precise cancellation of the lf kinetic energy and lf potential energy terms for the quadratic confinement potential . this mechanism provides a viable alternative to the conventional description of nonperturbative qcd based on vacuum condensates , and it eliminates a major conflict of hadron physics with the empirical value for the cosmological constant @xcite . invited talk , presented by sjb at the third workshop on the qcd structure of the nucleon ( qcd - n12 ) , bilbao , spain , october 22 - 26 , 2012 . this work was supported by the department of energy contract de
|
we show that ( a ) the conformal properties of anti - de sitter ( ads ) space , ( b ) the properties of a field theory in one dimension under the full conformal group found by de alfaro , fubini and furlan , and ( c ) the frame - independent single - variable light - front schrdinger equation for bound states all lead to the same result : a relativistic nonperturbative model which successfully incorporates salient features of hadronic physics , including confinement , linear regge trajectories , and results which are conventionally attributed to spontaneous chiral symmetry breaking .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
shot noise in a mesoscopic conductor is a consequence of the stochastic character of electron tunneling and of the discreteness of charge . unlike thermal noise , shot noise describes the non - equilibrium fluctuations of current ; therefore , the study of current fluctuations can provide further understanding of properties related to correlation mechanisms , internal energy scales or the carrier statistics which can not be obtained by measuring the average current @xcite . a well studied example of physical processes where electron correlations play a dominant role is the phenomenon of coulomb blockade . in a system of small tunneling junctions , due to the large electrostatic energy ( as compared to temperature or voltages ) , the electronic charge is transported one by one . this effect leads to many remarkable features in transport properties and has been a subject of extensive study for the last decades @xcite . as an example , the strong dependence of the current - voltage characteristics on the gate charge was exploited to use a single electron transistor ( set ) as a highly sensitive charge detector @xcite , and proposed as measuring apparatus of the charge state of a josephson quantum bit @xcite . since it leads to a strong correlation of consecutive tunneling events , coulomb blockade has turned out to manifest itself in a peculiar way on the current noise . such an effect has been studied both in the sequential tunneling @xcite and in the cotunneling regime @xcite . additional interest in studying noise in single electron devices comes from the recently proposed schemes that employ current fluctuation measurements to detect entanglement in solid state systems @xcite . an even richer scenario occurs when the coherence of charge carriers is maintained over a significant portions of the system . such a circumstance is encountered quite often when the tunneling junctions are superconducting . in this case , the charge carriers are cooper pairs , and their coherent tunneling across the junctions gives rise to a series of pronounced structures in the @xmath0-@xmath1 characteristics at sub - gap voltages @xcite . furthermore , as analyzed in ref . , the scattering of quasi - particles ( and consequently the shot noise ) in superconducting point contacts is significantly enhanced in the presence of the supercurrent produced by a coherent flow of cooper pairs . in this paper , we analyze a superconducting double tunnel junction device , operating in a suitably chosen bias voltage regime , such that one of the junctions of the set is on resonance for cooper pair tunneling ( the case where cooper pair resonance occurs on both junctions has been recently analyzed in ref . ) . the interplay between coherence and interaction is explored by sweeping the operating point of the device through the cooper pair resonance . we will show that the fluctuations of the charge on the central island are sensitive to both coulomb blockade and quantum coherence . more pronounced effects arise in the regime in which the rates of incoherent quasi - particle tunneling matches the frequency of coherent cooper - pair oscillation . this gives rise to an enhanced fluctuation of charge in the central - island and to a substantial suppression of the current noise . by investigating the statistics of the tunneling events , we show that the suppression in the shot noise is related to the deviation of the counting statistics from the poissonian distribution . the probability distribution of tunneling events exhibits a parity dependence and remains non - poissonian in a wide range of parameter values . the interplay between coherence and coulomb blockade affects the overall charge transport and is also clearly observed in the finite frequency behavior of the current noise . its power spectrum displays a sharp resonance peak at the josephson frequency , resulting from coherent oscillations between two quantum states . the work presented here applies to the setup used in a recent experiment @xcite to probe the coherent evolution of quantum states in a cooper pair box as well as in an earlier experiment @xcite on resonant cooper pair tunneling . in this paper we extend the results of ref . . the paper is organized as follows . in section [ long : sec2 ] , we introduce the model to describe the set transistor and we describe the relevant processes involved in the josephson - quasiparticle cycle . in the same section we also introduce the master equation , in its general form , used to obtain all the results of this work . the solution of the master equation is worked out to analyze in detail the fluctuations of the charge on the central electrode and , in particular , the spectrum of the charge fluctuation as measured by a detector coupled to the island ( section [ long : sec3 ] ) , the counting statistics for the tunneled charges ( section [ long : sec4 ] ) , and the shot noise of the current ( section [ long : sec5 ] ) . in section [ long : sec5 ] we discuss in some details the properties of the shot noise at zero frequency , making explicit the results already contained in the counting statistics . in the same section we also discuss the frequency dependence of the shot noise . section [ long : sec6 ] is devoted to the conclusions . the system we consider is a superconducting set ( see fig . [ fig01 ] ) , which consists in a small central electrode ( island ) connected by tunnel junctions to two leads and capacitively coupled to a gate electrode . the electrostatic energy of the island can be adjusted by controlling the gate voltage @xmath2 . a transport voltage @xmath1 is applied to the outer leads , determining the current flowing through the device . letting @xmath3 and @xmath4 be the gate and left and right junction capacitances , respectively , the electrostatic charging energy is given by @xmath5 where @xmath6 is the total capacitance of the island . the device operates in the coulomb blockade regime , i.e. with the charging energy much larger than both the josephson coupling energy @xmath7 and the thermal energy @xmath8 ( @xmath9 ) , but still much smaller than the superconducting gap @xmath10 which we suppose to be the largest energy scale in the problem . charge can pass through the tunnel barriers coherently and incoherently . in addition to quasi - particle tunneling , josephson effect allows to maintain coherence in the cooper - pair tunneling processes . this coherence does play an important role in transport as long as charge states which differ by one cooper pair are almost degenerate . the hamiltonian of the system is given by @xmath11 where @xmath12 ( @xmath13 ) is the bcs hamiltonian of the left ( l ) , right ( r ) lead and of the central island ( i ) . the tunneling hamiltonian @xcite is @xmath14 \,,\ ] ] where @xmath15 is the tunneling amplitude and @xmath16 ( @xmath17 ) creates ( annihilates ) a particle with momentum @xmath18 and energy @xmath19 in electrode @xmath20 . the variable @xmath21 is the superconducting phase difference at the left ( right ) junction and it is canonically conjugated to the number @xmath22 of electrons that have passed across the left ( right ) junction _ out of _ the central electrode ( @xmath23=i\delta_{jk}$ ] ) . finally , @xmath24 is the electrostatic energy , @xmath25 where @xmath26 is the number of excess electrons on the central island , while @xmath27 is the offset charge due to the applied voltages . for later convenience it is useful to define the part of the hamiltonian which accounts for the coherent dynamics of the macroscopic variable @xmath28 . it includes the charging and the josephson terms . by properly adjusting the bias and gate voltages , one can put either the right or left junction at such a resonance for cooper - pair tunneling . we consider the case of resonance across the left junction , and consequently we keep only the corresponding josephson coupling @xmath29 a quasi - particle tunneling _ into _ ( _ out of _ ) the island across the junctions leads to the transition @xmath30 ( @xmath31 ) of the island charge . the rates of these incoherent processes are given by the relation @xmath32 \frac{{\mathop{\mathrm{im}}\nolimits}i_\mathrm{qp}({{\mathcal e}}_{n,\pm}^{l / r})}{2e } \,,\ ] ] where @xmath33 , @xmath34 , @xmath35 , and @xmath36 is the quasi - particle tunneling current at the bias voltage @xmath37 in the absence of charging effects ( see , e.g. , ref . for an explicit expression of @xmath38 in terms of @xmath15 ) . since the set transistor operates in the charge regime ( @xmath39 ) , we can take advantage of the strong suppression of charge fluctuations to use the eigenstates of @xmath28 as basis states for the island . moreover , we focus on the bias regime @xmath40 where only the two charge states with @xmath41 and @xmath42 , are nearly degenerate . such a condition implies that quasi - particle tunneling only takes place from the central island toward the right electrode , while the left junction allows only for coherent cooper pair tunneling . furthermore , we suppose that the josephson energy of the right junction is negligible ( the corresponding term has already been omitted from @xmath43 , which is justified within the rotating wave approximation ) . all these conditions are met in the recent experiment by nakamura _ et al . _ @xcite , designed to probe the state of the island via the detection of the incoherent tunneling current . in this situation one can imagine that the coherent cooper - pair tunneling occurring across the left junction is interrupted from time to time by quasi - particle tunneling across the right junction , as sketched in fig.[fig01 ] ( b ) . due to the strong coulomb blockade , it suffices to keep the three charge states , @xmath44 , and two tunneling rates , @xmath45 and @xmath46 ; the other tunneling rates are exponentially suppressed . in order to simplify the notation , we assume that @xmath47 , which is a very good approximation in the regime we are interested in . for example , in the experiment of ref . , @xmath48 and @xmath49 . the transport properties of the system in the set - up described above can be well described in terms of two variables , either @xmath28 and @xmath50 or @xmath28 and @xmath51 ( @xmath26 ) . however , the quantum dynamics of these system is affected by the quantum noise due to the coupling to the environment provided by the fermionic bath . in order to describe this effect , we adopt a master equation approach , which has been widely used to describe quantum open systems @xcite . a master equation for the reduced density matrix @xmath52 is obtained by taking the trace over the fermionic degrees of freedom from the liouville equation ( @xmath53 ) @xmath54\ ] ] for the density matrix @xmath55 of the system plus environment . the resulting equation can then be written in the lindblad form as @xcite @xmath56 + \frac{1}{2}\sum_{n=1,2}\gamma_n\bigl [ 2l_n\,\rho(t)\,l_n^\dag - l_n^\dag l_n\,\rho(t ) - \rho(t)\,l_n^\dag l_n \bigr ] \,.\ ] ] here @xmath57 is a lindblad operator corresponding to the quantum jump @xmath31 and @xmath58 , i.e. , in the @xmath59-basis @xmath60 . the first term describes a purely phase - coherent dynamics , while the second one is responsible for both dephasing and relaxation due to the quasi - particle tunneling . the solution of the eq . behaves in distinct ways in the two limiting cases of strong and weak coupling with the quasi - particle reservoir . in the _ strong dephasing limit _ ( either @xmath61 or @xmath62 , see below ) , the dephasing time @xmath63 , which describes the decay of the off - diagonal elements of @xmath64 to their stationary values , is small compared to the relaxation time @xmath65 which sets the time - scale for the variation of the diagonal elements ( i.e. population of the charge states ) . the relaxation time is given by @xmath66 on the other hand , in the _ weak dephasing limit _ ( @xmath67 ) , there is no such a clear separation of time scales ; both the diagonal and off - diagonal elements vary over the same time scale @xmath68 . a first insight into the interplay among coherent cooper - pair tunneling , coulomb blockade , and incoherent quasi - particle tunneling can be obtained by examining the fluctuations of the charges on the island as a function of the quasi - particle tunneling rate @xmath69 , the gate voltage , and the josephson coupling energy . in this case , we only need to keep track of the variable @xmath28 , and thus define a reduced density matrix for the central island charge , @xmath70 , which satisfies an equation identical to eq . , but with @xmath71 now operating on the reduced ` @xmath28 ' space only @xcite . in the stationary state , the master equation has the solution ( @xmath72 ) [ long : sigma ] @xmath73 here @xmath74 measures the energy difference between the state with @xmath41 and @xmath42 charge on the island ; the cooper pair resonance corresponds to @xmath75 . from the stationary - state solution , we evaluate the characteristic function @xmath76 for the quantum variable @xmath28 . it is given by @xmath77 from @xmath78 one can evaluate all the statistical moments of the charge on the island ; in particular , we concentrate on the variance @xmath79 ^ 2}\ ] ] shown in fig . [ fig02 ] . of the charge on the island as a function of @xmath80 and @xmath81.,title="fig : " ] of the charge on the island as a function of @xmath80 and @xmath81.,title="fig : " ] an interesting point about the result in eq . is that the maximum of the variance is found at @xmath82 for @xmath75 . in other words , the fluctuation is enhanced when the decoherence time matches the time for the josephson coherent oscillations . moreover as @xmath83 increases ( up to @xmath84 ) , the optimal value of @xmath69 decreases as @xmath85 ; meaning that @xmath81 enhances the _ effective _ dephasing rates . these two features will appear more clearly when we discuss the statistics ( section [ long : sec4 ] ) and noise ( section [ long : sec5 ] ) of the transport across the junctions . an important quantity to consider is the fluctuation spectrum for the number of electron charge residing on the island . as already discussed by many authors , it is important to include the back - action of the measuring apparatus @xcite , which could be a set transistor capacitively coupled to the central island . we include the charge detector coupling via an hamiltonian term of the form @xmath86 where @xmath87 is a detector operator . assuming the correlation time for the detector to be the fastest time scale of the problem , we write ( here we follow averin s treatment @xcite ): @xmath88 the non - zero value of @xmath89 is the essential cause of the measurement back - action . indeed , a term proportional to @xmath89 enters the master equation eq.([long : mastereq1 ] ) , thus affecting the time evolution of the system variables . to effectively measure the charge number , we look at an output detector operator @xmath90 , which , in the linear regime , evolves as @xmath91 , ( see ref . ) . the response coefficient @xmath92 is determined by the imaginary part of the equilibrium correlation function @xmath93 . furthermore , @xmath92 can be related to @xmath89 so that we can write for the signal to noise ratio @xmath94 where it is assumed that the real part of the @xmath90@xmath87 correlator vanishes ( which is the most favorable case for a measurement ) . here @xmath95 is the charge number fluctuation spectrum evaluated as @xmath96 where the time evolution is obtained from a modified master equation including the back - action . note that here we use the symmetric correlation function since the island charge itself is coupled to the quasi - particle bath , so that the detector can also receive energy from the system . ) , with the indicated back - action rate values and with @xmath97 ( left ) and @xmath98 ( right).,title="fig : " ] ) , with the indicated back - action rate values and with @xmath97 ( left ) and @xmath98 ( right).,title="fig : " ] as shown in fig . ( [ fig03 ] ) , the spectrum displays a resonance peak at the josephson frequency which is broadened by the quasi - particle rate @xmath69 ( the peak is only visible in the weak dephasing regime , otherwise it is completely washed out independently of the value of @xmath89 ) . as the back - action rate @xmath89 increases , a maximum develops at zero frequency , which finally hides the resonance structure . this enhanced zero frequency noise results from incoherent transition induced by the detector coupling . near the maximum at @xmath99 , and for @xmath100 , the spectrum takes the approximate form @xmath101 we turn now to a description of the statistical distribution of the number of charges transmitted through the system during a period @xmath102 , @xcite . specifically , we will examine the probability @xmath103 that @xmath104 electrons have been transferred across the right junction during the interval @xmath105 $ ] . we note that @xmath106 where @xmath107 is the joint probability that @xmath51 electrons have passed across the right junction up to the time @xmath108 and @xmath109 electrons up to time @xmath110 . to obtain @xmath103 , we define a characteristic matrix @xmath111 defined so that @xmath112 is the characteristic function for @xmath103 . namely , @xmath113 where the trace is taken over the states @xmath114 . following the same procedure that led to eq . , one can show that @xmath115 satisfies the following master equation : @xmath116 + \frac{1}{2}\sum_{n=1,2}\gamma_n \left [ 2 e^{i \theta } l_n g_t l_n^{\dag } - g_t l_n^{\dag } l_n - l_n^{\dag } l_n g_t \right]\ ] ] with the initial condition @xmath117 . here we will consider two limiting cases for the solution , the strong and the weak dephasing limit ( see the discussion at the end of section [ long : sec2 ] ) . we find that in the strong dephasing case ( @xmath118 or @xmath119 ) @xmath120 \exp\left[- \frac{\gamma_\mathrm{r}\tau}{2 } ( 1-z^2)\right]\ ] ] while in the weak dephasing limit at resonance ( @xmath121 ) @xmath122}{f(z ) } \sinh\frac{\gamma\tau f(z)}{4 } \right ] \\ + \frac{\gamma}{e_j}(1-z){\mathop{\mathrm{im}}\nolimits}\sigma_{02}(t ) \left[\cosh\frac{\gamma\tau f(z)}{4 } + \frac{4z+1}{f(z)}\sinh\frac{\gamma\tau f(z)}{4 } \right ] \biggr\ } \exp\left(-\frac{3\gamma\tau}{4}\right ) \\ - \frac{\gamma}{e_j}(1-z)\left\ { { \mathop{\mathrm{im}}\nolimits}\sigma_{02}(t)\cos(e_j\tau ) - \frac{1}{2}\left[\sigma_{00}(t)-\sigma_{22}(t)\right]\sin(e_j\tau ) \right\ } \exp\left(-\frac{\gamma\tau}{2}\right ) \,,\end{gathered}\ ] ] where @xmath123 , and @xmath124 . in figs . [ fig04 ] and [ fig05 ] , the resulting statistics are shown for the weak and strong dephasing cases , respectively , in the transient state ( i.e. @xmath125 ) . obtained by numerically solving equation ( [ long : mastereq2 ] ) , in the case of @xmath126 , with initial condition @xmath127 . ] ) and at resonance ( @xmath75 ) . in this regime , the counting probability does not depend on the initial condition for @xmath64 ] at @xmath129 in the strong dephasing regime with @xmath130 for the three initial conditions a ) @xmath131 ( dashed line ) , b ) @xmath132 ( dotted line ) c ) @xmath133 ( continuous line ) .,width=226 ] as seen in eqs . and , @xmath115 and hence @xmath103 depend on the charge state of the island at time @xmath108 . this point is further illustrated in fig . [ fig06 ] where the counting statistics at short time is shown in the strong dephasing limit for various starting conditions @xmath134 . in the limit of strong dephasing , the counting statistics depends sensitively on the initial state @xmath134 . this has been exploited in ref . , where the measurement of the quantum state is performed with the system taken far from degeneracy . on the contrary , the dependence on the initial condition is quickly lost in the weak dephasing regime , since the strong josephson energy can rapidly produce a change in the state , before quasi - particles have any time to be produced . another important limit to consider is the stationary state ( @xmath135 ) , where physical properties do not depend on the initial preparation of the system . in the strong dephasing limit , eq . is reduced to the simple form @xmath136 \,.\ ] ] it gives the probability distribution function for the transmitted charges [ countstat ] @xmath137 @xmath138 shows a strong even - odd asymmetry : for even @xmath104 , the distribution is poissonian , but the probability that an odd number of electrons has passed is negligible . below we will see that this strong parity effect manifests itself as an enhancement of zero - frequency shot noise . we leave the physical interpretation of the parity effect until we discuss shot noise in section [ long : sec5 ] . in the weak dephasing limit , eq . is reduced to @xmath139\end{gathered}\ ] ] so that [ p1 ] @xmath140 where @xmath141 this distribution function shows a much weaker ( but still finite ) even - odd asymmetry than the previous case [ cf . eq . ( [ countstat ] ) ] . furthermore , the distribution clearly deviates from a poissonian function , indicating that the presence of the strong coherent tunneling of cooper pairs tends to correlate the quasi - particle tunneling events across the right junction . this is further reflected in the deviations of the current noise from the classical shot noise value ( see discussions in section [ long : sec5 ] ) . one may expect that for a long waiting time ( @xmath142 ) , implying very large numbers of tunneled charges , the distribution of @xmath104 should approach a gaussian . in particular , this becomes an exact result if the distribution is poissonian . in our case , on the other hand , we have [ long : p2 ] @xmath143 where @xmath144 is a gaussian distribution , @xmath145 \label{normal}\ ] ] with @xmath146 . the distributions for both even and odd @xmath104 are separately gaussian , but @xmath147 as a whole is not , since even - odd asymmetry is still present . in fig . [ fig07 ] we compare the stationary - state results for @xmath148 in the weak and strong dephasing limits with the gaussian distribution . at @xmath149 ( a ) @xmath150 and ( b ) @xmath151 . for a comparison , a normal distribution function given in eq . ( [ normal ] ) is also shown ( dashed line).,title="fig:",width=226 ] at @xmath149 ( a ) @xmath150 and ( b ) @xmath151 . for a comparison , a normal distribution function given in eq . ( [ normal ] ) is also shown ( dashed line).,title="fig:",width=226 ] at @xmath152 for @xmath153 ( filled circle ) . for a comparison , the poissonian distribution is also plotted ( empty circle ) . ] finally , it is interesting to understand what happens to the stationary counting probability @xmath154 in the intermediate regime , i.e. , when the dephasing rate is comparable to the josephson energy . unfortunately , an analytic expression is not available in this case ; the numerical results , however , are shown in fig.[fig08 ] , where one can see that the distribution function deviates significantly from a poissonian , being suppressed ( enhanced ) for odd ( even ) @xmath104 . a deeper insight into the transport process can be obtained in the frequency domain , from a careful analysis of the spectral power of current fluctuations . the zero - frequency shot noise could be directly determined by the second moment of the counting probability eq . ( [ serve ] ) , see ref . . here , however , we follow a different route which allows us to get the entire current spectrum . to this end , we define the noise spectrum as @xmath155 where @xmath156 and @xmath157 . the total current @xmath158 through the system is related to the _ tunneling _ currents @xmath159 across each junction by @xcite @xmath160 to simplify the evaluation of @xmath161 , it is convenient to introduce the spectral densities of currents flowing across the individual junctions and the cross correlation spectral powers . therefore , in a way analogous to eq . ( [ sdef ] ) , we write ( @xmath162 ) @xmath163 which allows to express the total shot noise spectrum in the form @xmath164 \,.\end{gathered}\ ] ] in the stationary state @xmath165 , so that @xmath166 in the zero - frequency limit . in the opposite limit ( @xmath167 ) , @xmath168 @xcite . in our case , the left junction is ( nearly ) at resonance for the cooper pair tunneling and therefore @xmath169 . in order to obtain @xmath170 we have to calculate two - time correlation functions . we follow the standard procedure based on the quantum regression theorem @xcite and define the auxiliary matrices [ long : chieta ] @xmath171 where the index @xmath172 runs over the left and right junctions ( @xmath173 ) . these auxiliary operators , @xmath174 and @xmath175 , satisfy a master equation of exactly the same form as eq . ( but with respect to @xmath102 instead of @xmath108 and , of course , with different initial conditions ) . their relevance can be understood by noticing that the correlation functions can be expressed directly in terms of their average values@xcite : @xmath176 @xmath177 and @xmath178 the problem is now reduced to ( a ) solving a master equations of the form given in eq . to get @xmath179 , @xmath180 , and @xmath181 for respective initial conditions , and ( b ) evaluating eqs . , , and to obtain the correlation functions . following this procedure and performing the fourier transforms of the resulting correlation functions , we find that in the stationary state @xmath182 @xmath183 and @xmath184 where we have used the bra - ket notations @xmath185 @xmath186 and @xmath187 from eqs . , , , and , it follows that the zero - frequency noise is given by @xmath188 in the strong dephasing limit ( @xmath189 ) , the second term in eq . becomes negligibly small , as it vanishes as @xmath190 . therefore , the zero - frequency shot noise is enhanced approximately by a factor @xmath191 compared with its classical value , @xmath192 . this can be understood in terms of the josephson quasi - particle ( jqp ) cycle @xcite . because of the fast quasi - particle tunneling across the right junction , each cooper pair that has tunneled into the central island breaks up immediately into quasi - particles , and quickly tunnels out . the charge is therefore transferred in units of @xmath193 ( compared with @xmath194 in classical charge transfer ) for each jqp cycle . this was already confirmed in the counting statistics of the transmitted charges . according to eq . , the probability that an odd number of electrons are transferred is zero and charges are transferred only in pairs . in the weak ( @xmath195 ) and moderate ( @xmath196 ) dephasing limits , the semiclassical jqp picture breaks down and we do not have shot noise enhancement any longer . in the limit @xmath195 , the period of oscillations of cooper pair is very short compared to the typical time for for quasi - particles to tunnel out of the central island . the system can be regarded as a single - junction circuit , where the quasi - particle tunneling events are independent . the fano factor @xmath197 , becomes much closer to unity in this limit . the small deviation from the poisson value is due to the fact that the quasi - particle tunneling events can not be considered as independent because coulomb blockade allows only one cooper pair to oscillate coherently across the left junction . therefore , the tunneling process corresponding to @xmath198 is likely to be followed by @xmath199 . it is clear that this behavior is related to the residual even - odd asymmetry we found in the counting statistics , eq . , even in the long waiting - time limit ( @xmath200 ) , eq . . with moderate dephasing ( @xmath196 ) , quasi - particle tunneling events across the right junction are strongly affected by the _ coherent _ oscillation of cooper pairs across the left junction . indeed , this effect gives rise to the significant deviation from the poissonian distribution of the tunneling statistics , eq . . most remarkably , it leads to a suppression of the shot noise . the strongest suppression , by a factor of @xmath201 , is achieved at resonance ( @xmath75 ) for @xmath153 , see fig . [ fig09 ] . this is reminiscent of the shot noise suppression in ( non - superconducting ) double - junction systems@xcite , whose maximal suppression is by factor @xmath202 for the symmetric junctions . we emphasize , however , that in the latter case , the coherence was not essential . in our case , on the contrary , the role of coherence becomes evident by noticing that the dip in fano factor disappears when moving away from the resonant condition as shown in fig . [ fig09 ] . . the dip in the noise is most pronounced at resonance ( @xmath75).,width=226 ] ) for @xmath203 , and ( b ) at a fixed weak dephasing rate ( @xmath204 ) for @xmath205.,title="fig : " ] ) for @xmath203 , and ( b ) at a fixed weak dephasing rate ( @xmath204 ) for @xmath205.,title="fig : " ] as a function of frequency @xmath206 in the ( a ) strong ( @xmath207 ) and ( b ) weak ( @xmath208 ) quasi - particle tunneling limits . for both plots , @xmath75 and @xmath209 were assumed.,title="fig : " ] as a function of frequency @xmath206 in the ( a ) strong ( @xmath207 ) and ( b ) weak ( @xmath208 ) quasi - particle tunneling limits . for both plots , @xmath75 and @xmath209 were assumed.,title="fig : " ] in figs . [ fig10 ] and [ fig11 ] we show the typical behavior of the finite - frequency noise spectrum in the ( a ) strong and ( b ) weak dephasing limits . it is interesting to notice that ( only ) in the weak dephasing limit ( @xmath195 ) , there is a resonance peak of the form @xmath210 where the resonance frequency is given by @xmath211 \,.\ ] ] clearly , the peak is an effect of coherent quantum oscillations between the two energy levels separated by @xmath212 , induced by the josephson effect across the left junction . as expected , the resonance peak is reduced in its height and broadened in its width with increasing @xmath69 . on the contrary , as @xmath81 increases , the peak gets sharper and the peak height increases quadratically with @xmath81 . however , this should not be confused with the zero - frequency case , where @xmath81 effectively enhances the decoherence effects . as @xmath81 increases , the josephson oscillation across the left junction becomes faster , and there are less chances that it is interrupted by the quasi - particle tunneling across the right junction . this , in turns , implies that the coherent oscillation is better defined and the spectral component ( especially @xmath213 ) at frequency @xmath214 is highly enhanced . for a vanishingly small quasi - particle tunneling rate , @xmath215 would approximately become a delta like function , centered at @xmath216 . however , one should not be misled by this result , since the noise is always proportional to the average current , which vanishes in this limit . it is worth mentioning here on the relation between this result and the description of the noise output from linear detector @xcite . in the setup considered in this work , the right electrode has the role of the detector of cooper pair oscillations ; since the total current in the circuit is due to quasi - particle tunneling ( i.e. it is a _ dissipative _ current ) , the output signal may be considered as classical . this has to be compared to the case of a detector measuring the charge on the island . there , the back - action of the detector was essential to produce an observable result . in the case of the current , instead , the `` detector '' is intrinsically part of the system and it couples to the observed quantity in an essentially non - linear way . in this paper we considered properties of the distribution of the transmitted charge in a superconducting set tuned close to a cooper pair resonance . the dominant process to the transport in the regime considered here , is the jqp cycle , a process in which coherent cooper pair oscillations are accompanied by ( incoherent ) quasi - particle tunneling . the interplay between the coherence and the strong coulomb blockade manifests itself in various ways both in the counting statistics and in the shot noise . we found two distinct regimes characterized by different ratios of the time scales for dephasing and relaxation , @xmath217 in the strong dephasing limit or @xmath218 in the opposite case of weak dephasing . a generic feature of the counting statistics , valid in both the regimes , is its even - odd asymmetry related to the fact that charge transport is mediated by the cooper pair tunneling . other properties are more pronounced in one of the two regimes . an example is the dependence of @xmath103 on the initial time @xmath108 . this is clearly visible in the strong dephasing limit while quickly lost in the weak dephasing regime since , because of the strong josephson energy , the state changes significantly before quasi - particles have any time to be produced . another important point is that the counting statistics is not poissonian , due to the relevance of correlations between different tunneling events . as a consequence the fano factor is different from the classical value . the maximal suppression of the zero - frequency shot noise is observed when the quasi - particle tunneling rate is comparable to the frequency scale of the coherent cooper pair oscillations . we finally investigated the shot noise at finite frequencies , which shows a resonance peak at the josephson oscillation frequency . this maximum can be interpreted as an effect of coherent quantum transitions between the two energy levels involved in the transport phenomena in the device . we thank d.v . averin , y. blanter , and j. siewert for very useful discussions . we acknowledge financial support from european community ( ist - fet - squbit ) and infm ( pais - tin ) . m .- s.c . acknowledges the support from the swiss - korean outstanding research efforts award program . d. v. averin , in _ `` exploring the quantum classical frontier : recent advances in macroscopic and mesoscopic quantum phenomena '' _ , eds . j. r. friedman and s. han , to be published ( cond - mat /0004364 ) ; d. v. averin , preprint ( 2002 ) , cond - mat/0202082 . d. loss and e.v . sukhorukov , phys . * 84 * , 1035 ( 2000 ) ; m .- s . choi , c. bruder , and d. loss , phys . b * 62 * , 13 569 ( 2000 ) ; f. plastina , r. fazio , and g. m. palma , phys . rev . b * 64 * , 113306 ( 2001 ) . a microscopic derivation is given in g .- ingold and y. v. nazarov , in _ single charge tunneling : coulomb blockade phenomena in nanostructures _ , edited by h. grabert and m. devoret ( plenum press , new york , 1992 ) .
|
we analyze charge tunneling statistics and current noise in a superconducting single - electron transistor in a regime where the josephson - quasiparticle cycle is the dominant mechanism of transport . due to the interplay between coulomb blockade and josephson coherence ,
the probability distribution for tunneling events strongly deviates from a poissonian and displays a pronounced even
odd asymmetry in the number of transmitted charges .
the interplay between charging and coherence is reflected also in the zero - frequency current noise which is significantly quenched when the quasi - particle tunneling rates are comparable to the coherent cooper - pair oscillation frequency . furthermore the finite frequency spectrum shows a strong enhancement near the resonant transition frequency for josephson tunneling .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the increasing demand for accurate control of quantum devices using high - fidelity control protocols@xcite , has stimulated interest in the study of the dynamics of quantum systems in response to slowly varying hamiltonian . moreover , rapid progress in the field of adiabatic quantum computing has fueled further interest in and need for more careful analysis of the dynamics of quantum systems whose parameters vary slowly in time.@xcite in addition , decoherence in any real quantum system sets a rigid constraint on the time interval during which a quantum protocol must be carried out , limiting all protocols to intermediate time intervals that are shorter than the decoherence time . at these intermediate time scales , both non - adiabatic corrections and coupling to the environment become equally important . the previous analysis@xcite of the qubit dynamics with time - dependent hamiltonians was based on the lindblad master equation@xcite that describes the interaction with environment in terms of dephasing and transition processes characterized by phenomenological decoherence rates . an alternative microscopic approach , formulated as a perturbative theory for a quantum system with a time - independent hamiltonian interacting with its environment , introduces the bloch redfield ( br ) master equation@xcite . if the hamiltonian of the system changes in time , the br approach has to be modified to properly account for a non - adiabatic corrections . in this paper we extend the br approach to account for slow evolution of the system hamiltonian in the presence of the environment . the main concept of the br theory is based on the identification of decoherence processes in terms of the matrix elements for transitions caused by environment in the eigenstate basis of the quantum system.@xcite for the hamiltonian that varies with time , one can still use a basis defined by eigenvectors of the hamiltonian@xcite , where the hamiltonian is always represented by a diagonal matrix @xmath0 , where the unitary transformation @xmath1 denotes a transition from the original basis to the eigenstate basis . time - dependence of @xmath1 produces an extra term in the time evolution of the quantum system that is effectively described by the new hamiltonian @xmath2 . this expression is not necessarily diagonal and another basis transformation is required . such series of diagonalization transformations can be continued indefinitely , but for slowly changing hamiltonian , the series can be truncated after a finite number of transformations neglecting terms of the higher order in time - derivatives of the parameters in the hamiltonian . in addition to changes in the effective spectrum of the system , matrix elements representing coupling between the quantum system and its environment are also modified , resulting in a redefinition of the transition rates for the system . we focus our analysis on the dynamics of a two - level quantum system a qubit or a spin-1/2 system in the presence of time - dependent field , which we refer to below as the control field of the qubit . we study the dynamical response of the transverse magnetization to quench velocity of the control field . the transverse magnetization measurements can provide the value of the berry curvature of a quantum system @xcite and , consequently , characterize topological properties of a ground state of the system . since any real qubit is always coupled to its environment , it is necessary to perform detailed analysis of the non - adiabatic dynamics of a qubit system in the presence of dissipation . to this end , we investigate the effect of pure dephasing and energy relaxation due to the ohmic bath on the qubit polarization . our results indicate that the decoherence suppresses the transient wiggles of the out - of - plane qubit projection , thereby bringing up the linear relation between the qubit response and the quench velocity . thus , the dissipation facilitates the berry curvature measurement based on the non - adiabatic response , proposed in ref . @xcite . furthermore , this study is also related to the measurement technique of the berry phase in qubits , which are based on an interference experiment where the drive parameter was changed slowly @xcite , see also refs . @xcite for theoretical analysis of the influence of environment on the berry phase . we also apply the modified br equation to the landau zener ( lz ) problem@xcite in a qubit coupled to environment at arbitrary temperature . the lz problem in a quantum system coupled to its environment has attracted significant interest recently , where the environment was considered either as a source of classical noise@xcite , or quantum fluctuations that cause transitions between qubit states @xcite , or pure dephasing@xcite . more recently , the lz interferometry has attracted a growing interest@xcite . here we focus on the role of quantum fluctuations in the environment that cause transitions between the eigenstates of the qubit in the lz problem . we argue that during the lz transition , the matrix elements of the coupling between the qubit and its environment must be considered in the basis of eigenstates of the full qubit hamiltonian and therefore , the matrix elements acquire an explicit time dependence due to rotation of the eigenstate basis in addition to straightforward dependence on the energy difference between the eigenstates . this treatment modifies the previous results of refs . @xcite and generalizes the results of refs . @xcite , where a similar basis transformation was naturally included in the calculations . we disregard the effect of the lamb stark shift on the qubit spectrum due to coupling to the environment , considered in ref . @xcite , since this can be included in the redefined control field of the qubit . we focus solely on the transition effects due to non - unitary evolution of the qubit density matrix . we consider the quantum fluctuations of the environment that are fixed along the direction of the control field at very long initial and final moments of the lz transition so that the matrix element that characterizes the transition between qubit states at long times is absent and environment produces dephasing only . for arbitrary direction of the fluctuating field , the transition remains effective over long time and will effectively bring the qubit to the ground state for zero temperature environment . we also consider `` dephasing '' coupling@xcite when the quantum fluctuations occur only in the direction parallel to the direction of the control field in the parameter space of the qubit hamiltonian . our result is in agreement with ref . @xcite of the same problem within lindblad master equation , in the limit of a high - temperature environment . this paper is organized as follows . in section [ sec2 ] , we present a formalism of the br equations in transformed basis for time - dependent hamiltonians . in section [ sec3 ] , we study the evolution of a qubit whose control field rotates in a plane with a constant magnitude and consider different directions of the environmental coupling field . in section [ sec4 ] , we consider the lz problem in the presence of zero and finite temperature environment and show that transition is dominated by thermal excitation of the qubit at finite temperatures . in section [ sec5 ] , we analyze the non - adiabatic effects within the lindblad formalism . we end with conclusions in section [ sec6 ] . we consider a spin coupled to a bath of harmonic oscillators . the full hamiltonian @xmath3 is a sum of the hamiltonian for the spin in the magnetic field @xmath4 @xmath5 the interaction hamiltonian of the spin with the environment @xcite @xmath6 and the bath hamiltonian @xmath7 here we assume that each environment oscillator interacts with the spin as a quantized magnetic field @xmath8 in the common direction @xmath9 , @xmath10 and @xmath11 are raising and lowering operators of the field . the reduced density matrix @xmath12 of the spin is determined by tracing out environment degrees of freedom of the full density matrix @xmath13 . the full density matrix satisfies the unitary master equation @xmath14\,.\end{aligned}\ ] ] there are several approaches to obtain the corresponding equations for time evolution of the reduced density matrix for the qubit . here we consider the limit of weak coupling of a qubit to the environment , when the density matrix is defined by the br equations,@xcite see also refs . @xcite where a diagrammatic technique was developed to treat the weak coupling to environment . the environmental effects are characterized by the spectral density function of the coupling @xmath15 . a generic spectral function has a power law dependence on energy at small energies , @xmath16 , and vanishes rapidly for energies above the ultraviolet cutoff @xmath17 . here , we consider the ohmic ( @xmath18 ) environment with exponential high - energy cutoff : @xmath19 where the dimensionless parameter @xmath20 defines the strength of coupling between the qubit and its environment and @xmath17 is the cutoff . we restrict ourself to the weak coupling limit , @xmath21 . in general , the effect of weak environment on the qubit dynamics is twofold . on one hand , the qubit hamiltonian is renormalized by the environment modes with @xmath22 , known as the lamb and stark effects . on the other hand , when we integrate out the environmental degrees of freedom , we also obtain non - unitary terms in the evolution of the quantum system . both of these effects are accounted for by the br equation@xcite for the qubit density matrix @xmath23 . we first consider the case of a constant external magnetic field along @xmath24 direction , @xmath25 . then , the br equation has the following form in the eigenstate basis [ eq : brall ] @xmath26 we obtained the above equations within secular approximation that neglects fast oscillating terms with frequencies larger than the decoherence rates . the equation in the matrix form , eq . , determines the evolution of diagonal elements of the density matrix . the relaxation and excitation rates , @xmath27 and @xmath28 , are defined by the spectral density @xmath29 at the energy corresponding to the energy difference between two states of the qubit : [ eq : gamma_re ] @xmath30 and @xmath31 $ ] is the planck s function . the factor @xmath32 indicates that only the component of the fluctuating environment field that is perpendicular to the direction of the control field @xmath33 gives rise to the qubit flip processes . the off - diagonal elements of the density matrix are characterized by the decoherence rate @xmath34 and pure dephasing rate @xmath35 given by @xmath36 the decoherence stems from two processes the qubit flip processes with rate @xmath37 , and pure dephasing which is not responsible for energy transitions at low frequency with rate @xmath38 . the only source of pure dephasing is the fluctuating fields of the environment along the external field @xmath33 , hence the factor @xmath39 in the definition of the pure dephasing term , @xmath40 . the renormalization of the qubit hamiltonian by the environment due to the lamb or stark effects are determined by the imaginary part of the environmental correlation function , as discussed in ref . explicitly , the renormalized qubit energy @xmath41 is @xmath42 where @xmath43 denotes the cauchy principal value . below , we assume that the control field @xmath33 already includes renormalization effects from the environment . the goal of this paper is to investigate the features of the qubit evolution originating from decoherence characterized by rates @xmath44 and @xmath45 , respectively . the significance of the effect of the lamb and stark shifts on the evolution of the qubit was demonstrated in ref . @xcite in the context of the lz problem . we note that the qubit density matrix can be defined in terms of the magnetization in @xmath46 , @xmath47 and @xmath48 directions as @xmath49 then the br equations , eq . , acquire a more common form of the bloch equations [ eq : blochall ] @xmath50 the above br equations were obtained in the basis of qubit eigenstates . in case when the control field @xmath4 changes in time , we perform transformation @xmath51 of the basis that keeps the qubit hamiltonian diagonal . this basis is commonly referred to as adiabatic . the corresponding transformation has two consequences . the first consequence of @xmath51 transformation is that the hamiltonian in the new basis acquires an extra term originating from the time dependence of the transformation @xmath51 . thus , the qubit hamiltonian in the new basis is @xmath52 the resulting hamiltonian still may remain non - diagonal due to the berry connection term , @xmath53 . we can introduce a new transformation @xmath54 that diagonalizes the right hand side of , but this transformation generates a new term @xmath55 and the `` diagonalization '' series of transformations @xmath56 does not stop for an arbitrary time evolution of @xmath4 , because the berry connection terms appearing in each consecutive diagonalization transformation acquires an extra time derivative . however , for slow time evolution , the series of transformations can be truncated by the first one or two transformations . since the br treatment of environmental effect requires anyway that the system changes in time slower than the rates given by eqs . and in the master equation , the truncation to a limited number of transformations @xmath56 under slow evolution of @xmath4 is justified . also , in a special case of constant rotation of @xmath4 in a plane , the second transformation @xmath57 is time - independent and transformation series stops after this second basis rotation . the second consequence of the basis transformations is the modified interaction term in that the coupling between the qubit and its environment @xmath58 is modified from the initial coupling operator @xmath59 to the environment field by the transformation matrix @xmath60 . this transformation changes the corresponding `` projection '' factors @xmath61 in eqs . as well as the spectral weights @xmath29 . modification of the coupling between the qubit and its environment , introduced by eq . , swaps components of the fluctuating field responsible for the pure dephasing and transition processes . for example , in case of a fixed external field @xmath62 , fluctuations along @xmath63 give rise to pure dephasing and do not cause transition processes between qubit eigenstates . however , as @xmath4 rotates while @xmath9 remains in @xmath63 direction , the fluctuating component along field @xmath4 is the only one responsible for the dephasing with the corresponding rate proportional to the spectral weight of its low - frequency fluctuations @xmath64 , while the component of the fluctuating field perpendicular to @xmath33 will produce qubit flip processes with the rate characterized by the spectral weight of fluctuating field with the energy equal to the energy of qubit flip @xmath65 . the second unitary transformation further mixes matrix elements of the coupling to environment representing qubit flip processes and pure dephasing . below , we present explicit expressions for the rates in eqs . for two special cases of evolution of @xmath4 for different types of environment . we focus on the effect of qubit flip processes due to environment and assume that @xmath66 in most numerical solutions . we note that the pure dephasing produced by the low frequency noise of the environment can be successfully described in terms of fluctuations of the classical field and may also include non markovian time correlations that are omitted in the br approach . effects of classical noise were discussed in refs . @xcite for the lz transition and in refs . @xcite for berry phase measurements . we first consider a qubit with the hamiltonian characterized by a time - dependent field in @xmath67 plane : @xmath68 . by definition , @xmath69 for @xmath70 . the transformation to adiabatic basis is defined by : @xmath71 and the resulting qubit hamiltonian has the form @xmath72 here , the second term is responsible for the non - diagonal form of the hamiltonian for time - dependent rotation angle @xmath73 and causes the resultant field to point out of the rotation plane of @xmath4 . this hamiltonian has eigenvalues @xmath74 and eigenvectors , which are different from the vectors of the adiabatic basis . the latter two represent spin states in the ( @xmath67 ) plane with @xmath75 . on the contrary , the qubit in the ground state @xmath76 of the hamiltonian has a non - zero expectation value of the polarization @xmath77 in the direction perpendicular to the ( @xmath67 ) plane of the control field @xmath33 : @xmath78 in the limit of slow rotations , @xmath79 , this result is consistent with a more general expression that connects a generalized force @xmath80 to time - dependent parameters @xmath81 of the hamiltonian through the berry curvature @xmath82 as @xcite @xmath83 comparing eq . and eq . , we identify @xmath84 , @xmath85 and @xmath86 . explicitly , the coefficient of the linear term in the rate of change of the magnetic field , _ i.e. _ @xmath87 , is the berry curvature @xmath88 . indeed , this value of the berry curvature gives the berry phase @xmath89 for one full rotation of the control field in the @xmath90 plane after its integration over the half - sphere , @xmath91 . this relation holds for an isolated qubit controlled by field @xmath4 , assuming that @xmath4 is a slowly varying function of time with continuous higher derivatives . however , if the rotation of the control field @xmath33 starts instantaneously with constant angular velocity @xmath92 , _ i.e. _ @xmath93 , the rotation is equivalent to a quantum quench in the representation of eq . from @xmath94 to @xmath95 . the qubit that was initially in the ground state of the original time - independent hamiltonian , @xmath96 , is in the superposition of eigenstates of the new hamiltonian and exhibits precession around new direction of the effective field @xmath97 . this precession causes oscillations of @xmath98 around its average value given by eq . . in this section we demonstrate that a qubit coupled to a zero - temperature environment relaxes towards the lower eigenstate of hamiltonian and for long time limit after the rotation started , the qubit state obeys eq . . for rotation with constant angular velocity @xmath99 , the transformed hamiltonian , eq is time independent and can be diagonalized by the second basis transformation @xmath100 the qubit hamiltonian in a new basis after a full transformation @xmath101 becomes fully diagonal with time - independent eigenvalues : @xmath102 we can apply the br equation for the qubit density matrix , where the rates in eq . are defined by the interaction term @xmath103 , eq . , with @xmath104 replaced by its transformation under @xmath105 according to eq . . the result of the @xmath105 transformation depends on the original orientation of the vector @xmath9 in the qubit space . below , we consider three orientations of @xmath9 . we note that for the limit @xmath106 considered in this section , the shift of eigenvalues of hamiltonian and modification of the coupling to environment by the second transformation @xmath107 is not significant and can be disregarded to the lowest order in @xmath99 . ( color online ) time dependence of the out - of - plane polarization , @xmath108 , at zero temperature of environment for @xmath109 ( solid line ) and @xmath110 for fluctuating environment field out of the plane of rotation , @xmath111 . the pure dephasing rate is zero , @xmath66 . the frequency of rotation of the control field is @xmath112 . ] we first consider the case when the coupling between the qubit and its environment is determined by the vector @xmath113 perpendicular to the plane of rotation of the external field @xmath4 . for time independent hamiltonian , this coupling causes qubit flip processes and the corresponding decoherence rates are defined by the environment spectral function at the excitation energies equal to the qubit energy splitting . for time - dependent hamiltonian with rotating @xmath4 , we have to write the qubit coupling operator @xmath114 in the rotated basis that diagonalizes the original hamiltonian . as we discussed above , the transformation is a product of two consecutive transformations . the first transformation , @xmath51 to the adiabatic basis does not change the coupling operator @xmath115 . the second transformation results in @xmath116 here , the first term represents the qubit flip process , while the second term preserves the qubit orientation and causes pure dephasing . the corresponding rates in the br equations are given by [ eq : ratesa ] @xmath117 , \\ \gamma_{e } & = \frac{\cos^{2}\eta}{2}j(w ) n(w),\\ \gamma_{2 } & = \frac{\gamma_{r}+\gamma_{e}}{2 } + \frac{\sin^{2}\eta}{2}j_0,\end{aligned}\ ] ] with @xmath118 and @xmath119 defined by eqs . and . the qubit dynamics is characterized by the relaxation and excitation rates proportional to the spectral function @xmath120 of environment at energy @xmath118 , these rates appear with factor @xmath121 and recover the case of the qubit with a time - independent hamiltonian with @xmath122 when only environment modes in resonance with the qubit contribute to the qubit dynamics . at finite @xmath99 , however , the pure dephasing mechanism arises after transformation @xmath123 and originates from the low frequency modes of the environment with spectral density @xmath64 . the pure dephasing rate contains factor @xmath124 which is small for slow rotation with @xmath106 . ( color online ) time dependence of the out - of - plane polarization , @xmath108 , at various temperatures of environment : @xmath125 ( solid line ) , @xmath126 ( dashed line ) and @xmath127 ( dotted line ) for fluctuating environment field out of the plane of rotation , @xmath111 . the pure dephasing rate is zero , @xmath66 . the frequency of rotation of the control field is @xmath112 . the coupling to environment @xmath110 . ] the bloch equations eq . with rates given by eq . can be solved to get the qubit density matrix @xmath128 in the secondly rotated basis . in conjunction with the initial condition , the time evolution of @xmath108 is then obtained by @xmath129 $ ] . first , we provide an exact analytical solution by choosing the initial state to be a thermal state @xmath130 . defining @xmath131 , the initial condition for the bloch equation becomes @xmath132 , @xmath133 and @xmath134 . integrating the bloch equation with the above initial condition yields @xmath135 where @xmath136 and we assumed @xmath66 . in the long times limit , @xmath137 , @xmath108 reaches its stationary state solution @xmath138 regardless of the form of the initial state . the significance of this expression is that the dynamical transverse response of the qubit subject to a rotating magnetic field is a consequence of the geometric phase effect in the sense that the stationary value @xmath139 does not depend on the strength of the coupling to environment . therefore , @xmath139 is purely geometrical and immune to quantum zero - temperature fluctuations of the environment . next , in order to get the numerical solution of the br equations we utilize standard integration methods for a system of linear differential equations with time - dependent coefficients . alternatively , we obtain the same results using the br functions of the qutip package @xcite with a proper adjustment to the system hamiltonian and the interaction term , see eqs . and , for time dependence of the eigenstate basis , as presented in figs . [ fig:3a_my_vs_t_variosgammatzero ] and [ fig:3a_my_vs_t_for_temp ] . we verified that the results shown in the plots are identical to numerical integration of the br equations with the rates given by eqs . . in both plots , the initial condition of the density matrix is chosen to be the ground state at @xmath140 when @xmath141 . we obtain plots consistent with the analytical result , eq . , for the thermal state of the density matrix at @xmath140 . in fig . [ fig:3a_my_vs_t_variosgammatzero ] , we present the time evolution of @xmath108 for several values of the coupling to the environment . from the plot it is clear that the role of the environment is to suppress transient wiggles of @xmath77 and to bring the system to the steady state , defined by eq . with @xmath142 . however , the transverse magnetization is fragile to thermal fluctuations , since these fluctuations create excitation to the higher energy state . the result is shown in fig . [ fig:3a_my_vs_t_for_temp ] , where we fix @xmath20 and plot @xmath108 for different temperatures @xmath143 . we note that since the dephasing rate , @xmath144 grows with the temperature , the oscillations decay faster for higher temperatures . also , at finite temperatures , the spin has nonzero probability to stay in the excited state , the asymptote of @xmath145 is reduced in agreement with eq . . ( color online ) time dependence of the out - of - plane polarization , @xmath108 , at zero temperature of environment , for @xmath110 ( solid line ) and @xmath146 ( dashed line ) for fluctuating environment field in the plane of rotation , @xmath147 . the pure dephasing rate is zero , @xmath66 . the frequency of rotation of the control field is @xmath112 . the relaxation is reduced for time intervals when @xmath148 . ] we now consider the qubit interacting with environment field in the plane of rotation . we take @xmath149 and for @xmath150 the coupling to the environment results in pure dephasing and is characterized by the low frequency spectral density @xmath64 . as @xmath33 rotates , the effect of environment alternates between pure dephasing and qubit transitions between eigenstates . we obtain this variation in qubit flip and dephasing rates already after applying transformation @xmath151 to the interaction hamiltonian of the qubit and environment , eq . . however , for rotating @xmath4 we have to take into account the gauge term @xmath152 in eq . by applying the second transformation @xmath123 to @xmath153 . we obtain @xmath154 that contains matrix elements for qubit flip processes at any moment of time . the corresponding rates in the bloch redfield equations are [ eq : ratesz ] @xmath155 , \\ \gamma_{e } & = \frac{g(t)}{2}j(w ) n(w),\\ \gamma_{2 } & = \frac{\gamma_{r}+\gamma_{e}}{2 } + j_0 \cos^{2}\eta\cos^{2}\omega t , \end{aligned}\ ] ] where @xmath156 and thus the qubit flip rates are nonzero as a function of time . the evolution of the qubit in this case corresponds to precession of a spin in the magnetic field with initial state distinct from its new ground state after the quench . namely , its dynamics will correspond to suppression of off - diagonal elements of its density matrix with the rate @xmath157 and equilibration of the diagonal elements of @xmath158 with rates @xmath159 . we emphasize that in this case all decoherence rates are time - dependent . we calculate time - dependence of @xmath108 by numerically solving the br equations with the rates given by eq . . we present the result of integration in fig . [ fig:3b_z_zerotemp_variousgamma ] for two different values of @xmath20 at zero temperature and find clear evidence that the decoherence rates are roughly one half smaller compared to the result of previous subsection for the same value of @xmath20 . meanwhile , in fig . [ fig:3b_z_variostemp ] we fix @xmath20 and plot @xmath108 for different temperatures . at time longer than the relaxation time @xmath160 , @xmath108 becomes constant with its value @xmath161 , see eq . ( color online ) time dependence of the out - of - plane polarization , @xmath108 , for @xmath146 and @xmath125 ( solid line ) , @xmath126 ( dashed line ) and @xmath127 ( dotted line ) in case when fluctuating environment field is in the plane of rotation , @xmath147 . the pure dephasing rate is zero , @xmath66 . the frequency of rotation of the control field is @xmath112 . ] we also consider a somewhat artificial scenario when the coupling vector @xmath162 in eq . rotates together with the external field @xmath4 . e.g. _ when @xmath4 is realized as two quadratures of microwave pulse driving a qubit and the environment is described by longitudinal quantum fluctuations of the pulse . ] for a stationary hamiltonian this environment does not produce qubit flip processes and results in pure dephasing , when the diagonal elements of the density matrix do not change and only off diagonal elements decrease with time . in case when the direction of the control field rotates with frequency @xmath99 , the basis transformation term in eq . introduces qubit flip processes for this coupling with the rates in eqs . given by [ eq : gammas_lf ] @xmath163 , \\ \gamma_{e } & = \frac{\sin^2\eta}{2}j(w ) n(w),\\ \gamma_{2 } & = \frac{\gamma_{r}+\gamma_{e}}{2 } + \cos^{2}\eta j_0 . \end{aligned}\ ] ] for slow rotation @xmath106 , we have @xmath164 and qubit flip processes are small . in this case , dephasing will suppress precession on time scale @xmath165 , and further equilibration of the system occurs on a longer time scale @xmath166 . we describe the evolution of a qubit coupled to high temperature environment using a dephasing lindblad model in sec . [ sec5 ] . ( color online ) time dependence of the out - of - plane polarization , @xmath108 , at zero temperature of environment for a qubit coupled to a damped harmonic oscillator with damping rate @xmath167 and coupling constant between the qubit and environment @xmath168 . coupling vector @xmath169 ( solid line ) and @xmath170 ( dash - dotted line ) . for comparison , the solution for the bloch redfield equation is presented ( dashed line ) with @xmath171 and @xmath66 . the rotation angular velocity is @xmath112 . ] in this subsection we consider the interaction of a qubit with a single damped quantum harmonic oscillator . this model can be used to describe environment with a sharp spectral function @xmath29 . the interaction part of the hamiltonian is similar to eq . : @xmath172 and the single - mode hamiltonian of the oscillator is @xmath173 . we describe dissipation of the oscillator using the lindblad relaxation operators for the full density matrix @xmath174 of the qubit and the oscillator system : @xmath175 -\kappa \left ( \hat{a}^\dagger \hat{a } \bar \rho+ \bar \rho \hat{a}^\dagger \hat{a}- 2 \hat{a}\bar \rho \hat{a}^\dagger \right)\ ] ] this equation is a standard lindblad master equation with time dependent hamiltonian . the difference with the previous calculations of this section is that we keep a full quantum mechanical treatment of the qubit interaction with the oscillator and perform all transformations of the qubit basis for the full hamiltonian of the qubit and the oscillator . at the same time , we assume that the lindblad superoperator for the relaxation of the harmonic oscillator , represented by the last term in eq . , is not affected by these transformations . we evaluate the qubit projection perpendicular to the rotation plane of the control field as a function of time . [ fig:3c_zerotemp ] shows the comparison between calculation of bloch - redfield equations and damped quantum oscillator with different coupling directions at zero temperature . all three curves saturate at universal value @xmath176 . it is worth pointing out that the @xmath170 coupling results in time - dependent transition rates that are at minimum when @xmath177 and at maximum when @xmath122 , as one can conclude from the amplitude of oscillations of @xmath108 for @xmath178 . effectively , the overall relaxation is slower than that of the case @xmath169 and the amplitude of oscillating @xmath77 at @xmath179 decays insignificantly . the calculations at finite temperature @xmath180 are plotted in fig . [ fig:3c_temp0p5 ] and in all cases @xmath139 is consistent with eq . . ( color online ) time dependence of the out - of - plane polarization , @xmath108 , at environment temperature @xmath126 for a qubit coupled to a damped harmonic oscillator with damping rate @xmath167 and coupling constant between the qubit and environment @xmath168 . coupling vector @xmath169 ( solid line ) and @xmath170 ( dash - dotted line ) . for comparison , the solution for the bloch redfield equation is presented ( dashed line ) with @xmath171 and @xmath66 . the rotation angular velocity is @xmath112 . ] in this section we consider the landau zener transition in a qubit coupled to its environment . the external field in the qubit hamiltonian has the following form @xmath181 , where @xmath182 is the minimal level separation and @xmath183 characterizes the rate at which the hamiltonian changes . for the landau zener problem , the qubit is initially in the ground state @xmath76 with the density matrix @xmath184 . the task is to find the probability of the system to be in the excited state @xmath185 which is given by @xmath186 . effects of the environment on qubit s dynamics can be separated into pure dephasing of the qubit state during the lz process and inelastic qubit flips . when we consider a qubit coupled to its environment that causes qubit flip processes , we have to be careful with the formulation of the lz problem . indeed , the lz process is formally infinitely long and the qubit flip processes accompanied by the energy exchange will result in equilibration of the qubit system with its environment . in particular , for the zero temperature environment , the qubit will relax to the ground state even if it was temporarily excited during the lz process . for environment at finite temperature , the qubit state will tend to thermal state @xmath187 with @xmath188 . but as formally @xmath189 for long times @xmath190 , the qubit will relax to the ground state and we find @xmath191 . previous considerations , see _ e.g. _ @xcite , predicted @xmath192 for the ohmic environment with large high - frequency cutoff in the environment modes . but in this case the problem looses its meaning since the lz transition is shadowed by trivial relaxation of a quantum system to its ground state by releasing its energy to the environment . one can reformulate the problem in terms of finite time lz process , which may be experimentally relevant situation in some cases . alternatively , one can assume that the environment spectral function has a relatively low cutoff at high frequencies @xmath193 and the relaxation is absent after time @xmath194 . here , we consider a special orientation of the coupling vector with environment when @xmath195,where @xmath63 is defined by @xmath196 . in this situation , the relaxation processes becomes weak at long times @xmath197 . this type of coupling is expected to be dominant in qubits with relatively long energy relaxation times , but with short dephasing time due to dominant coupling with the fluctuating field parallel to the qubit field along @xmath63 . we utilize the bloch redfield approach to the problem of landau zener transitions in the presence of environment with @xmath198 . in principle , we need to write the br equations in the basis where the transformed qubit hamiltonian is diagonal after an infinite series of basis transformations given by @xmath199 , which can be an infinite series . however , under the condition @xmath200 , the series of basis transformations can be limited by @xmath201 . ( color online ) projection of a qubit state during the landau zener process on the bloch sphere on @xmath202 state in the diabatic basis ( dash - dotted line ) and on the ground state in the adiabatic basis ( solid line ) and the `` improved '' eigenstate basis ( dashed line ) . in the diabatic basis the projection of the qubit state shows long oscillations with amplitude decreasing as a power law in time , while the eigenstate projections quickly reach the asymptotic value . level - crossing speed @xmath203 and no coupling to environment . ] the first transformation changes the representation from diabatic basis of states @xmath204 and @xmath202 along @xmath63 to the adiabatic basis of the ground , @xmath76 , and excited , @xmath185 , states , where the hamiltonian is diagonal . the first transformation matrix @xmath51 has the same form as in eq . except the rotation angle @xmath73 , which is now defined as @xmath205 the transformed hamiltonian in the adiabatic basis has the form@xcite @xmath206 the second transformation is chosen to diagonalize matrix @xmath207 and has the form @xmath208 the hamiltonian in this `` improved eigenstate '' basis has the form [ eq : h0u2lz ] @xmath209 without dissipation , the lz problem is equivalent in all three representations , with a properly written hamiltonian , i.e. , eq . for the diabatic basis , eq . for the adiabatic basis , and eq . for `` improved eigenstate '' basis . in all representations , the qubit follows the appropriate instantaneous control field @xmath210 , but since this field is time - dependent , the qubit deviates from the instantaneous direction of @xmath210 and acquires an additional precession around the control field . when the original field eventually reaches its final direction , @xmath150 at @xmath211 , the direction of the control field becomes time independent and the qubit simply precesses around @xmath63 with a non - zero projection of its state on the excited state , given by the known expression@xcite @xmath212 note that in fig . [ fig:3spheres ] this precession remains in all three considered representations , but the overall trajectories are smoother in the transformed representations . as we look at the projection of the qubit state on the `` excited state '' @xmath213 in the appropriate basis , see fig . [ fig : lzidealpz ] , the oscillations decrease faster in the transformed representations , because the control field @xmath4 aligns faster with its final direction . we also note that since the control field remains aligned with its initial direction longer in transformed basis , the numerical computation can run over shorter time intervals thus making computation faster and more accurate . next , we take into account interaction with the environment within the bloch redfield approach . the coupling to the environment is modified in the diagonal basis of the hamiltonian , see eq . and ref . @xcite . under the markovian approximation and to the second order in the coupling to environment , we obtain the corresponding br equations in the form [ eq : brlz ] @xmath214 where @xmath215 and @xmath216 are given by eq . . the rates for the above equations are [ eq : brrateslz ] @xmath217 , \label{eq : gammarlz } \\ \gamma_{e } & = \frac{g_{lz}(t)}{2}j(w(t ) ) n(w(t)),\\ \gamma_{2 } & = \frac{\gamma_{r}+\gamma_{e}}{2 } + j_0 \cos^{2}\eta\cos^{2}\theta(t ) , \end{aligned}\ ] ] where @xmath218 is a function of time dependent basis rotation angles @xmath73 and @xmath219 defined by eqs . and . we note that the above equations for br rates are given by truncation of transformation series of interaction hamiltonian , eq . , up to the second order , @xmath220 . therefore , the rates are defined within @xmath221 accuracy . the unitary evolution described by either @xmath207 or @xmath222 has no approximations and is valid for arbitrary values of @xmath183 . we emphasize that once the basis transformation gives rise to non - zero decoherence rates , the qualitative results are similar regardless of our choice of the br rates in the basis obtained after either @xmath223 or @xmath224 transformations . the rates in the @xmath223 basis are given by eq . with @xmath225 . we now discuss solution of eq . . ( color online ) the probability of occupation of the excited state in the landau zener transition in the @xmath226 basis . the temperature of environment is zero , @xmath125 , the level velocity is @xmath203 . we assume that the dephasing is absent , @xmath66 . the asymptotic curve for @xmath227 is given by eq . with a proper choice of integration constant @xmath228 ] we first consider the zero temperature environment and set @xmath229 to focus solely on qubit flips rather than dephasing . we numerically integrate the br equation and plot the probability of the system to be in the excited state @xmath230 as a function of time in fig . [ fig : lz_t0 ] for @xmath110 . for numerical integration , we used both direct integration of linear differential equations and the qutip s package for numerical solution of the bloch redfield equations @xcite , obtaining identical results . as the qubit levels go over the avoided crossing , the probability of the qubit to be in the excited state increases , roughly following the same function of time as @xmath231 for an isolated qubit , @xmath232 . as the levels further depart from each other , the relaxation of the qubit from the excited state becomes the dominant process in the qubit dynamics , and @xmath231 monotonically decreases and becomes constant once the level separation @xmath233 exceeds the ultraviolet cutoff @xmath17 , or @xmath194 and the qubit is effectively decoupled from the environment . in fig . [ fig : lz_t0 ] we compare the behavior of @xmath231 for different values of @xmath17 . for finite ultraviolet cutoff @xmath234 , the probability @xmath231 saturates for @xmath235 . for @xmath236 , the probability @xmath231 slowly decreases for all @xmath237 . to evaluate this suppression , we can utilize eqs . in the asymptotic regime for @xmath238 , when @xmath239 . we write [ eq : rho11lzt0 ] @xmath240 where we used the relaxation rate @xmath27 from eq . . the latter equation demonstrates that even for environment with @xmath178 , the relaxation on long times scales is important . formally , the power law dependence of @xmath231 on time originates from the slow converging integral @xmath241 due to linearly increasing environment spectral function @xmath29 with energy . with a proper choice of integration constant c , we obtain a good agreement between computed @xmath231 in fig . [ fig : lz_t0 ] and asymptote , defined by eq . . this power law dependence stops and reaches a fixed value @xmath242 when the qubit level separation exceeds the environment ultra - violet cutoff at times @xmath194 . we evaluate the long time asymptotic value of @xmath243 by taking into account the high energy cutoff in the environment spectral function , eq . . we obtain @xmath244 where @xmath245 is the euler s constant , the integration constant @xmath246 and factor @xmath247 describes suppression of the excited state due to slow relaxation while qubit level separation increases from its minimum @xmath182 to values above the cutoff energy @xmath17 , see appendix a for the derivation of eq . . equations are valid for @xmath248 . for larger values of @xmath20 , one has to take into account the renormalization of qubit hamiltonian when the off - diagonal matrix element in the original hamiltonian @xmath249 is given by the following self - consistent relation@xcite @xmath250 with solution @xmath251 . hence the relaxation rate is@xcite @xmath252 where @xmath253 is the gamma function . the integration over time with @xmath254 gives@xcite @xmath255 notice that in the limit @xmath248 , @xmath256 , the relaxation rate @xmath27 reduces to @xmath257 in agreement with the relaxation rate in eq . . similarly , eq . becomes eq . ( color online ) the probability of occupation of the excited state , @xmath231 in the landau zener transition in the @xmath226 basis at finite temperature of environment for @xmath203 , @xmath110 and @xmath66 . the solid lines represent solutions of rate equations that show good agreement with the br equations at higher temperatures . ] at finite temperatures , the excitation and relaxation rates may exceed @xmath258 terms for strong enough coupling of the qubit to its environment and slow drive @xmath183 . in this case , we disregard @xmath216 terms in eq . and the diagonal elements of the density matrix satisfy the rate equations . since the rate equations preserve the trace of the density matrix , @xmath259 , with @xmath260 , we introduce @xmath261 and obtain the differential equation for @xmath262 : @xmath263 the initial condition is @xmath264 for @xmath265 . while we can write a formal solution to eq . , the solution is not well defined due to logarithmic divergence of @xmath266 for the spectral function @xmath267 without a cutoff . we present the result of numerical solution of eqs . and the rate equations in fig . [ fig : lz_temprateeq ] . we notice that for higher temperatures , these two solutions are indistinguishable because the thermal effects dominate only in short time scales @xmath268 such that the time window is long enough for the qubit to be thermalized and its off - diagonal elements of density matrix vanish . integrating eq . over @xmath190 yields the following solution of @xmath269 : @xmath270 the integral over time @xmath190 is understood as thermal activation processes with rate @xmath271 and integral in the exponent can be considered as contribution of relaxation processes after thermalization . for weak coupling @xmath248 and not very high temperatures @xmath272 , the integral in the exponential is a slow function of @xmath190 . therefore , we can replace the lower bound of the integration by @xmath140 . we obtain @xmath273 in the limit of low temperatures @xmath274 @xmath275 and in the limit of higher temperatures @xmath276 @xmath277 where @xmath278 is defined by eq . . the details of the derivation of the above equations are presented in appendix [ app : a ] . we remind that eqs . are valid when the rate equations are a good approximation to the br equations . in this case , the transition of the system to the excited state is a consequence of incoherent excitation by environment of the qubit , and is not the coherent phenomenon that leads to the excitation in the landau zener transition of an isolated quantum system . however , the excitation processes only happen when the adiabatic eigenstates of the qubit have a non - zero matrix elements with the coupling to environment , the latter happens when the `` control field '' @xmath279 is not parallel to the environment field which happens during time @xmath280 , when the excitation rate can be estimated as @xmath281 , resulting in the excitation probability @xmath282 , _ cf . _ to eq . as the level separation @xmath283 exceeds temperature , only relaxation process remains that causes transitions to the ground state . the effect of this relaxation is represented by the exponential factor in eqs . and , _ _ to eq . ( color online ) transition probability @xmath242 as a function of environment temperature @xmath284 , at different values of coupling between the qubit and the environment for @xmath285 . level - crossing speed @xmath203 , the high energy cutoff for the environment is @xmath286 and @xmath66 . we take @xmath287 . ] from the above analysis , we conclude that a finite temperature of the environment leads to the `` equilibration '' between the ground and excited states of the qubit , and as temperature increases , the probability of the transition to the excited state in the lz process increases monotonically , cf this behavior is demonstrated in fig . [ fig : pinfty_t ] , where @xmath242 is shown as a function of @xmath284 for several values @xmath20 of coupling between the qubit and its environment . we also note that the temperature effects appear at @xmath288 , at smaller @xmath284 , values of @xmath242 are characterized by the excitation through unitary evolution with the subsequent relaxation . when we consider @xmath242 as a function of coupling @xmath20 for several values of @xmath284 , we observe a more complicated behavior . for @xmath125 , shown by the solid line in fig . [ fig : pinfty_alpha ] , the transition probability @xmath242 monotonically decreases from its value @xmath289 , eq . , as @xmath20 increases , in agreement with eq . . at finite temperatures , @xmath242 increases for smaller values of @xmath20 , as the excitation process becomes more efficient and provides extra boost for transitions to the excited state in addition to that produced by unitary dynamics . however , this boost is only a linear function of @xmath20 , see eqs . and , and at stronger values of @xmath20 the exponential dependence of @xmath278 on @xmath20 results in decreasing @xmath242 as @xmath20 increases . ( color online ) transition probability @xmath242 as a function of the coupling parameter of the qubit and the environment , @xmath20 , at different environment temperatures for @xmath285 . level - crossing speed @xmath203 , the high energy cutoff for the environment is @xmath286 and @xmath66 . we take @xmath290 . ] as a function of environment temperature @xmath284 for @xmath291 , at different values of drive velocity . the high energy cutoff for the environment is @xmath286 and @xmath66 . the solid lines represent solutions of rate equations eq . . we take @xmath287 . ] we also consider the environment that produces fluctuating field along the direction of the control field , @xmath292 , in the landau zener problem . the decoherence rates in the br equations are given by [ eq : gammarlzlong ] @xmath293 , \\ \gamma_{e } & = \frac{\sin^2\eta } { 2}j(w(t ) ) n(w(t)),\\ \gamma_{2 } & = \frac{\gamma_{r}+\gamma_{e}}{2 } + j_0 \cos^{2}\eta.\end{aligned}\ ] ] for this configuration of coupling between the qubit and environment , the matrix elements for transitions between different eigenstates of the qubit caused by the environment are small and the qubit flip rates @xmath294 are proportional to @xmath295 and vanish fast for @xmath296 as @xmath297 . such fast decrease of the qubit flip rates in time simplifies either numerical or analytical integration of the br equation and makes @xmath242 independent from the high - energy cutoff @xmath17 . in particular , for finite temperatures , when the br equations can be reduced to the rate equations , time evolution of @xmath261 is given by eq . with @xmath298 . the general solution of the rate equation takes similar form to eq . : @xmath299 performing time integration in eq . gives for @xmath300 : @xmath301 for high temperatures , @xmath302 , we obtain ( see appendix b ) @xmath303.\ ] ] as we mentioned above , the results in eqs . and are independent from the cutoff energy @xmath17 . equation shows that @xmath273 vanishes in the low temperature limit , unless we take into account non - adiabatic unitary evolution of the quantum state in the lz problem . in the limit of high temperatures @xmath276 , but still weak coupling , @xmath304 , we obtain the linear dependence of @xmath273 on @xmath284 : @xmath305 which follows from eq . . since simple form of @xmath273 can not be obtained in the intermediate temperature regime , we numerically calculate the solution of rate equation as well as that of bloch - redfield equation for comparison , see fig . [ fig : pinfty_vs_t_long ] . when the level - crossing speed @xmath183 is small enough , the transition is mainly due to thermalization at short times and energy relaxation at longer times . in this regime , the rate and br equations are in a very good agreement , as demonstrated in fig . [ fig : pinfty_vs_t_long ] for @xmath306 . however , as the level crossing speed increases , the non - adiabatic unitary evolution also contributes to the transition to the excited state increasing the probability for a system to be in the excited state . since the non - adiabatic unitary evolution is not incorporated in the rate equations , the equations underestimate the probability of the excitation in the lz process , compare the solid and dashed curves in fig . [ fig : pinfty_vs_t_long ] for @xmath203 . we compare the results obtained from the br equations in the case of longitudinal coupling with the theory based on the lindblad equation for pure dephasing operators . for both problems , the qubit hamiltonian can be parametrized by the control field @xmath307 , where @xmath283 is the magnitude of the control field equal to the qubit level separation . the corresponding equation for the density matrix in the adiabatic basis has the form : [ eq : l - lz ] @xmath308+\frac{i\dot \theta}{2}[\sigma_y,\rho]+\frac{\gamma}{2 } ( \sigma_z\rho\sigma_z-\rho).\ ] ] in the component form the above equation is @xmath309 these equations are similar to eqs . , but because they are not written in the eigenstate basis , the last two equations contain extra terms . time derivatives of diagonal terms contain the off - diagonal terms of the density matrix multiplied by the quantity characterizing the off - diagonal part of the hamiltonian , @xmath310 . time derivatives of the off - diagonal components of the density matrix have the terms identical to those in eqs . and the extra terms characterized by the diagonal matrix elements and parameter @xmath310 . in this section we again consider the two cases : ( 1 ) the qubit rotation with a constant angular velocity @xmath311 , i.e. @xmath312 , and @xmath313 ; ( 2 ) the lz problem with @xmath314 and @xmath315 . when the control field rotates in @xmath90 plane , @xmath316 , the effective hamiltonian is time independent . to make a comparison with the calculation of br equations , one can look for a quasi - stationary state solution of the density matrix at time scale @xmath317 with ansatz that the off - diagonal elements are @xmath318 . we disregard @xmath319 terms for @xmath320 and take @xmath321 . then , we have @xmath322 , @xmath323 and the out of plane qubit projection is@xcite @xmath324 we argue , however , that the above expression does not hold for authentic steady state , @xmath325 , at longer times and for general configuration of the initial conditions . we present the result of numerical integration of the lindblad equations in fig . [ fig : my - l ] for @xmath112 and @xmath326 . in our calculation , we consider the case when the qubit is prepared in the ground state prior to rotation for @xmath70 . when the rotation starts , the hamiltonian acquires extra terms @xmath327 and the qubit exhibits a precession around new direction of the control field . this precession is reduced by the decoherence with rate @xmath328 and the oscillatory component in @xmath108 vanishes for times @xmath317 . at longer times , the diagonal matrix elements start changing as well and the system will eventually relax to @xmath329 and @xmath330 . the reason for this behavior is that at long times , the diagonal elements acquire significant changes even though these changes have small factor @xmath319 . in the language of the br equation , the lindblad pure dephasing operator contains relaxation and excitation components in the eigenstate basis of the transformed hamiltonian @xmath331 and @xmath332 , which is the high temperature limit because it does not distinguish processes with absorption or emission of environment excitations . correspondingly , the density matrix reaches the high - temperature limit with equal probabilities of occupation of eigenstates of the qubit hamiltonian @xmath333 this asymptotic behavior is consistent with the result obtained from the numerical solution of the lindblad equation , shown in fig . [ fig : my - l ] . ( color online ) polarization @xmath108 as a function of time @xmath190 for dephasing lindblad evolution . the decoherence rate @xmath326 and rotation velocity @xmath112 . after the rotation starts , polarization shows an oscillatory behavior originating from the qubit precession , at longer times the precession stops and the qubit relaxes to unpolarized state according to eq . . ] ( color online ) transition probability @xmath242 as a function of daphasing rate @xmath334 for different level crossing speeds @xmath335 . solid lines are numerical solution of the lindblad equation , eq . , and dashed lines are given by eq . . ] the expression for landau zener problem to the lowest order in @xmath183 can be obtained from the explicit form of the lindblad equation with @xmath283 given by eq . and @xmath336 . we assume that the changes in the system are slow and disregard @xmath337 and @xmath338 in eqs . . then we find @xmath339 $ ] and @xmath340^*$ ] . substituting these expressions to eq . , we obtain : @xmath341 \nonumber \\ & = \frac{1}{2}\left[1-\exp\left(-\frac{\pi v}{2\delta^2}r\left(\frac{\gamma}{\delta}\right)\right)\right],\end{aligned}\ ] ] where @xmath342 in the limit @xmath343 , we recover the result of ref . @xcite : @xmath344 at small decoherence rate and slow drive , @xmath345 , we take @xmath346 and reproduce the previous result , eq . , if we identify @xmath347 . the agreement between eqs . and has a simple interpretation . the lindblad equation can be viewed as the high temperature limit of the br equation for the ohmic environment@xcite . the lindblad equation is written in the basis that does not completely diagonalize the hamiltonian operator , and when we rewrite this equation in the basis diagonalizing matrix @xmath348 , we arrive to the collapse operators that represent transition processes between the eigenstates with equal excitation and relaxation rates @xmath349 . it is the excitation processes that cause transitions of the system to the excited state with the population of an excited state @xmath242 in accordance with eq . . to account for finite temperatures , the lindbladian operators are to be written in the eigenstate basis of the `` dressed '' hamiltonian , see ref . . large decoherence rate , @xmath350 , suppresses the off - diagonal elements of the density matrix , and effectively reduces the excitation and relaxation rates @xmath351 . as a result , the qubit is more likely to stay in its ground state without experiencing an excitation during the lz avoided level crossing . the maximum of @xmath352 is reached at @xmath353 . we compare eq . ( dashed lines ) with the result of numerical integration of the lindblad equation ( solid lines ) in fig . [ fig : l - lz ] . we observe that at stronger decoherence rate , when the off - diagonal unitary terms in the evolution of the density matrix can be neglected in comparison with the decoherence terms , @xmath354 in the lz problem , the two solutions are equivalent . in conclusion , we have presented a detailed analysis of the dynamics of an open quantum system in the presence of time - varying control field . specifically , we applied the bloch - redfield formalism to a spin-1/2 system whose hamiltonian varies slowly with time and investigated two problems . in the first problem , we studied the response of a qubit to a rotating control field of the qubit with a fixed magnitude . we noted that when the qubit basis is transformed to keep the effective hamiltonian in the diagonal form , which is required for proper perturbative analysis of the coupling between the qubit and its environment , the transformed hamiltonian acquires extra gauge terms . the gauge terms result in the modification of the qubit environment coupling and are related to the renormalization of the mass and friction terms due to changing parameters of the hamiltonian , cf . the exact form of the renormalization depends on a particular orientation of the control field with respect to the fluctuating environment field . we have illustrated this scenario by considering different orientations of the environment field : ( 1 ) control field and fluctuations are always perpendicular to each other , and the corresponding relaxation rates are time - independent ; ( 2 ) control and fluctuation fields are parallel only at some moments of time , in which case the relaxation rates significantly oscillate in time ; ( 3 ) fluctuations are always along the direction of the control field , then the relaxation rates are small in the parameter given by the ratio of the rotation velocity and level separation . our analysis offers a clear evidence of robustness of topological features against external noises . to see this one needs to consider a long time limit where the qubit density matrix reaches a steady state solution that at zero temperature coincides with the ground state of the effective hamiltonian . when this ground - state qubit configuration is looked at in the original laboratory basis , the qubit has a constant projection in the direction perpendicular to the plane of rotation and the magnitude of the projection is proportional to the product of rotation velocity of the control field and the berry curvature of the qubit ground state . in the long time limit , this response is unaffected by the environmental coupling field , at least for zero temperature environment . this relation of the response at long times and the berry curvature can be utilized as a practical method for measurements of the chern number@xcite of a quantum system . we also considered an environment with a very sharp spectral function . we represent this environment by a quantum harmonic oscillator that has internal relaxation . in this case we solve the lindblad master equation for the system of coupled qubit and oscillator and find that the results are qualitatively similar to the solution of the br equation with properly chosen relaxation rates . in the second example , we revisited the landau zener problem . in this case , the modification of the matrix elements for transitions between eigenstates of the qubit hamiltonian is essential , even though it was not always taken into account.@xcite the eigenstate basis that is necessary to use in treatment of interaction of the qubit with its environment is also convenient for numerical evaluation because in this basis the system behavior during the landau zener level crossing is represented by a smooth function that quickly reaches its long - time asymptotic value . for a qubit weakly coupled to the environment , the evolution , long after the level crossing , reduces to suppression of the off - diagonal elements of the density matrix and relaxation of the excited state to the ground state , the latter is accurately described by the rate equations . for the fluctuating field along the asymptotic direction of the control field , the relaxation rate decreases as the level separation increases due to suppression of the matrix elements of qubit transition between eigenstates caused by the environment . however , this suppression is not sufficient to cut the relaxation in the long time limit , and the relaxation results in a power law decay of the excited state , until the separation between the qubit states exceeds the ultra - violet cutoff of the environment . at finite temperature , in addition to enhancement of decoherence rates for the qubit , the excitation processes produce transitions from the ground to the excited qubit states , eventually increasing the probability for the qubit to appear in the excited state after the transition . the br equations accurately describe the crossover for the landau zener transition in an isolated quantum system , eq . , with unitary evolution , to the open system at arbitrary temperature , see sec . [ sec4 ] . furthermore , we compare the results obtained from the generalized br equations with that from the lindblad master equation . in particular , we focused on the case of pure dephasing lindblad superoperators,@xcite that are equivalent to the longitudinal coupling of the environment ( fluctuating field of the environment is along the control field ) . we found that the two results are consistent in the high temperature limit , when the lindblad and br equations are equivalent , but application of the lindblad equation for a system coupled to low temperature environment may result in unphysical solutions . finally , we note that the generalization of the bloch redfield equations can be applied to accurately evaluate the fidelity of quantum gates . by taking into account proper modification of the transition and dephasing rates caused by time - varying parameters in the hamiltonian , optimization techniques for gate operations can be further improved . similarly , the br equations for time - dependent hamiltonian are also required for accurate description of protocols for adiabatic quantum computing and the berry phase measurement in recent experiments . @xcite we thank i. aleiner , a. glaudell , f. nori , a. polkovnikov , s. shevchenko and a. levchenko for fruitful discussions . the work was supported by nsf grants no . dmr-1105178 and dmr-0955500 , aro and lps grant no . w911nf-11 - 1 - 0030 . here we evaluate the integral in eq . . notice that while the integral over @xmath355 in the exponent , @xmath356 originates on long interval from @xmath357 to @xmath358 , the second integral converges for time @xmath359 , for not very large temperatures , we can replace the low limit of integration in eq . by @xmath140 . in this case , we have @xmath360 where @xmath361 , @xmath362 , @xmath363 with @xmath364 and @xmath365 . first , let us change the integration variable @xmath366 such that @xmath367 and the integral in the exponential then reads @xmath368 this integral can be evaluated in two cases . first , we consider the low temperature limit @xmath369 , in which the hyperbolic cotangent @xmath370 . therefore , the integral is obtained @xmath371,\end{aligned}\ ] ] where @xmath372 is the 0th order modified bessel function of the second kind with the following asymptotes : @xmath373 for @xmath374and @xmath375 for @xmath376 , @xmath377 is the euler constant . as the result , for @xmath274 , we have @xmath378.\ ] ] the first term can be disregarded for @xmath274 . at higher temperatures , there is a stronger contribution to @xmath379 originating from short time interval @xmath359 . we can estimate this contribution as @xmath380 we emphasize that this is the contribution which we do not evaluate correctly when replace eq . by eq . . therefore , we can treat the above expression for @xmath381 as the boundary of applicability of our approximation , indicating that transition from eq . to is justified not for very high temperatures , such that @xmath382 . next , we evaluate the integral @xmath383 as before , we first consider the low temperature limit , @xmath274 , in which we approximate @xmath384\simeq \exp(-s / t)$ ] . then the integral becomes @xmath385 in the high temperature limit , we utilize @xmath384\simeq t / s$ ] , and we obtain @xmath386 this equation is valid for high temperature limit @xmath276 , provided that our substitution of eq . by is justified , or @xmath272 . to sum up , we evaluated @xmath273 in the limits of low and moderately high temperatures . the results are presented by eqs . and . for the longitudinal coupling , the transition probability @xmath242 in limit of low temperatures @xmath387 can be evaluated similarly to the calculations in appendix a. we replace eq . , where the integral over time @xmath190 converges fast for @xmath359 , by the following expression @xmath388 where in the last integral we take the lower limit of integration to zero and @xmath389 . in the above expression , @xmath361 , @xmath362 , @xmath363 with @xmath390 and @xmath365 . similarly , let us change the integration variable @xmath366 such that @xmath367 . the integral @xmath391 then reads @xmath392 we note that this integral converges fast and the high - energy cutoff of the environment can be omitted . similarly , the integral over @xmath393 can be rewritten as @xmath394 in the high temperature limit , we follow a different approach . we assume that the environment is at high temperature and the relaxation rates are enhanced by factor @xmath395 . in this case , we also have a fast convergence of integrals @xmath396 at @xmath397 and for @xmath276 , we can simplify the rate equation to @xmath398 this equation can be integrated to find @xmath262 with initial condition @xmath399 , and used to define @xmath400 : @xmath401 for @xmath343 , we obtain @xmath402 arriving to eq . .
|
we study the dynamics of a two - level system described by a slowly varying hamiltonian and weakly coupled to the ohmic environment .
we follow the bloch
redfield perturbative approach to include the effect of the environment on qubit evolution and take into account modification of the spectrum and matrix elements of qubit transitions due to time - dependence of the hamiltonian .
we apply this formalism to two problems .
( 1 ) we consider a qubit , or a spin-1/2 , in a rotating magnetic field .
we show that once the rotation starts , the spin has a component perpendicular to the rotation plane of the field that initially wiggles and eventually settles to the value proportional to the product of angular rotation velocity of the field and the berry curvature .
( 2 ) we re - examine the landau
zener transition for a system coupled to environment at arbitrary temperature .
we show that as temperature increases , the thermal excitation and relaxation become leading processes responsible for transition between states of the system .
we also apply the lindblad master equations to these two problems and compare results with those obtained from the bloch redfield equations .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
non equilibrium phenomena are on the agenda in modern statistical physics , soft condensed matter and biophysics . open systems driven far from equilibrium are ubiquitous . a classical case is driven navier stokes turbulence , other cases are driven lattice gases , growing interfaces , growing fractals , etc . unlike equilibrium physics where the boltzmann - gibbs scheme applies , the ensemble is not known in non equilibrium . here the problem is defined in terms of a numerical algorithm , a master equation , or a langevin equation . an interesting class of non equilibrium systems exhibit scale invariance . one example is diffusion limited aggregation ( dla ) driven by the accretion of random walkers yielding a growing scale invariant fractal with dimension @xmath0 in 2d . another case is a growing interface driven by random deposition or propagating in a random environment . here the width of the growing front @xmath1 conforms to the dynamical scaling hypothesis @xmath2 , where @xmath3 is the size of the system and @xmath4 and @xmath5 scaling exponents ; @xmath4 characterizing the roughness and @xmath5 describing the dynamical crossover to the stationary profile @xcite . in the present paper we focus on the kardar - parisi- zhang ( kpz ) equation which describes an intrinsic non equilibrium problem and plays the same role as the ginzburg - landau functional in equilibrium physics . the kpz equation was introduced in 1986 in a seminal paper by kardar , parisi and zhang @xcite . it has the form @xmath6 and purports to describe non equilibrium aspects of a growing interface ; see refs . @xcite . here @xmath7 is the height of an interface at position @xmath8 and time @xmath9 , the linear diffusion term @xmath10 , characterized by the diffusion coefficient @xmath11 , represents a surface tension , the nonlinear growth term @xmath12 , characterized by @xmath13 , is required to account for the lateral growth , @xmath14 is an imposed constant drift , and the random aspects , i.e. , the random deposition of material or the random character of the medium , are encoded in the noise @xmath15 . the noise is assumed to be locally correlated in space , and time , its strength characterized by @xmath16 . despite its simple form the kpz equation is difficult to analyze and many aspects remain poorly understood @xcite . apart from its intrinsic interest the kpz equation is also related to fundamental issues in _ turbulence _ and _ disorder_. introducing the local slope field @xmath17 the kpz equation takes the form of a burgers equation driven by conserved noise , @xmath18 in the noiseless case setting @xmath19 and regarding @xmath20 as a velocity field , the non linear term appears as the convective term in the navier - stokes equation and eq . ( [ burgers ] ) has been used to model aspects of _ turbulence_. in the 1d case the relaxation of the velocity field takes place subject to a transient pattern formation composed of domain walls and ramps with superimposed diffusive modes . in the driven case ( [ burgers ] ) was studied earlier in the context of long time tails in hydrodynamics @xcite . on the other hand , applying the non linear cole - hopf transformation the kpz equation maps to the cole - hopf equation ( ch ) @xmath21 a linear diffusion equation driven by multiplicative noise . the ch equation has a formal path integral solution @xcite which can be interpreted as an equilibrium system of directed polymers ( dp ) with line tension @xmath22 in a quenched random potential @xmath23 ; a model system in the theory of _ disorder _ which has been studied using replica techniques @xcite . the kpz equation lives at a critical point and conforms to the dynamical scaling hypothesis . for the height correlations we have @xmath24 here @xmath4 , @xmath5 , and @xmath14 are the roughness exponent , dynamic exponent , and scaling function , respectively . to extract scaling properties the initial analysis of the kpz equation was based on the dynamic renormalization group ( drg ) method , previously applied to dynamical critical phenomena and noise - driven hydrodynamics @xcite . an expansion in powers of @xmath13 in combination with a momentum shell integration yield to leading order in @xmath25 the drg equation @xmath26 , with beta - function @xmath27 . here @xmath28 is the effective coupling strength and @xmath29 the logarithmic scale parameter @xcite . the emerging drg phase diagram is depicted in fig . [ fig1 ] . . in @xmath30 the drg flow is towards the strong coupling fixed point @xmath31 . above the lower critical dimension @xmath32 there is an unstable kinetic transition line , separating a rough phase from a smooth phase . ] before discussing the phase diagram we note two further properties of the kpz equation . first , subject to a galilean transformation the equation is invariant provided we add a constant slope to @xmath33 and adjust the drift @xmath14 , i.e. , @xmath34 note that the slope field @xmath20 and the diffusive field @xmath35 transform like @xmath36 and @xmath37 $ ] , respectively . the galilean invariance implies the scaling law @xmath38 relating @xmath4 and @xmath5 @xcite ; the galilean invariance is a fundamental dynamical symmetry specific to the kpz equation , delimiting the universality class . second , a fluctuation - dissipation theorem is operational in 1d since the stationary fokker - planck equation admits the explicit solution @xcite @xmath39 . \label{stat}\end{aligned}\ ] ] this distribution shows that the slope @xmath40 fluctuations are uncorrelated and that the height field @xmath41 performs a random walk . consequently , from eq . ( [ corr ] ) we infer the roughness exponent @xmath42 and from the scaling law ( [ scal ] ) the dynamic exponent @xmath43 . in other words , the scaling exponents associated with the strong coupling fixed point are exactly known in 1d , see fig . [ fig1 ] ; moreover , results for the scaling function can be obtained by loop expansions @xcite . the lower critical dimension is @xmath32 . below @xmath32 the drg flow is towards the strong coupling fixed point . above @xmath32 the drg equation yields an unstable kinetic phase transition line as indicated in fig . [ fig1 ] . for @xmath13 below a critical coupling strength @xmath44 the drg flow is towards @xmath45 , corresponding to the linear edwards - wilkinson ( ew ) equation , the kpz equation for @xmath46 , yielding @xmath47 and @xmath48 , note that the scaling law is not operational for @xmath45 ; for @xmath49 the drg flow is towards a non perturbative strong coupling fixed point . a drg analysis to all order in @xmath25 yields @xmath47 and @xmath50 on the transition line and a singularity at the upper critical dimension @xmath51 . mode coupling techniques also give @xmath51 , whereas a directed polymer analysis yields @xmath52 @xcite . in order to disentangle the properties of the kpz equation , both regarding scaling and otherwise , there is a need for alternative methods . in the present paper we summarize a non perturbative weak noise approach which we have pursued in recent years @xcite . the working hypothesis here is to focus on the noise strength @xmath16 as the determining parameter rather that the non linearity @xmath13 and ensuing drg analysis or mapping to dp . for @xmath53 the interface relaxes subject to a transient pattern formation ; for @xmath54 the interfaces initially decays but is eventually driven into a stationary fluctuating state ; the crossover time diverging for @xmath55 . in the next section we summarize the scheme that will allow us to access the weak noise regime in a non perturbative fashion . the weak noise scheme takes as its starting point a generic langevin equation @xmath56 determining the problem we wish to analyze . here @xmath57 is a multi - dimensional random variable , @xmath58 a general non linear drift , and @xmath23 additive white noise correlated with strength @xmath16 . in order to implement a weak noise approximation we consider the equivalent fokker - planck equation for the distribution @xmath59 @xmath60 interpreting @xmath61 as a momentum operator , @xmath62 as an effective wave function , and @xmath16 as an effective planck constant , eq . ( [ fp ] ) has the form of an imaginary time schrdinger equation and it is natural to introduce the well - known wkb or eikonal approximation @xmath63 . \label{wkb}\end{aligned}\ ] ] to leading order in @xmath16 the action then obeys a principle of least action @xmath64 as expressed by the hamilton - jacobi equation @xmath65 with canonical momentum @xmath66 . the hamiltonian ( energy ) takes the the form @xmath67 , \label{ham}\end{aligned}\ ] ] yielding the coupled equations of motion @xmath68 the action associated with an orbit from @xmath69 to @xmath57 in time @xmath70 is @xmath71= \frac{1}{2}\int_{x_1,0}^{x , t}dt p(t)^2 . \label{act}\end{aligned}\ ] ] the weak noise recipe is clear . we solve the equations of motion ( [ eqm ] ) and identify an orbit from @xmath69 to @xmath57 in time @xmath70 with @xmath72 as an adjusted variable . the orbits lie on constant @xmath73 manifolds . evaluating the action @xmath74 for a specific orbit the wkb approximation ( [ wkb ] ) yields the transition probability from @xmath69 to @xmath57 in time @xmath70 . assuming @xmath75 for small @xmath57 the phase space has the generic structure depicted in fig . [ fig2 ] . to the final @xmath57 in transition time @xmath70 lies on the energy manifold @xmath76 . in the long time limit the orbit migrates to the zero - energy manifold composed of a transient submanifold for @xmath77 , corresponding to the noiseless case , and a stationary manifold for @xmath78 , corresponding to the noisy case . the submanifolds intersect in the saddle point ( sp ) , determining the markovian behavior . ] the present variationally based weak noise scheme , dates back to onsager . in more recent formulations it corresponds to the saddlepoint contribution ( optimal path ) in the functional martin - siggia - rose scheme , the weak noise scheme applies directly to the kpz equation in the cole - hopf formulation ( [ cheq ] ) . extending the scheme in order to incorporate multiplicative noise , @xcite , the wkb scheme yields the equations of motion , action , and distribution @xmath79+k_0 ^ 2w^2p,~~ \frac{\partial p}{\partial t}=-\nu[\nabla^2 w - k^2w]-k_0 ^ 2pw^2 , \label{eqmo } \\ & & s(w , t)=(k_0 ^ 2/2)\int^{w , t}d \vec r dt(wp)^2,~~p(w , t)\propto\exp[-s(w , t)/\delta],\end{aligned}\ ] ] with parameters @xmath80 and @xmath81 . on the transient and stationary manifolds @xmath77 and @xmath82 the equations of motion reduce in the static case to the diffusion and non linear schrdinger equations @xmath83 admitting localized spherically symmetric solutions @xmath84 and @xmath85 for large @xmath86 , respectively . in terms of the height field @xmath87 and slope field @xmath17 we obtain the fundamental localized static growth modes for large @xmath86 @xmath88 the growth modes are characterized by the amplitude or charge @xmath89 , determined by the imposed drift @xmath14 in the kpz equation . the growth mode with positive charge @xmath90 , @xmath91 lives on the transient noiseless @xmath77 manifold and corresponds to a cone or dip in @xmath33 and a constant positive monopole in @xmath20 . the mode carries zero action , zero energy @xmath92=0 , and zero momentum @xmath93 ; the distribution @xmath94 associated with the mode is of @xmath95 . the growth mode with negative charge @xmath96 , @xmath97 is associated with the stationary noisy manifold @xmath82 and corresponds to an inverted cone or tip in @xmath33 and a constant negative monopole in @xmath20 . this modes carries a finite action @xmath98 yielding the distribution @xmath99 $ ] . in the 1d case the growth modes correspond to right hand and left hand domain walls , @xmath100 . the right hand domain wall is the viscosity broadened shock wave in the deterministic burgers equation ; the left hand domain wall is noise induced , in the present scheme characterized by a finite @xmath72 , see refs . @xcite . in fig . [ fig3 ] we have depicted the static growth modes in the height and slope fields . applying the galilean transformation ( [ gal ] ) we can boost the static modes and obtain the propagating growth modes @xmath101 the moving localized growth modes are the fundamental elementary excitations incorporating the non linear aspects of the kpz equation . in a quantum field theory context the growth modes correspond to instantons or solitons . by means of the propagating localized growth modes ( [ gm ] ) we construct a global solution of the field equations ( [ eqmo ] ) . the galilean invariance ( [ gal ] ) determines the matching of the modes and we obtain the dilute network solution @xmath102 here @xmath103 is the assignment of charges and @xmath104 the initial positions . this network construction correspond to a multi - instanton solution in quantum field theory . as time evolves the modes propagate and the velocities adjust to constant values given by the self - consistent equation @xmath105 superimposed on the propagating network of growth modes is a spectrum of linear extended diffusive modes with dispersion @xmath106 , following from a linear analysis of the field equations ( [ eqmo ] ) . for large @xmath8 the slope field @xmath107 . in order to ensure the boundary condition of a flat interface at large distances we impose the neutrality condition @xmath108 ; note that this boundary condition still allows for a local offset of @xmath33 and the propagation of facets or steps . during time evolution the dynamical network propagates across the system . imposing periodic or bouncing boundary conditions increments are added to @xmath33 and the interface grows , see also ref . @xcite . from an analysis of eqs . ( [ net1 ] ) and ( [ net2 ] ) it follows that the growth modes with positive charge exerts an attraction of the other modes , whereas the negatively charged growth modes repel the other modes . although our analysis so far only applies to a dilute network , it follows tentatively from a numerical simulations of eqs . ( [ net1 ] ) and ( [ net2 ] ) that the modes form dipoles and that the long time stable network configuration is composed of a gas of propagating dipoles , i.e. the pairing of monopole growth modes with opposite charges . in terms of the slope field a dipole mode with charges @xmath109 and @xmath110 has the form @xmath111 $ ] , @xmath112 . the mode propagate with velocity @xmath113 and carries energy , momentum and action , @xmath114 , @xmath115 , and @xmath116 . in terms of the height field @xmath117 $ ] the mode in depicted in fig . asymptotically the height field is flat corresponding to a vanishing slope field . the dipole mode corresponds to a propagating local defect or deformation . in 1d the dipole mode corresponds in the slope field to a matched right hand and left hand domain wall propagating across the system , see fig . the scaling issues for the kpz equation remain unsettled except in 1d where the fluctuation - dissipation theorem ( [ stat ] ) and the scaling law ( [ scal ] ) give access to the strong coupling fixed point with scaling exponents @xmath42 and @xmath118 , see fig . [ fig1 ] . in higher d there has been many attempts to access the strong coupling fixed point both on the basis of drg , mode coupling , directed polymers ( dp ) , and numerically ; however , the strong coupling features remain elusive @xcite . the present weak noise method is not a scaling approach but rather a many body description of a growing interface . nevertheless , the method allows a discussion of some of the scaling features . since the scheme is consistently galilean invariant the scaling law ( [ scal ] ) is automatically obeyed . the roughness exponent @xmath4 is associated with the static correlations @xmath119 and requires the static distribution @xmath120 , @xmath121 $ ] , @xmath122 . the stationary distribution is associated with the zero - energy stationary manifold which in general is difficult to identify for a system with many degrees of freedom . in 1d in terms of the slope field @xmath40 and associated noise field @xmath72 the hamiltonian takes the form @xmath123 $ ] and it follows that the stationary zero - energy manifold is given by @xmath124 , yielding a hamiltonian density as a total differential . correspondingly , the stationary action @xmath125 and the distribution @xmath126 $ ] in accordance with ( [ stat ] ) , implying @xmath42 . since galilean invariance is built in the exponent @xmath43 follows automatically . however , we can also infer @xmath43 from an independent argument based on the dispersion law for the low lying gapless excitations . the elementary excitations are the right hand and left hand domain walls . a composite quasi particle or dipole mode satisfying the boundary condition of vanishing slope can be constructed by paring two domain walls . the dipole mode propagates with energy @xmath127 and momentum @xmath128 , where @xmath129 is the dipole amplitude . using the analogy between the stochastic formulation and quantum mechanics , i.e. , the canonical quantization of the weak noise scheme with @xmath16 as an effective planck constant , and using the spectral representation @xmath130 , @xmath131 is a form factor , we note that the bottom of the gapless dipole energy spectrum @xmath132 implies @xmath133 , where @xmath43 . in higher @xmath134 we have not been able to identify the stationary zero - energy manifold and the exponent @xmath4 . also , in order to determine @xmath5 , for example from the form of the time dependent correlations @xmath135 , we need both the transition probabilities and the stationary distribution , i.e. , by definition @xmath136 . in the weak noise scheme @xmath137 $ ] which requires a detailed analysis of the dynamical network and the associated action . in the dipole sector we can , however , present some preliminary scaling results . since the propagating dipole mode according to ( [ actgm ] ) carries action @xmath138 and propagates with velocity @xmath139 the mode moves the distance @xmath140 in time @xmath70 . expressing @xmath74 in the form @xmath141 we infer the single dipole distribution @xmath142 $ ] and , correspondingly , the dipole mean square displacement @xmath143 with hurst exponent @xmath144 ; note that the dynamic exponent @xmath145 . in the stochastic representation the dipole mode thus performs anomalous diffusion . in @xmath146 we have @xmath147 , @xmath148 , in agreement with a formal dp result @xcite . in @xmath30 we obtain @xmath149 and the exact result @xmath43 , i.e. , the dipole modes exhaust the spectrum . in @xmath32 we have @xmath150 and @xmath47 , i.e. , ordinary diffusion . in @xmath151 we obtain @xmath152 and @xmath153 , the dipole mean square displacement falls off logarithmically @xmath154 , however , @xmath5 diverges at variance with accepted drg and dp results . in @xmath155 we have @xmath156 and the mean square displacement is arrested . below @xmath32 ( the lower critical dimension ) the dipole modes superdiffuse , above @xmath32 we have subdiffusion . these scaling results only refer to the dipole sector . the last issue is the much discussed upper critical dimension for the kpz equation @xcite . the weak noise approach allows a non scaling argument for the existence of a critical dimension . above @xmath155 the negative growth mode as a bound state solution to the non linear schrdinger equation ( [ nse ] ) ceases to exist this implies that the dynamical network representation of a growing interface ceases to be valid . this result follows from a numerical analysis of the non linear schrdinger equations ( [ nse ] ) but can also be inferred by an algebraic proof based on derrick s theorem . first , introducing @xmath157 , @xmath158 , and @xmath159 we infer from ( [ nse ] ) the identity @xmath160 . second , deducing ( [ nse ] ) from a variational principle @xmath161 with @xmath162 and performing a constrained minimization @xmath163 , @xmath164 , @xmath165 , @xmath166 and @xmath167 we have the second identity @xmath168 . finally , requiring @xmath169 the identities imply @xmath170 , q.e.d . in this paper we have presented a short review of a recently developed asymptotic weak noise approach to the kardar - parisi - zhang equation . the scheme provides a many body description of a growing interface in terms of a dynamical network of growth modes . the growth modes are the elementary building blocks and their propagation accounts for the kinetic growth . kinetic transitions are determined by an associated dynamical action , replacing the customary free energy landscape . superimposed on the network is a gas of diffusive modes . in 1d the dispersion laws delimit the universality classes : in the kpz case the gapless domain wall modes yield @xmath43 , the diffusive modes being subdominant ; in the ew case the domain walls are absent and the gapless diffusive modes yield @xmath47 . in higher d the scaling results based on the weak noise method are still subject to scrutiny . finally , we mention that the weak noise method has also been applied to the noise - driven ginzburg - landau equation , a finite - time - singularity model , and dna bubble dynamics @xcite . 99 a. -l . barabasi and h. e. stanley , _ fractal concepts in surface growth _ ( cambridge university press , 1995 ) ; j. krug and h. spohn , _ solids far from equilibrium ; kinetic roughening of growing surfaces : fractal concepts in surface growth _ ( cambridge university press , 1992 ) ; j. krug , adv . 46 * , 139 ( 1997 ) m. kardar , g. parisi and y. c. zhang , phys . lett . * 56 * , 889 ( 1986 ) ; e. medina , t. hwa , m. kardar and y. c. zhang , phys . a * 39 * , 3053 ( 1989 ) t. halpin - healy and y. c. zhang , phys . 254 * , 215 ( 1995 ) d. forster , d. r. nelson and m. j. stephen , phys . * 36 * , 867 ( 1976 ) ; phys . a bf 16 , 732 ( 1977 ) m. kardar and y. c. zhang , phys . 58 * , 2087 ( 1987 ) ; m. kardar , nucl . b bf 290 , 582 ( 1987 ) h. c. fogedby , phys . lett . * 94*,195702 ( 2005 ) ; phys . e * 73 * , 031104 ( 2006 ) ; phys . e * 68 * , 026132 ( 2003 ) ; phys . e59 , 5065 ( 1999 ) ; phys . e57 , 49431 ( 1998 ) ; phys . 80 , 1126 ( 1998 ) l. onsager and s. machlup , phys . rev . * 91 * , 1505 ( 1953 ) , ibid 1512 ; p c. martin , e. d. siggia and h. a. rose , phys . rev . a * 8 * , 423 ( 1973 ) ; r. baussch , h. k. janssen and h. wagner , z. phys . b * 24 * , 113 ( 1976 ) d. a. huse , c. l. henley and d. s. fisher , phys . lett . * 55 * , 2924 ( 1985 ) ; e. frey and u. c. tuber , phys . e * 50 * , 1024 ( 1994 ) ; e. frey , u. c. tuber and t. hwa , phys . e * 53 * , 4424 ( 1996 ) ; k. j. wiese , j. stat . phys . * 93 * , 143 ( 1998 ) ; f. colaiori and m. a. moore , phys . lett . * 86 * , 3946 ( 2001 ) ; m. lssig , phys . rev . lett . * 80 * , 2366 ( 1998 ) ; nucl . b * 448 * , 559 ( 1995 ) ; m. lssig and h. kinzelbach , phys . lett . * 78 * , 903 ( 1997 ) ; p. le doussal and k.j . wiese , phys . e * 72 * , 035101 ( 2005 ) h. c. fogedby , j. hertz and a. svane , europhys . lett . * 62 * , 795 ( 2003 ) ; h. c. fogedby and v. poutkaradze , phys . e * 66 * , 021103 ( 2002 ) ; hans c. fogedby and ralf metzler , phys . lett . * 98 * , 070601 ( 2007 ) .
|
we review a recent asymptotic weak noise approach to the kardar - parisi - zhang equation for the kinetic growth of an interface in higher dimensions .
the weak noise approach provides a many body picture of a growing interface in terms of a network of localized growth modes . scaling in 1d
is associated with a gapless domain wall mode .
the method also provides an independent argument for the existence of an upper critical dimension .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the laughlin wave function @xcite is a paradigm as a variational wave function @xcite . its construction is simple , yet its implication is rich . it not only explains the most robust fractional quantum hall effects @xcite , but also embodies many exotic notions like fractionally charged excitations , fractional statistics , and composite - fermions / bosons @xcite . its simplicity and fertileness make it very intriguing , and significant efforts have been dedicated to its nature @xcite . besides the conventional way of studying correlation functions , a more recent approach is studying the entanglement in the wave function @xcite . in ref . @xcite , bipartite entanglement entropy in the laughlin wave function has been studied both analytically and numerically . a noteworthy result is that , by studying the scaling behavior of the entanglement entropy , the topological entropy corresponding to the so - called `` total quantum dimension '' @xcite is extracted , confirming the fact that the laughlin wave function is a topologically ordered state . however , a most natural way of quantifying the entanglement in the laughlin wave function has not yet been explored so far . it is well known that unlike the integer quantum hall effect , for the fractional quantum hall effect , the coulomb interaction between the electrons is essential . for a non - interacting fermionic system , the multi - particle eigenstates are very simple they are slater determinants @xcite , the simplest kind of fermionic wave function satisfying the antisymmetry condition . for a generic interacting fermionic system , the interaction mixes up the slater determinants and a generic eigenstate @xmath2 can no longer be written in the form of a slater determinant @xcite , no matter how the single - particle orbitals are chosen . specifically , as we shall prove below , the laughlin wave function can not be written as a single slater determinant . now , from the point of view of approximation theory @xcite , a natural question is , how close can it be approximated by a slater determinant ? this question leads to the notion of the optimal slater approximation of a fermionic wave function @xcite , namely , the slater determinant which maximize the overlap @xmath3 where @xmath4 denotes a slater determinant . denote the maximal value of @xmath5 as @xmath6 . it takes values from the interval @xmath7 $ ] . apparently , the larger @xmath6 is , the closer the original wave function @xmath2 is to a slater determinant , and it is reasonable to say that the less the fermions are entangled with each other . in particular , if @xmath8 , the wave function is a slater determinant and the entanglement is zero . therefore , a measure of entanglement can be defined as @xmath9 this is a geometric measure of entanglement in the sense that it involves only the notion of inner product between two states in a hilbert space . it is a natural fermionic generalization of the geometric measure of entanglement for spin systems @xcite , or more precisely , composite systems whose hilbert space is the tensor product of that of its components . the point is that , for spin systems , a generic wave function has no symmetry and one uses separable states to approximate it ; while here for a system of identical fermions , the target wave function is always antisymmetric , and hence the simpler wave function used to approximate it should also be antisymmetric . we thus need to antisymmetrize a separable state and in doing so we get a slater determinant if not zero . we note that while the geometric entanglement of spin systems have been extensively studied ( e.g. , see refs . @xcite and references therein ) , it is much less discussed for identical particles , although some works on symmetric spin states @xcite can be translated into the bosonic language too . the reason might be that the extra symmetry or antisymmetry constraint makes the optimization problem seemingly more complicated @xcite . despite this difficulty , however , recently some progress has been made on the fermionic case @xcite . in particular , in ref . @xcite , an efficient algorithm to construct the optimal slater approximation of an arbitrary fermionic wave function was brought up . it is this algorithm that enables us to study the geometric entanglement in the laughlin wave function . before proceeding to do so , we note that the advantage of the geometric entanglement to bipartite entanglements is that , it is intrinsic , because it involves no partition , neither of the space nor of the particles . by avoiding partitioning the space ( the modes ) , it is representation - independent ; by avoiding partitioning the particles into two groups , it treats identical particles identically , i.e. , on an equal footing . in contrast , the schmidt decomposition based bipartite entanglement depends on the partition of the system , which is inevitably arbitrary . the algorithm we will utilize was already detailed in ref . @xcite . here we sketch it briefly for the sake of completeness , and tailor it somehow for the specific problem . the laughlin wave function of @xmath10 electrons , in the dimensionless form , is ( @xmath11 ) @xmath12 here @xmath13 is a positive odd integer and @xmath14 is a normalization factor . only the @xmath15 states are of interest the @xmath16 state is simply a slater determinant @xcite . we shall focus on the @xmath17 and @xmath18 states . this is on the one hand due to the limitation of numerical resources and on the other hand because of the fact that the states with higher values of @xmath19 are less good as variational wave functions @xcite . by construction , each particle in the laughlin wave function ( [ lwf ] ) is in the lowest landau level . more specifically , @xmath20 here @xmath21 is an @xmath22-tuple with @xmath23 . in the second line , we have introduced the single - particle orbitals ( @xmath24 ) @xmath25 which are an orthonormal basis of the lowest landau level . by ( [ lwf2 ] ) , we see that @xmath26 is supported by the @xmath27-dimensional single - particle hilbert space of @xmath28 actually , it is readily seen that these first @xmath29 @xmath30 s are also the natural orbitals @xcite of @xmath31 and each of them has a nonzero population . because an @xmath22-particle slater - determinant wave function has exactly @xmath22 populated natural orbitals ( each of population 1 ) , we see that the laughlin wave function with @xmath32 is not a slater determinant . the fact that @xmath26 is supported by a finite dimensional single - particle hilbert space greatly facilitates the numerical calculation . the @xmath22-electron hilbert space is spanned by the @xmath33 slater determinants @xmath34 with @xmath35 being an @xmath22-tuple . the laughlin wave function can be expanded in terms of these slater determinants as @xmath36 this expansion ( to determine the coefficients @xmath37 ) is highly nontrivial . but fortunately , some efficient algorithms , which we employ here , have been given in ref . @xcite . a generic slater determinant is supported by an @xmath22-dimensional subspace @xmath38 of @xmath39 . suppose @xmath40 is an orthonormal basis of @xmath41 . they are linearly related to the orthonormal basis @xmath42 of @xmath43 by a @xmath44 matrix @xmath45 , i.e. , @xmath46 the condition that @xmath47 be orthonormal requires that @xmath48 the @xmath49 identity matrix . the slater determinant constructed out of @xmath47 is @xmath50 choosing another basis of @xmath41 will yield the same wave function up to a global phase . the strategy to maximize the modulus of the overlap between @xmath51 and the laughlin wave function @xmath26 is very simple . we start from some initial orbitals @xmath47 chosen randomly , and then fix all but one of them and try to optimize it with respect to the rest orbitals . for example , let us fix orbitals @xmath52 and try to find an optimal @xmath53 with respect to them . we note that the inner product between @xmath4 and @xmath26 can be written as ( @xmath54 ) @xmath55 here from the first line to the second line , we have used the fact that the wave function @xmath56 is antisymmetric and thus the @xmath57 terms summing over @xmath58 are all equal actually . from the second line to the third line , the integrals over @xmath59 were performed , and we have defined a single - particle state @xmath60 equation ( [ inner1 ] ) is in the form of the inner product of @xmath53 and @xmath61 . by the cauchy - schwarz inequality , we immediately see that the optimal @xmath62 should be proportional to @xmath63 . here an accidental and fortunate fact is that , @xmath61 is orthogonal to all @xmath64 because of the antisymmetry property of @xmath26 . hence , when we update @xmath62 , the orthonormal condition ( [ mm ] ) between the @xmath65 s is still maintained . the problem is then reduced to calculating @xmath61 . using ( [ expansion ] ) and ( [ gvsf ] ) , it is straightforward to get the compact formula @xmath66 we can then calculate @xmath63 using the laplace expansion @xcite for the determinant , and update the orbital @xmath53 , or more specifically , update the first column of @xmath67 . by doing so , the overlap @xmath5 increases . now the point is that , while @xmath53 has been chosen as the optimal one with respect to the rest @xmath68 orbitals , @xmath69 is not necessarily optimized with respect to its accompanying orbitals , namely @xmath70 . therefore , we can turn to @xmath71 and update it in a similar way . this procedure can be repeated by sweeping across the @xmath22 orbitals ( the @xmath22 columns of @xmath67 ) in a circular way . the overlap increases after each update and surely will converge as it is upper bounded by unity . to make sure that we hit the global maximum instead of being trapped by some local one , the program has to be run dozens of times with different initial values of @xmath45 . in fig . [ m3m5 ] , we present the geometric entanglement @xmath72 against the number of electrons , both for @xmath17 and @xmath73 . each data point is obtained with at least 30 different initial values of @xmath45 , and the columns of @xmath45 are updated at least for 40 rounds . out of the final values of @xmath5 , the maximal one is then taken as @xmath74 . we note that the @xmath17 case is surprisingly regular . first , all the points lie very close to a straight line . this linear behavior is not obvious at all , as we know that for the maximally entangled dicke state , in which half of the spins are up and half are down , @xmath75 is not linear in the number of spins but logarithm @xcite . second , even if @xmath75 is found to be linear asymptotically in the large @xmath10 limit ; in the small @xmath10 limit , deviation from this behavior is expected . that is , there should be some crossover regime . however , the linear behavior extends well down to the smallest possible value of @xmath76 . the @xmath73 case is less regular . however , again there is no apparent crossover regime at the lower limit of @xmath77 . the pattern of the data points suggests using the linear least squares approximation to fit the data . that is , @xmath78 we get for @xmath17 , @xmath79 and for @xmath73 , @xmath80 in ref . @xcite , it is found that for some topologically ordered systems such as the toric code , double semion , color code , and quantum double models , the geometric entanglement , like the bipartite entanglement based on space partition , consists of a term proportional to @xmath22 plus a constant term called the topological entropy . if this is also the case for the laughlin wave function , we would have the constant term @xmath81 as @xmath82 for @xmath17 , and @xmath83 for @xmath84 . however , this is not the case as shown by ( [ m3 ] ) and ( [ m5 ] ) . actually , even the trend is in the wrong direction . in conclusion , we have used an iterative algorithm to find the optimal slater approximation of the laughlin wave function and in turn to calculate the geometric entanglement in it . to the best of our knowledge , this is the first time that the geometric entanglement in a strongly interacting fermionic system , which is of great physical interest , is calculated . the importance of the laughlin wave function warrants such a study . we find that the geometric entanglement @xmath75 scales linearly with the electron number @xmath10 . while this is not so surprising , it is unexpected that the linear behavior extends well down to the lower limit of @xmath85 . at the time being , a full understanding of this finding is still lacking . but , it is in alignment with the fact that the laughlin wave function has a very short correlation length ( it is on the order of the magnetic length ; or unity by our notation ) @xcite , which means that the @xmath85 case is already close to the thermodynamic limit . the linear behavior of the geometric entanglement prompted us to compare the constant term with the expected topological entropy . as it turns out , they do not agree . this is in contrast to ref . @xcite , where the topological entropy can be successfully extracted from the scaling behavior of the geometric entanglement . our numerical result as a negative case then indicates that the relation between the geometric entanglement and the topological entropy is subtle . this is reasonable , as the geometric entanglement involves no spatial partition , to which the topological entropy pertains , at least in its original derivation @xcite . we are grateful to k. yang , f. pollman , and t .- c . wei for stimulating discussions . this work is supported by the fujian provincial science foundation under grant no . 2016j05004 , nsf of china under grant no . 11504061 , the china postdoctoral science foundation , and the foundation of lcp . note that we are talking about the generic case . interaction does not necessarily mean the total wave function can not be written as a slater determinant . for example , for @xmath22 fermions in @xmath86 orbitals , the wave function is always a slater determinant , although superficially it is the linear superposition of @xmath86 slater determinants . see appendix a in ref . @xcite or lemma 1 in ref . @xcite for more details .
|
we study numerically the geometric entanglement in the laughlin wave function , which is of great importance in condensed matter physics .
the slater determinant having the largest overlap with the laughlin wave function is constructed by an iterative algorithm .
the logarithm of the overlap , which is a geometric quantity , is then taken as a geometric measure of entanglement .
it is found that the geometric entanglement is a linear function of the number of electrons to a good extent .
this is especially the case for the lowest laughlin wave function , namely the one with filling factor of @xmath0 .
surprisingly , the linear behavior extends well down to the smallest possible value of the electron number , namely , @xmath1 .
the constant term does not agree with the expected topological entropy .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
two dimensional @xmath0 principal chiral models defined by the continuum lagrangian @xmath8 are the simplest quantum field theories sharing with non - abelian gauge theories the property of asymptotic freedom and whose large n limit is a sum over planar diagrams . due to the existence of higher - order conservation laws , multiparticle amplitudes are factorized , and exact @xmath9 matrices have been proposed @xcite . the resulting bound state mass spectrum is represented by @xmath10 and the bound state of @xmath11 particles transforms as the totally antisymmetric tensor of rank @xmath11 . the mass-@xmath5 parameter ratio has been computed , and the result is @xcite @xmath12 a `` standard '' lattice version of principal chiral models is obtained by choosing the action @xmath13\;\;\ ; , \;\;\;\;\;\beta\;=\;{1\over nt}\;\;\ ; , \label{latt - act}\ ] ] whose properties have been investigated by several authors @xcite especially by strong - coupling and mean field methods . numerical simulations have also been performed ( at @xmath14 ) , most recently by dagotto - kogut @xcite and hasenbusch - meyer @xcite . as a preliminary step within a more general program whose ultimate goal is performing the numerical @xmath15 expansion of matrix - valued field theories , we decided to explore the properties of principal chiral models at larger - than - usual values of @xmath4 . in particular we wanted to investigate the following issues : - region of applicability , accuracy and @xmath4 dependence of the strong coupling series . - onset of scaling , with special attention to the interplay between thermodynamical ( peak in the specific heat ) and field theoretical ( dip in the @xmath3-function ) effects . - check of conjectured exact results ( especially mass ratios ) by monte carlo measurements in the scaling region . - role of coupling redefinitions in the widening of the asymptotic scaling regions . to this purpose , we performed a variety of strong coupling and weak coupling calculations , and a number of numerical simulations for different values of @xmath4 , and especially at @xmath7 , where the mass spectrum is sufficiently non - trivial ( two independent mass ratios can be measured and compared with prediction ) , and @xmath16 effects should be already significantly depressed . in the present paper we only report on our analytical results , without offering any details on the derivations , that will be presented elsewhere . we found that the most convenient approach to strong coupling is the character expansion . free energy character expansion for @xmath17 chiral models to twelfth order and mass gap expansion to fifth order were presented in ref . the formal extension of these series to @xmath18 is easily achieved with the abovementioned precision for @xmath19 . paying some attention in order to avoid double - counting , @xmath20 can be also obtained by the same technique . we found explicit representations of the coefficients of the @xmath18 character expansion in the strong coupling regime in terms of bessel functions , by generalizing the technique discussed in refs . these representations are exact up to @xmath21 . as consequence we could compute the @xmath18 free energy to twelfth order in @xmath3 for @xmath19 in two dimensions @xmath22 \beta^8 \ , + \ , \left[56 + { 8n^2 ( 5n^2 - 2)\over ( n^2 - 1)^2 } \right]\beta^{10 } \nonumber \\ & & + \left [ 248 + { 8n^2(35n^2 - 17)\over ( n^2 - 1)^2 } + { 2n^2(14n^6 - 11n^4 + 8n^2 - 2)\over ( n^2 - 1)^4 } + { 16n^4(9n^4 - 26n^2 + 8)\over 3(n^2 - 1)^2 ( n^2 - 4)^2}\right ] \beta^{12 } \nonumber \\ & & + \ ; o\left ( \beta^{14}\right)\;\;+\;\;\ ; 4{n^{n-2}\over n!}\beta^n \,+\ , \left [ 8 { n^{n-1}\over n!}- 4{n^n\over ( n+1)!}\right]\beta^{n+2 } \nonumber \\ & & + \;\left [ 2{n^{n+2}\over(n+2 ) ! } + 4{n^{n+1}\over ( n-1)n ! } - 8{(n+2)n^n\over ( n+1 ) ! } + 24{n^{n-1}\over n ! } + 4{n^{n-2}\over ( n-2)!}\right]\beta^{n+4 } \ , + \ , o\left ( \beta^{n+6}\right)\;. \label{scs1}\end{aligned}\ ] ] in the case @xmath7 an analysis of the @xmath23 and @xmath21 contributions led to the result @xmath24 the internal energy ( per link ) density @xmath25 is immediately obtained from the previous results by @xmath26 these results have been used to draw the strong coupling curves in our figures and compare very well with numerical simulations in the region @xmath27 . short weak coupling series for the free - energy density of @xmath17 and @xmath18 chiral models were presented in ref . @xcite . we calculated the energy density up to three loops finding @xmath28\ ; \rangle \;=\ ; { n^2 - 1\over 8n^2\beta}\left [ 1\,+\,{a_1\over\beta}\,+\ , { a_2\over \beta^2}\,+\, ... \;\right ] \label{wcenergy}\ ] ] where @xmath29 @xmath30 and @xmath31 being numerical constants : @xmath32 and @xmath33 . asymptotic scaling requires the ratio of any dimensional quantity to the appropriate power of the two loop lattice scale @xmath34 to go to a constant as @xmath35 . @xmath36 and @xmath37 are the first universal coefficients of the expansion of the @xmath3-function : @xmath38 @xmath39 evaluation of the ratios of @xmath5 parameters requires a one loop calculation in perturbation theory , which leads to @xcite : @xmath40 in order to get a more accurate description of the approach to asymptotic scaling we performed the change of variables suggested by parisi @xcite , defining a new temperature @xmath41 proportional to the energy : @xmath42 notice that the corresponding specific heat is , by definition , constant . the ratio of @xmath43 , the @xmath5 parameter of the @xmath44 scheme , and @xmath45 is easily obtained from the two loop term of the energy density : @xmath46 we encountered the usual ( and yet unexplained ) phenomenon of a much better convergence to asymptotic scaling for quantities plotted as functions of @xmath44 @xcite . we tried to check for a perturbative explanation of this phenomenon by computing the first perturbative correction to the two loop lattice scale @xmath47\;\;\ ; , \label{asy_corr}\ ] ] in the standard and the @xmath44 scheme , which requires the calculation of the three - loop term of the @xmath3-function in both schemes . in the standard scheme we found @xmath48 \label{b2l}\ ] ] where @xmath49 @xcite . the equivalence of the @xmath50 chiral model to the @xmath51 @xmath52 model allows a check of this equation , indeed for @xmath53 it must give ( and indeed it does ) the same @xmath54 of the standard lattice @xmath51 @xmath52 model @xcite . the @xmath3-function of the @xmath44 scheme can be written in the form @xmath55 where @xmath56 is the specific heat and @xmath57 must be considered as a function of @xmath41 . expanding perturbatively eq . ( [ betae ] ) and using eq . ( [ wcenergy ] ) one finds @xmath58 as one may easily verify , the linear corrections to the two loop lattice scale in eq . ( [ asy_corr ] ) are small and of the same order of magnitude ( although of opposite sign ) . they can not therefore explain the failure of the first and the success of the second scheme with respect to achieving asymptotic scaling . we believe that the origin of this phenomenon is fully non - perturbative , and it can presumably be traced to the phenomenologically apparent correlation existing between the peak in the specific heat and the dip in the lattice @xmath3-function : the non - perturbative variable transformation that flattens the peak manages to fill the dip , in a theoretically yet uncontrolled way . we performed monte carlo simulations of the lattice @xmath0 chiral models for a wide range of values of @xmath4 ( in particular @xmath59 ) and @xmath3 . summaries of the runs are presented in tables [ n3-table],[n6-table],[n9-table ] and [ n15-table ] . in our simulations we implemented the cabibbo - marinari algorithm @xcite to upgrade @xmath18 matrices by updating its @xmath60 subgroups . in most cases , we chose to update the @xmath61 diagonal subsequent @xmath60 subgroups of each @xmath18 matrix variable by employing the over - heat - bath algorithm @xcite ( for the `` heat bath '' part of it we used the kennedy - pendleton algorithm @xcite ) . an important class of observables of the @xmath0 chiral models can be constructed by considering the group invariant correlation function @xmath62\;\rangle\;\;\;.\ ] ] we define the correlation function @xmath63 from the second moment of the correlation function @xmath64 . on the lattice @xmath65\;\;\ ; , \label{xig}\ ] ] where @xmath66 is the fourier transform of @xmath64 . the inverse mass gap @xmath67 is extracted from the long distance behavior of the zero space momentum correlation function constructed with @xmath64 . moreover we measured the diagonal wall - wall correlation length @xmath68 to test rotation invariance . @xmath69 should reproduce in the continuum limit the mass of the fundamental state . the first definition of correlation length @xmath63 offers the advantage of being directly measurable , while the calculation of @xmath67 requires a fit procedure . on the other hand , since @xmath63 is an off - shell quantity an analytical prediction exists only for the inverse mass - gap ( eq . ( [ mass - lambda ] ) ) . in tables [ n3-table],[n6-table],[n9-table ] and [ n15-table ] we present data for the energy density @xmath25 , the specific heat @xmath70 , the magnetic susceptibility @xmath71 defined from the correlation function @xmath64 , the correlation length @xmath63 , the dimensionless ratios @xmath72 and @xmath73 , respectively for @xmath59 . we carefully checked for finite size effects . it turned out that for @xmath74 the finite size systematic errors in evaluating infinite volume quantities should be safely smaller than 1% , which is the typical statistical error of our data . in figs . [ en_n6 ] and [ en_n9 ] we show the energy density versus @xmath3 respectively at @xmath7 and @xmath75 . there the strong coupling series up to twelfth order in @xmath3 and the weak coupling one up to third order in @xmath76 are drawn . as in other asymptotically free models , at all values of @xmath4 the specific heat shows a peak , connecting the two different asymptotic behaviors : monotonically increasing in the strong coupling region and decreasing at large @xmath3 . in figs . [ asy_n3],[asy_n6],[asy_n9 ] and [ asy_n15 ] @xmath77 is plotted respectively for @xmath59 with the corresponding @xmath78 order strong coupling series ( except for @xmath14 ) . increasing @xmath4 , the peak moves slightly towards higher @xmath3 values ( @xmath79 at @xmath7 , @xmath80 at @xmath81 ) , and becomes more and more pronounced . we found the position of the peak to be more stable at large @xmath4 when plotting @xmath77 versus @xmath63 , as in fig . [ specheat ] . notice that , increasing @xmath4 , the specific heat around the peak does not show any apparent convergence to a finite value , which might be an indication of a ( first order ? ) phase transition at @xmath82 . the @xmath1 order ( @xmath78 order ) strong coupling series of the energy ( specific heat ) are in quantitative agreement ( within our statistical errors ) for @xmath83 , and in qualitative agreement up to the peak of the specific heat , whose position should give an estimate of the strong coupling convergence radius . tests of scaling , based on the stability of dimensionless physical quantities ( for example , the ratio @xmath72 ) and rotation invariance ( checking that @xmath84 ) , showed that , within our statistical errors , the scaling region is reached already at small correlation lengths , i.e. for @xmath85 . fitting data in the scaling region to a constant we found @xmath86 notice that scaling is observed even before the peak of the specific heat . since strong coupling series should be effective in this region , it might be possible to calculate continuum physical quantities by strong coupling techniques . in order to investigate this issue work to extend the strong coupling series is in progress . we checked asymptotic scaling according to the two loop formula ( [ asysc ] ) by analyzing @xmath87 . in figs . [ asy_n3],[asy_n6],[asy_n9 ] and [ asy_n15 ] we show the corresponding data respectively for @xmath59 . at all values of @xmath4 we observe the usual dip in the @xmath3-function , which is , again , more and more pronounced when increasing @xmath4 . since @xmath88 we compare @xmath89 directly with eq . ( [ mass - lambda ] ) ( using also eq . ( [ lratio1 ] ) ) , whose predictions , @xmath90 are represented by dashed lines in the figures . notice that the monte carlo data are much larger than the predicted values , while the first perturbative corrections in eq . ( [ asy_corr ] ) are , in all cases , about 20% at @xmath91 . furthermore , data show a similarity with the behavior of the specific heat , strengthening the idea of a strong correlation between the two phenomena . the approach to asymptotic scaling gets an impressive improvement using the @xmath44 scheme . in figs . [ asy_n3],[asy_n6],[asy_n9 ] and [ asy_n15 ] we also plot @xmath92 now data approach the correct value , and the discrepancies are even smaller than the linear correction calculated in sec . [ numericalresults ] ( which is about 15% at @xmath91 ) . so flattening the peak of the specific heat by performing the coupling redefinitions @xmath93 , the dip of the @xmath3-function disappears . we believe this to be the key point of the success of the @xmath44 scheme in widening the asymptotic scaling region . the peak of the specific heat should be explicable in terms of complex @xmath3-singularities of the partition function close to the real axis @xcite . the sharpening of the peak with increasing @xmath4 would indicate that the complex singularities get nearer and nearer to the real axis , pinching it at @xmath82 where a phase transition is expected . such singularities should also cause the abrupt departure from the weak coupling behavior . then a coupling transformation eliminating the peak should move the complex @xmath3-singularities away from the real axis , and therefore improve the approach to asymptotic scaling . from the monte carlo data and the exact result ( [ mass - lambda ] ) we can extract the effective @xmath5-parameters @xmath94 and @xmath95 . fig . [ ll ] and [ le ] show respectively the ratios @xmath96 and @xmath97 , where @xmath98 and @xmath99 are the corresponding two loop functions : @xmath100 . similarly to the specific heat , the effective @xmath5-parameter @xmath94 does not give evidence of convergence at large @xmath4 . on the contrary @xmath95 appears to approach a finite function @xmath101 , which is well approximated by the two loop formula . in conclusion , scaling and asymptotic scaling ( in the @xmath44 scheme ) are observed at all values of @xmath4 considered , even around the peak of the specific heat . it is interesting to notice that , even though the behavior of the specific heat with respect to @xmath4 suggests the existence of a phase transition at @xmath82 , the above scenario is apparently stable at large @xmath4 . we studied the mass spectrum at @xmath7 , where eq . ( [ massratios ] ) predicts the existence of two independent mass ratios . in order to extract the other two independent mass values besides the fundamental one , we considered the following operators : @xmath102 @xmath103 having respectively the same transformation properties of the two and three particle bound states . the mass values @xmath104 and @xmath105 were determined from the large distance behavior of the zero space momentum correlation functions constructed with the above operators . in practice we found distances @xmath106 to be large enough to fit the data to the expected exponential behavior . in table [ mass - table ] and in fig . [ mass - fig ] we present the data for the ratios @xmath107 , @xmath108 and @xmath109 , analyzed using the jackknife method . they show good scaling . fitting them to a constant we found @xmath110 this result confirms , within statistical errors of about 1% , the conjectured exact result ( [ massratios ] ) , which predicts @xmath111
|
lattice @xmath0 chiral models are analyzed by strong and weak coupling expansions and by numerical simulations .
@xmath1 order strong coupling series for the free and internal energy are obtained for all @xmath2 .
three loop contributions to the internal energy and to the lattice @xmath3-function are evaluated for all @xmath4 and non - universal corrections to the asymptotic @xmath5 parameter are computed in the `` temperature '' and the `` energy '' scheme .
numerical simulations confirm a faster approach to asymptopia of the energy scheme . a phenomenological correlation between the peak in the specific heat and the dip of the @xmath3-function is observed .
tests of scaling are performed for various physical quantities , finding substantial scaling at @xmath6 . in particular , at @xmath7 three different mass ratios are determined numerically and found in agreement , within statistical errors of about 1% , with the theoretical predictions from the exact s - matrix theory .
# 1currentlabel@secnum>0 .[#1 ] two dimensional @xmath0 chiral models on the lattice dipartimento di fisica delluniversit and i.n.f.n . , i-56126 pisa , italy
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the last decade has witnessed to significant experimental progresses on the spectrum and decay products of the hadrons containing heavy quarks . these progresses have been stimulated the theoretical interests on the spectroscopy of these baryons via various methods ( for some of them see @xcite and references therein ) . for a better understanding of the heavy flavor physics , it is also necessary to gain deeper insight into the radiative , strong and weak decays of the baryons containing a heavy quark . for some of the related studies , see @xcite and references therein . the strong coupling constants are the main ingredients of the strong interactions of the heavy baryons . to improve our understanding on the strong interactions among the heavy baryons and other hadrons and gain knowledge about the nature and structure of the participated particles , one needs the accurate determinations of these coupling constants . in the present study , we calculate the strong coupling constants @xmath4 and @xmath5 within the framework of the qcd sum rule @xcite as one of the most powerful and applicable tools to hadron physics . these coupling constants are relevant in the bottom and charmed mesons clouds description of the nucleon which may be used to explain exotic events observed by different collaborations . besides , in order to exactly determine the modifications in the masses , decay constants and other parameters of the @xmath1 and @xmath3 mesons in nuclear medium we should immediately consider the contributions of the baryons @xmath9}$ ] and @xmath10}$ ] in the medium produced by the interactions of @xmath1 and @xmath3 mesons with the nucleon , viz . @xmath11 hence , we need to know the exact values of the strong coupling constants @xmath4 , @xmath5 , @xmath12 and @xmath13 entering the born term in the calculations @xcite . note that , among these couplings , we have only one approximate prediction for the strong coupling @xmath5 in the literature calculated at zero transferred momentum square taking the borel masses in the initial and final channels as the same @xcite . we shall also refer to a pioneering work @xcite , which estimates the strong coupling constant @xmath14 . here we should also stress that our work on the calculation of the strong coupling constants @xmath12 and @xmath13 is in progress . the layout of this article is as follows . the next section presents the details of the calculations of the strong coupling constants under consideration . in section 3 , we numerically analyze the sum rules obtained and discuss the results . the purpose of the present section is to give the details of the calculations of the coupling form factors @xmath15 and @xmath16 . the values of these form factors at @xmath17}^2 $ ] give the strong coupling constants among the participating particles . to fulfill this aim , the starting point is the usage of the following three - point correlation function : @xmath18}(0)~ \bar{j}_{\lambda_b[\lambda_c]}(x)\right)|0{\rangle},\end{aligned}\ ] ] where @xmath19 denotes the time ordering operator and @xmath20 is transferred momentum . the three - point correlation function contains interpolating currents that can be written in terms of the quark field operators as : @xmath21}(x)&=&\varepsilon_{abc}u^{a^t}(x)c\gamma_5d^{b}(x)b[c]^{c}(x ) , \nonumber \\ j_{n}(y)&=&\varepsilon_{ijk}\big(u^{i^t}(y)c\gamma_{\mu}u^{j}(y)\big)\gamma_{5}\gamma_{\mu}d^{k}(y ) , \nonumber \\ j_{b[d]}(0)&=&\bar{u}(0)\gamma_5b[c](0),\end{aligned}\ ] ] where @xmath22 is the charge conjugation operator . the calculation of the three - point correlation function is made via following two different ways . in the first way , which is called as hadronic side , one calculates it in terms of the hadronic degrees of freedom . in the second way , which is called as ope side , it is calculated in terms of quark and gluon degrees of freedom using the operator product expansion in deep euclidean region . these two sides are then matched to obtain the qcd sum rules for the coupling form factors . we apply a double borel transformation with respect to the variables @xmath23 and @xmath24 to both sides to suppress the contributions of the higher states and continuum . the calculation of the hadronic side of the correlation function requires its saturation with complete sets of appropriate @xmath25 $ ] , @xmath26 $ ] and @xmath27 hadronic states having the same quantum numbers as their interpolating currents . this step is followed by performing the four - integrals over @xmath28 and @xmath29 , which leads to @xmath30}\mid b[d](q)\rangle \langle \lambda_{b}[\lambda_{c}](p ) \mid \bar{j}_{\lambda_{b}[\lambda_{c}]}\mid 0\rangle } { ( p^2-m_{\lambda_{b}[\lambda_{c}]}^2)(p^{\prime^2}-m_{n}^2)(q^2-m_{b[d]}^2 ) } \nonumber \\ & \times&\langle n(p^{\prime})b[d](q)\mid \lambda_{b}[\lambda_{c}](p)\rangle+\cdots~,\end{aligned}\ ] ] where @xmath31 represents the contributions coming from the higher states and continuum . the matrix elements in this equation are parameterized as @xmath32}\mid 0\rangle&=&\lambda_{\lambda_b[\lambda_c]}\bar{u}_{\lambda_b[\lambda_c]}(p , s ) , \nonumber \\ \langle 0 \mid j_{b[d]}\mid b[d](q)\rangle&=&i\frac{m_{b[d]}^2f_{b[d]}}{m_u+m_{b[c ] } } , \nonumber \\ \langle n(p^{\prime})b[d](q)\mid \lambda_b[\lambda_c](p)\rangle&=&g_{\lambda_bnb[\lambda_c nd]}\bar{u}_n(p^{\prime},s^{\prime})i \gamma_5 u_{\lambda_b[\lambda_c]}(p , s),\end{aligned}\ ] ] where @xmath33 and @xmath34}$ ] are the residues ; and @xmath35 and @xmath36}$ ] are the spinors for the nucleon and @xmath37 $ ] baryon , respectively . in the above equations , @xmath38}$ ] is the leptonic decay constant of @xmath26 $ ] meson and @xmath39}$ ] is the strong coupling form factor among @xmath37 $ ] , @xmath27 and @xmath26 $ ] particles . the use of eqs . ( [ matriselement1 ] ) in eq . ( [ physide1 ] ) is followed by summing over the spins of the @xmath27 and @xmath37 $ ] baryons , i.e. @xmath40}(p , s)\bar{u}_{\lambda_b[\lambda_c]}(p , s)&= & \not\!p+m_{\lambda_b[\lambda_c]}~.\end{aligned}\ ] ] as a result , we have @xmath41}^2f_{b[d]}}{m_{b[c]}+m_u}\frac{\lambda_n\lambda_{\lambda_b[\lambda_c ] } g_{\lambda_bnb[\lambda_c nd]}}{(p^2-m_{\lambda_b[\lambda_c]}^2)(p^{\prime^2}-m_n^2)(q^2-m_{b[d]}^2 ) } \nonumber \\ & \times&\big\{(m_nm_{\lambda_b[\lambda_c]}- m_{\lambda_b[\lambda_c]}^2)\gamma_5+(m_{\lambda_b[\lambda_c]}-m_n)\not\!p\gamma_5+\not\!q\not\!p\gamma_5-m_{\lambda_b[\lambda_c ] } \not\!q\gamma_5\big\}\nonumber\\ & + & \cdots~.\end{aligned}\ ] ] the final form of the hadronic side of the correlation function is obtained after the application of the double borel transformation with respect to the initial and final momenta squared , viz . @xmath42}^2f_{b[d]}}{m_{b[c]}+m_u}\frac{\lambda_n\lambda_{\lambda_b[\lambda_c ] } g_{\lambda_bnb[\lambda_cnd]}}{(q^2-m_{b[d]}^2)}e^{-\frac{m_{\lambda_b[\lambda_c]}^2}{m^2 } } e^{-\frac{m_n^2}{m^{\prime^2 } } } \nonumber \\ & \times&\big\{(m_nm_{\lambda_b[\lambda_c]}- m_{\lambda_b[\lambda_c]}^2)\gamma_5+(m_{\lambda_b[\lambda_c]}-m_n)\not\!p\gamma_5+\not\!q\not\!p\gamma_5-m_{\lambda_b[\lambda_c ] } \not\!q\gamma_5\big\}\nonumber\\&+&\cdots~,\end{aligned}\ ] ] where @xmath43 and @xmath44 are borel mass parameters . the ope side of the correlation function is calculated in deep euclidean region , where @xmath45 and @xmath46 . to proceed , the explicit expressions of the interpolating currents are inserted into the correlation function in eq . ( [ correlationfunc1 ] ) . after contracting out all quark pairs via wick s theorem we get @xmath47}^{h\ell}(-x ) \nonumber \\ & -&\gamma_5\gamma_{\mu}s^{cj}_{d}(y - x)\gamma_{5}cs_{u}^{ai^t}(y - x)c\gamma_{\mu}s^{bh}_{u}(y ) \gamma_{5}s_{b[c]}^{h\ell}(-x ) \bigg\}~,\end{aligned}\ ] ] where @xmath48}(x)$ ] represents the heavy quark propagator which is given by @xcite @xmath49}^{i\ell}(x)&=&\frac{i}{(2\pi)^4}\int d^4k e^{-ik \cdot x } \left\ { \frac{\delta_{i\ell}}{\!\not\!{k}-m_{b[c ] } } -\frac{g_sg^{\alpha\beta}_{i\ell}}{4}\frac{\sigma_{\alpha\beta}(\!\not\!{k}+m_{b[c]})+ ( \!\not\!{k}+m_{b[c]})\sigma_{\alpha\beta}}{(k^2-m_{b[c]}^2)^2}\right.\nonumber\\ & & \left.+\frac{\pi^2}{3 } \langle \frac{\alpha_sgg}{\pi}\rangle \delta_{i\ell}m_{b[c ] } \frac{k^2+m_{b[c]}\!\not\!{k}}{(k^2-m_{b[c]}^2)^4 } + \cdots\right\ } \ , , \end{aligned}\ ] ] and @xmath50 and @xmath51 are the light quark propagators and are given by @xmath52 + \cdots \ , .\end{aligned}\ ] ] the substitution of these explicit forms of the heavy and light quark propagators into eq . ( [ correlfuncope1 ] ) is followed by the usage of the following fourier transformations in @xmath53 dimensions : @xmath54^n}&=&\int\frac{d^dt}{(2\pi)^d}e^{-it\cdot(y - x)}~i~(-1)^{n+1}~2^{d-2n}~\pi^{d/2}~ \frac{\gamma(d/2-n)}{\gamma(n)}\big(-\frac{1}{t^2}\big)^{d/2-n } , \nonumber \\ \frac{1}{[y^2]^n}&=&\int\frac{d^dt^{\prime}}{(2\pi)^d}e^{-it^{\prime}\cdot y}~i~(-1)^{n+1}~2^{d-2n}~\pi^{d/2}~ \frac{\gamma(d/2-n)}{\gamma(n)}\big(-\frac{1}{t^{\prime^2}}\big)^{d/2-n}.\end{aligned}\ ] ] then , the four-@xmath28 and four-@xmath29 integrals are performed in the sequel of the replacements @xmath55 and @xmath56 . as a result , these integrals turn into dirac delta functions which are used to take the four - integrals over @xmath57 and @xmath58 . finally the feynman parametrization and @xmath59^{\alpha-\beta-2}},\end{aligned}\ ] ] are used to perform the remaining four - integral over @xmath60 . the correlation function in ope side is obtained in terms of different structures as @xmath61 where each @xmath62 function includes the contributions coming from both the perturbative and non - perturbative parts and can be written as @xmath63 the imaginary parts of the @xmath64 functions give the spectral densities @xmath65 appearing in the last equation , viz . @xmath66 $ ] . as examples , we present only the explicit forms of the spectral functions @xmath67 and @xmath68 corresponding to the dirac structure @xmath69 , which are obtained as @xmath70}m_us^{\prime^2}}{64\pi^4(q^2-m_{b[c]}^2 ) } \theta[l_1(s , s^{\prime},q^2)]+\int_{0}^{1}dx \int_{0}^{1-x}dy \frac{1}{64\pi^4u^3 } \nonumber \\ & \times&\big[2m_{b[c]}^4x^2\big(1 + 3x^2-y+6xy-4x)\big)+m_{b[c]}^3x\big(3m_d u(2x-1 ) + m_u(3 + 2x^2 \nonumber \\ & -&3y-5x-2xy)\big)+2m_{b[c]}^2x\big(s(12x^4+y^2-y-30x^3 + 36x^3y-6x+20xy \nonumber \\ & -&13xy^2 + 24x^2 -55x^2y+24x^2y^2)+q^2xy(18x-24xy+7y-12x^2 - 6 ) \nonumber \\ & + & s^{\prime}y(12x^3 + 7y-4y^2 - 27x^2 + 36x^2y+18x-43xy+24xy^2 - 3 ) \big ) \nonumber \\ & + & 2s^2u^2x\big(10x^3 + 6x-15xy+2y-16x^2 + 20x^2y\big)+2q^4x^2y^2\big(10x^2 - 7y \nonumber \\ & -&16x+20xy+6\big)+2s^{\prime^2}y^2u^2 \big(10x^2 - 3y-12x+20xy\big)-4q^2s^{\prime}xy^2\big(10x^3 \nonumber \\ & + & 9y-5y^2 - 24x^2 + 30x^2y+18x-39xy+20xy^2 - 4\big)+2suy \big(q^2x(32x^2 \nonumber \\ & -&40x^2y-20x^3 - 2y-13x+22xy+1)+s^{\prime}(20x^4 - 48x^3 + 60x^3y - y+y^2 \nonumber \\ & -&8x+27xy-18xy^2 + 36x^2 - 86x^2y+40x^2y^2)\big)+3m_{b[c]}m_u u\big(q^2x(x+2y \nonumber \\ & -&3xy-1)+sux(3x-1)+s^{\prime}u(3xy - x - y)\big ) -m_{b[c]}m_u\big(q^2x(3x^2y-3x^2 \nonumber \\ & + & 7y-4y^2 + 6x-10xy-3xy^2 - 3)-sux(3x^2-y-6x-6xy+3 ) \nonumber \\ & -&3s^{\prime}u(x^2y - x^2+y -y+x-3xy - xy^2)\big ) \big]\theta[l_2(s , s^{\prime},q^2 ) ] \bigg\},\end{aligned}\ ] ] and @xmath71}^2-q^2)}\big[2m_{b[c]}m_dm_u\langle \bar{d}d\rangle+\big(m_{b[c]}(3m_u^2 - 3m_dm_u-2s^{\prime } ) \nonumber \\ & + & m_d(4m_u^2+s - s^{\prime})+2m_us^{\prime } \big)\langle \bar{u}u\rangle\big]-\langle \alpha_s\frac{g^2}{\pi}\rangle\bigg[\frac{m_{b[c]}m_uq^2s^{\prime^2}}{192\pi^2(q^2-m_{b[c]}^2)^4 } \nonumber \\ & -&\frac{9m_{b[c]}s^{\prime}(m_d+m_u)+2s^{\prime}(q^2 - 2s+5s^{\prime})}{1152\pi^2(q^2-m_{b[c]}^2)^2}- \frac{m_{b[c]}(m_d-3m_u)}{128\pi^2(q^2-m_{b[c]}^2)}\bigg ] \nonumber \\ & -&m_0 ^ 2\langle \bar{d}d\rangle \frac{3m_{b[c]}+4m_d}{96\pi^2(m_{b[c]}^2-q^2)}+m_0 ^ 2\langle \bar{u}u\rangle \frac{9m_{b[c]}+3m_d-7m_u}{96\pi^2(m_{b[c]}^2-q^2)}\bigg\}\theta[l_1(s , s^{\prime},q^2 ) ] \nonumber \\ & + & \int_{0}^{1}dx \int_{0}^{1-x}dy\bigg\{\frac{1}{8\pi^2u}\big[\langle \bar{d}d\rangle\big(m_{b[c]}-2m_{b[c]}x - m_uu+m_d(3x-1)(y+u)\big ) \nonumber \\ & + & \langle \bar{u}u\rangle\big(m_{b[c]}-2m_{b[c]}x-4m_du-2m_u(y-3xy-3xu)\big)\big ] + \langle \alpha_s\frac{g^2}{\pi}\rangle \frac{1}{96\pi^2u^3 } \nonumber \\&\times&\big[3u^2(3x-1)(y+u)+xy(1-y+x(3x+6y-4))\big ] \bigg\}\theta[l_2(s , s^{\prime},q^2 ) ] , \nonumber \\\end{aligned}\ ] ] where @xmath72}^2x+sx - sx^2+s^{\prime}y+q^2xy - sxy - s^{\prime}xy - s^{\prime}y^2 , \nonumber \\ u&=&x+y-1,\end{aligned}\ ] ] with @xmath73 $ ] being the unit - step function . as we previously mentioned , the qcd sum rules for the strong form factors are obtained by matching the hadronic and ope sides of the correlation function . as a result , for @xmath69 structure , we get @xmath74}(q^2)=-e^{\frac{m_{\lambda_b[\lambda_c]}^2}{m^2}}e^{\frac{m_n^2}{m^{\prime^2}}}~ \frac{(m_{b[c]}+m_u)(q^2-m_{b[d]}^2)}{m_{b[d]}^2f_{b[d]}\lambda_{\lambda_{b}[\lambda_c]}^{\dag}\lambda_n(m_nm_{\lambda_{b}[\lambda_c]}- m_{\lambda_b[\lambda_c]}^2 ) } \nonumber \\ & \times & \bigg\{\int^{s_0}_{(m_{b[c]}+m_{u}+m_{d})^2}ds\int^{s'_0}_{(2m_u+m_d)^2}ds^{\prime}e^{-\frac{s}{m^2 } } e^{-\frac{s^{\prime}}{m^{\prime^2 } } } \big[\rho_1^{pert}(s , s^{\prime},q^2)+\rho_1^{non - pert}(s , s^{\prime},q^2)\big]\bigg\}~,\nonumber\\\end{aligned}\ ] ] where @xmath75 and @xmath76 are continuum thresholds in @xmath37 $ ] and @xmath27 channels , respectively . this section contains the numerical analysis of the obtained sum rules for the strong coupling form factors including their behavior in terms of @xmath77 . for the analysis , we use the input parameters given in table 1 . [ table1 ] .input parameters used in calculations . [ cols="^,^",options="header " , ] the analysis starts by the determination of the working regions for the auxiliary parameters @xmath43 , @xmath78 , @xmath75 and @xmath76 . these parameters , which arise due to the double borel transformation and continuum subtraction , are not physical parameters so the strong coupling form factors should be almost independent of these parameters . being related to the energy of the first excited states in the initial and final channels , the continuum thresholds are not completely arbitrary . the continuum thresholds @xmath79 and @xmath76 are the energy squares which characterize the beginning of the continuum . if we denote the ground states masses in the initial and final channels respectively by @xmath80 and @xmath81 , the quantities @xmath82 and @xmath83 are the energies needed to excite the particles to their first excited states with the same quantum numbers . the @xmath82 and @xmath83 are well known for the states under consideration @xcite , where they lie roughly between @xmath84 and @xmath85 . these values lead to the working intervals of the continuum thresholds as @xmath86~\mbox{gev$^2$}\leq s_0\leq34.5[6.7]~\mbox{gev$^2$}$ ] and @xmath87 for the strong vertex @xmath88 $ ] . in the determination of the working regions of borel parameters @xmath43 and @xmath78 , one considers the pole dominance as well as the convergence of the ope . in technique language , the upper bounds on these parameters are obtained by requiring that the pole contribution exceeds the contributions of the higher states and continuum , i.e. the condition [ nolabel ] ^_s_0ds^_s_0ds^e^- e^-_i(s , s^,q^2 ) ^_s_minds^_s_minds^e^- e^-_i(s , s^,q^2 ) < 1/3 , should be satisfied , where for each structure @xmath89 , @xmath90}+m_{u}+m_{d})^2 $ ] and @xmath91 . the lower bounds on @xmath43 and @xmath78 are obtained by demanding that the contribution of the perturbative part exceeds the non - perturbative contributions . these considerations lead to the windows @xmath92~\mbox{gev$^2$}\leq m^2\leq 20[6]~\mbox{gev$^2$}$ ] and @xmath93 for the borel mass parameters corresponding to the strong vertex @xmath88 $ ] in which our results have weak dependencies on the borel mass parameters ( see figures 1 - 2 ) . as a function of the borel mass @xmath43 at average values of continuum thresholds . * right : * @xmath94 as a function of the borel mass @xmath44 at average values of continuum thresholds . , title="fig:",width=302 ] as a function of the borel mass @xmath43 at average values of continuum thresholds . * right : * @xmath94 as a function of the borel mass @xmath44 at average values of continuum thresholds . , title="fig:",width=302 ] . , title="fig:",width=302 ] . , title="fig:",width=302 ] as a function of @xmath95 at average values of the continuum thresholds and borel mass parameters . * right : * @xmath96 as a function of @xmath95 at average values of the continuum thresholds and borel mass parameters . , title="fig:",width=302 ] as a function of @xmath95 at average values of the continuum thresholds and borel mass parameters . * right : * @xmath96 as a function of @xmath95 at average values of the continuum thresholds and borel mass parameters . , title="fig:",width=302 ] now , we use the working regions of auxiliary parameters as well as values of other input parameters to find out the dependency of the strong coupling form factors on @xmath95 . our numerical calculations reveal that the following fit function well describes the strong coupling form factors in terms of @xmath95 : @xmath97}(q^2)=c_1\exp\big[-\frac{q^2}{c_2}\big]+c_3,\end{aligned}\ ] ] where the values of the parameters @xmath98 , @xmath99 and @xmath100 for different structures are presented in tables [ fitparam ] and [ fitparam1 ] for @xmath101 and @xmath6 , respectively . in figure 3 , we depict the dependence of the strong coupling form factors on @xmath95 at average values of the continuum thresholds and borel mass parameters for both the qcd sum rules and fitting results . from this figure , we see that the qcd sum rules are truncated at some points at negative values of @xmath95 and the fitting results coincide well with the sum rules predictions up to these points . the values of the strong coupling constants obtained from the fit function at @xmath102}^2 $ ] for all structures are given in table [ couplingconstant ] . the errors appearing in the results are due to the uncertainties of the input parameters and those coming from the calculations of the working regions for the auxiliary parameters . from table 4 , we see that all structures except that @xmath69 lead to very close results . we also depict the average of the coupling constants under consideration , obtained from all the structures used , in table 4 . @xmath103 @xmath104 @xmath105 at this stage , we compare our result of the coupling constant @xmath5 obtained at @xmath7 with that of ref . @xcite for the dirac structure @xmath106 . at @xmath7 , we get the result @xmath107 for this structure , which is consistent with the prediction of @xcite , i.e. , @xmath108 within the errors . to summarize , we have calculated the strong coupling constants @xmath4 and @xmath5 in the framework of the three - point qcd sum rules . our results can be used in the bottom and charmed mesons clouds description of the nucleon which may be used to explain exotic events observed by different experiments . the obtained results can also be used in analysis of the results of heavy ion collision experiments like @xmath109 at fair . these results may also be used in exact determinations of the modifications in the masses , decay constants and other parameters of the @xmath1 and @xmath3 mesons in nuclear medium . this work has been supported in part by the scientific and technological research council of turkey ( tubitak ) under the research project 114f018 . d. w. wang , m. q. huang , phys . d 68 , 034019 ( 2003 ) . z. g. wang , eur . phys . j. c 54 , 231 ( 2008 ) . f. o. duraes , m. nielsen , phys . b 658 , 40 ( 2007 ) . x. liu , h. x. chen , y. r. liu , a. hosaka , s. l. zhu , phys . rev . d 77 , 014031 ( 2008 ) . d. w. wang , m. q. huang , c. z. li , phys . d 65 , 094036 ( 2002 ) . n. mathur , r. lewis , r. m. woloshyn , phys . d 66 , 014502 ( 2002 ) . d. ebert , r. n. faustov , v. o. galkin , phys . d 72 , 034026 ( 2005 ) . m. karliner , h. j. lipkin , phys . b 660 , 539 ( 2008 ) . m. karliner , b. keren - zur , h. j. lipkin , j. l. rosner , arxiv:0706.2163 ( 2007 ) . j. l. rosner , phys . d 75 , 013009 ( 2007 ) . t. m. aliev , k. azizi , a. ozpineci , nucl . b 808 , 137 ( 2009 ) . b. julia - diaz , d. o. riska , nucl . phys . a 739 , 69 ( 2004 ) . s. scholl , h. weigel , nucl . phys . a 735 , 163 ( 2004 ) . a. faessler et . al , phys . rev . d 73 , 094013 ( 2006 ) . b. patel , a. k. rai , p. c. vinodkumar , frascati physics series vol . xlvi ( 2007 ) , arxiv : 0803.0221 . c. s. an , nucl . phys . a 797 , 131 ( 2007 ) , erratum - ibid ; a 801 , 82 ( 2008 ) . t. m. aliev , a. ozpineci , m. savci , phys . rev . d 65 , 096004 ( 2002 ) . t. m. aliev , k. azizi , a. ozpineci , phys . d 79 , 056005 ( 2009 ) . f. s. navarra , m. nielsen , phys . b 443 , 285 ( 1998 ) . huang , h .- x . chen , s .- l . zhu , phys . d 80 , 094007 ( 2009 ) . wang , eur . j. a 44 , 105 ( 2010 ) ; phys . d 81 , 036002 ( 2010 ) . k. azizi , m. bayar , a. ozpineci , phys . rev . d 79 , 056002 ( 2009 ) . t. m. aliev , k. azizi , m. savci , phys . b 696 , 220 ( 2011 ) . h .- y . cheng , c .- k . chua , phys . d 75 , 014006 ( 2007 ) . a. khodjamirian , ch . klein , th . mannel , y .- m . wang , jhep 1109 , 106 ( 2011 ) . e. hernandez , j. nieves , phys . rev . d 84 , 057902 ( 2011 ) . k. azizi , m. bayar , y. sarac and h. sundu , phys . d 80 , 096007 ( 2009 ) . t. gutsche , m. a. ivanov , j. g. korner , v. e. lyubovitskij , p. santorelli , arxiv:1410.6043 ( 2014 ) . m. a. shifman , a. i. vainshtein , v. i. zakharov , nucl . b147 , 385 ( 1979 ) ; nucl . b 147 , 448 ( 1979 ) . l. j. reinders , h. rubinstein and s. yazaki , phys . 127 , 1 ( 1985 ) . k. a. olive et al . ( particle data group ) , chin . c , 38 , 090001 ( 2014 ) . a. khodjamirian , `` b and d meson decay constant in qcd '' , lecture 1 at 3rd belle analysis school " , sept . 22 , 2010 , kek . b. i. eisenstein et al . ( cleo collab . ) , phys . d78 , 052003 ( 2008 ) . k. azizi , n. er , eur . j. c74 , 2904 ( 2014 ) . b. l. ioffe , prog . 56 , 232 ( 2006 ) . v. m. belyaev , b. l. ioffe , sov . jetp 57 , 716 ( 1982 ) ; phys . b 287 , 176 ( 1992 ) .
|
we investigate the strong vertices among @xmath0 , nucleon and @xmath1 meson as well as @xmath2 , nucleon and @xmath3 meson in qcd .
in particular , we calculate the strong coupling constants @xmath4 and @xmath5 for different dirac structures entered the calculations . in the case of @xmath6 vertex , the result is compared with the only existing prediction obtained at @xmath7 . # 1#2#3 @xmath8 # 1#2#3 @xmath8 0^*0 5_5 o _ ^0 _ pacs number(s ) : 13.30.-a , 13.30.eg , 11.55.hx
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
ion - water clusters represent the transition between an ion in the gas phase and a fully hydrated ion . hence these systems have been extensively studied to understand the thermochemistry of hydration and to infer how bulk properties emerge from properties of ions in clusters @xcite . most notably , attempts have been made to infer the hydration free energy of an isolated ion , a thermodynamic descriptor of the nonideal interactions between the ion and water , by analyzing the thermochemistry of forming ion - water clusters @xmath0_n}$]@xcite , where x is the ion . thermochemical data alone is inadequate for inferring the structural characteristics of hydration . in this regard , theoretical calculations of cluster formation have proven useful . ( for example , see refs . . ) moreover , such calculations together with hydration free energy calculations of the clusters and water ligands have been used for estimating the hydration free energy of single ions . in calculating the hydration free energies , almost always a continuum model for the bulk solvent is assumed and either one @xcite or an ensemble @xcite of gas phase configurations of the cluster is used for obtaining the hydration free energy of the cluster . in some studies the gas - phase cluster is also allowed to relax in response to the model bulk medium @xcite . an important development in the statistical mechanics of hydration has been the quasichemical organization @xcite of the potential distribution theorem . this approach provides a rigorous , statistical mechanical framework to relate the thermodynamics of solute - water clusters to the bulk hydration of the solute . in practical implementations , approximations to the quasichemical theory are invariably made . in the _ primitive _ quasichemical approximation , the adjective _ primitive _ indicates neglecting the role of the bulk medium on the local clustering reaction the thermochemistry of ion - water clustering in the ideal gas phase , obtained using standard quantum chemistry approaches , is coupled with estimates of the response of the bulk medium , treated either as a dielectric continuum @xcite or as a discrete molecular solvent @xcite . within primitive quasichemical theory , the hydration free energy is estimated by varying the number of water molecules in the cluster and finding the optimal coordination number that minimizes the hydration free energy of the ion . this cluster variation approach has proven successful in establishing the hydration structure of small , hard ions such as li@xmath1(aq ) @xcite , na@xmath1(aq ) @xcite , and be@xmath2(aq ) @xcite ; for these cases , the optimal coordination structure predicted by primitive quasichemical theory is in good agreement with the most probable coordination observed in _ ab initio _ molecular dynamics simulations ( aimd ) . similar agreement is also seen for mg@xmath2 , ca@xmath2 , and some of the transition metal dications @xcite . given the then limitations of small systems , short simulation times , and uncertain quality of the underlying electro density functionals in aimd simulations ( for example , refs . ) , primitive quasichemical theory proved useful in cross - checking the simulation results itself . ( the predictions for h@xmath1@xcite and ho@xmath3@xcite are also in fair agreement with aimd simulations , but consensus remains elusive@xcite . these systems also challenge both theory and simulations because of the need to describe nuclear quantum effects . ) for the soft k@xmath1 ion , primitive quasichemical predicts an optimum coordination with four ( 4 ) water molecules @xcite . but the results from aimd simulations are less conclusive . some studies identify an inner - coordination number of four and a second outer - population with two additional water molecules within the nominal first hydration shell of the ion @xcite , but others make no such distinction @xcite . beside k@xmath1 , it has also been well appreciated that for some of the halides , optimum gas - phase clusters can show intermolecular bonding between the coordinating shell water molecules @xcite , a feature that is not usually observed in the coordination structure for the hydrated ion in the liquid . thus the primitive quasichemical approach for these ions has not been successful in reproducing the hydration structure and thermodynamics . the limitations in describing the hydration of soft cations and anions suggests that the bulk medium plays an important role in the hydration structure of these ions . developing the framework to understand this effect is the objective of this article . in section [ sc : theory ] , we present the quasichemical theory , elucidate the role of the medium , and highlight the physical consequences of the primitive quasichemical approximation . in our previous work @xcite we showed the importance of occupancy number variations of water in an empty coordination sphere for understanding an ion s coordination structure . developing those ideas further , here we find that the bulk medium promotes a better packing of solvent molecules around the ion , leading to a decrease in the contribution to the hydration free energy due to interactions between the ion and the solvent molecules within the coordination sphere . the medium stabilizes configurations which are otherwise not observed in isolated ion - solvent clusters . without proper account of this effect , the predicted optimal coordination state is typically lower than the optimal coordination observed in simulations , and appreciating this effect is necessary in calculating hydration thermodynamics as well . we present only the main elements of the quasichemical theory ; more extensive discussions are available elsewhere @xcite . we define the coordination sphere of radius @xmath4 around the ion , @xmath5 , and by so doing , separate the local , chemically intricate ion interactions with water molecules ( the solvent ligands ) within the coordination sphere from the interaction of the ion with the medium ( the bulk ) outside the coordination shell . the chemical potential can then be written as @xcite @xmath6 where @xmath7 , @xmath8 is the boltzmann constant and t is the temperature . @xmath9 is the probability of finding zero ( 0 ) solvent ligands within @xmath4 . ( the position of the water molecule is defined by the position of the oxygen atom . the ion in the @xmath10 hydration state has @xmath11-ligands within @xmath4 . ) the contribution to the free energy due to ion - bulk interactions when the ion is in the @xmath12 state is @xmath13 . the probability of finding zero ( 0 ) solvent ligands within @xmath4 when the ion - water interactions are turned off is @xmath14 . this factor accounts for the free energy of creating an empty coordination sphere in the solvent and thus accounts for the packing ( steric ) contributions to @xmath15 . the quasichemical form is obtained by noting that @xmath9 is specified by chemical equilibria of the form @xcite @xmath16}_0({\rm aq } ) + n{\rm h_2o}({\rm aq } ) & \rightleftharpoons & { \rm \alpha[h_2o]}_n({\rm aq } ) \ , , \label{eq : ionaqcoord}\end{aligned}\ ] ] where @xmath17_n$ ] denotes the ion plus @xmath11-water cluster within the coordination sphere and ` aq ' indicates that the clustering reaction is in the presence of the bulk medium . given the equilibrium constant of the above reaction is @xmath18 and the density of water molecules is @xmath19 , we immediately obtain @xcite @xmath20 note that the excess chemical potential of the cluster @xmath17_0 $ ] is just @xmath21 ( for ease in writing , we will denote @xmath22 by @xmath23 in all subscripts . ) approximating eq . [ eq : lnx0 ] by its maximum term , we get @xmath24 @xmath18 is given by the ratio of partition functions of the cluster to individual molecules and can be challenging to calculate with full account of the surrounding medium @xcite . but the equilibrium constant @xmath25 of the clustering reaction @xmath16}_0({\rm g } ) + n{\rm h_2o}({\rm g } ) & \rightleftharpoons & { \rm \alpha[h_2o]}_n({\rm g } ) \ , , \label{eq : iongcoord}\end{aligned}\ ] ] in the ideal gas phase is more amenable to theoretical calculations . in eq . [ eq : iongcoord ] , ` g ' indicates that the solvent outside the coordination shell is non - interacting ( or ideal ) ; that is , the reaction is performed in the ideal gas phase . since the free energy change for eqs . [ eq : ionaqcoord ] and [ eq : iongcoord ] , with appropriate choice of standard concentrations , are just @xmath26 and @xmath27 , respectively , we find that @xmath28 the first two terms on the right hand side represent the hydration free energies of the clusters containing 0 waters and @xmath11 waters , respectively , and @xmath29 is the hydration free energy of a water molecule . a similar development can be pursued when the coordination shell is empty , which is equivalent to the case when the ion and solvent do not interact . denoting the noninteracting solute by @xmath30 , the analog of eq . [ eq : knbykn0 ] is @xmath31 note that @xmath14 is related to @xmath32 @xcite exactly like @xmath9 is related to @xmath18 ( eq . [ eq : lnx0 ] ) , and equilibria analogous to eqs . [ eq : ionaqcoord ] and [ eq : iongcoord ] specify @xmath32 and @xmath33 , respectively . from eqs . [ eq : qcapprox ] and [ eq : knbykn0 ] we obtain : @xmath34 eq [ eq : qcapprox2 ] not only permits the calculation of @xmath35 , but it also suggests an approach to identify the optimal coordination state . for different @xmath11 , we can compute @xmath25 ; often highly descriptive forcefields , including _ ab initio _ potentials , are used to model the thermochemistry of the gas - phase reaction . the presence of the bulk medium outside the coordination shell is then corrected _ a posteriori _ by adding the free energy of hydrating the cluster and subtracting the free energy of transferring the requisite number of water molecules to the gas phase from the liquid . a continuum or dielectric model of the solvent is often assumed for this purpose . the coordination state that minimizes @xmath36 is then identified as the optimal coordination state @xcite . in the _ primitive _ quasichemical approximation , the configuration of the gas phase cluster is not allowed to relax in computing the hydration free energy . to better understand this approximation , we first identify the role of the bulk medium on local ion - water clustering . the ratio of equilibrium constants @xmath18 and @xmath32 was earlier@xcite shown to be @xmath37 where @xmath38 is the free energy of forming the ion - water cluster in the presence of the bulk and @xmath39 is hydration free energy of the ion in its @xmath40 coordination state . specifically , @xmath41 where @xmath42 is the interaction energy of the ion with the rest of the medium , @xmath43 indicates averaging when the ion and the medium are thermally uncoupled ( denoted by the subscript 0 ) , and @xmath44 indicates that only cases with exactly @xmath11 solvent molecules within the coordination shell are considered in averaging . by parsing the interaction energy @xmath42 into a local piece , @xmath45 , obtained by considering ion - interactions with the @xmath11 coordinating water molecules , and the remaining long - range piece , @xmath46 , we have @xmath47 where we have made use of the rule of averages @xcite , in rewriting the second factor on the right ; @xmath48 indicates averaging such that the @xmath11-coordination shell molecules are thermally coupled with the ion , and the bulk medium is uncoupled from the ion but is coupled with the @xmath11-water molecules . identifying the first factor on the right in eq . [ eq : munint ] by @xmath49 and the second factor by @xmath50 , we have @xmath51 we emphasize that @xmath49 accounts for the contribution to the chemical potential due to the interaction of the ion with solvent ligands within the coordination shell _ in the presence of the medium outside _ , and @xmath52 accounts for the contribution due to the interaction of the bulk medium with the ion when there are @xmath11 solvent ligands present inside the coordination shell . note that when @xmath53 , @xmath54 ( eq . [ eq : quasichemical ] ) . thus we find that @xmath55 ( our earlier work@xcite had an error in factoring @xmath56 . the error is discussed in appendix [ sc : appa ] for completeness . ) now if the medium outside the coordination shell was an ideal gas , @xmath57 from eq . [ eq : knbyknt1 ] and [ eq : knbyknt2 ] , we get @xmath58 and this together with eqs . [ eq : knbykn0 ] , [ eq : knbykn0 t ] , and [ eq : knbyknt3 ] gives @xmath59 finally , substituting in eq . [ eq : qcapprox2 ] and noting eq . [ eq : hardion ] , we get @xmath60 equation [ eq : frozen ] clearly identifies three different roles of the solvent medium and also helps us induce the physical consequences of the _ primitive _ quasichemical approach . the quantity @xmath61 is the change in the free energy due to the local ion solvent interaction upon coupling the gas - phase cluster with the bulk medium . it accounts for the energetics associated with the relaxation of the cluster in the presence of the medium . utilizing gas phase geometries completely ignores this effect . since the configuration of clusters with a high coordination state can be expected to be more sensitive to the presence of the bulk medium than configurations with lower coordination states , @xmath61 is expected to be larger for higher coordination states than for lower ones . the quantity @xmath62 accounts for the interaction of the bulk medium with the ion when there are @xmath11 solvent ligands inside the coordination shell . since the configuration of the cluster will change in the presence of the bulk medium , this quantity is also inadequately described when we do not allow the gas - phase cluster to relax . we do expect , however , that since only long range ion - solvent interactions contribute to @xmath62 , this quantity can be well captured by dielectric models of hydration . the final term is the hydration free energy of an @xmath11-water cluster without the ion . it consists of packing interactions which are also typically ignored in the _ primitive _ quasichemical approach . liquid water and ion water systems are studied under nvt conditions using metropolis monte carlo simulations@xcite . the cubic simulation cell comprises 306 water molecules for the pure water system ; ion - water system consists of an additional ion which is held fixed at the center . the box volume ( @xmath63 ) is adjusted such that the number density in each case is 33.33 nm@xmath64 . water is modeled with the spc / e potential@xcite . ion parameters and the magnitude of coordination radii are taken from our earlier study@xcite . electrostatic interactions are modeled by the generalized reaction field ( grf ) approach@xcite . both lennard - jones and electrostatic interactions were truncated at @xmath65 . as in our earlier work@xcite , na@xmath1 , k@xmath1 , and f@xmath3 ions were studied . since the trends for f@xmath3 follow those of the smaller na@xmath1 ion , for clarity we only present results comparing na@xmath1 and k@xmath1 . the parameters for the ions were obtained from ref . . simulations are carried out for 6@xmath6610@xmath67 sweeps . each sweep consists of one attempted translation or rotation of each water molecule . the first 3@xmath6610@xmath67 are set aside for equilibration . during the equilibration phase the maximum angular deflection and linear displacement of the water molecule is optimized to yield an acceptance ratio of 0.3 . the optimized values are then held fixed for the next 3@xmath6610@xmath67 sweeps . configurations of the system are saved every 10 sweeps for analysis . a similar simulation strategy was used for simulating ion - water and neat water clusters in the absence of the bulk medium . the solvent ligands comprising the cluster are restricted to a sphere of radius @xmath4 . clusters were simulated for 6@xmath6610@xmath68 sweeps with the first 3@xmath6610@xmath68 sweeps used for equilibration and the next 3@xmath6610@xmath68 sweeps used for production . during the production phase configurations of the clusters were saved every 10 sweeps for analysis . we calculate @xmath69 , the free energy of inserting an ion within a cluster in presence of the external solvent , for @xmath70 coordination states . for the ions studied here , the local chemical contribution @xmath71 is fully accounted for by this coordination number@xcite and thus these are the states of most interest . @xmath69 is obtained indirectly by calculating the response of the medium , @xmath62 , and subtracting it from @xmath39 , the hydration free energy of the ion in the @xmath40 coordination state ( eq . [ eq : mundecomp ] ) . values of @xmath39 were taken from our earlier work @xcite . to obtain @xmath62 , we simulate an ion water system in which the number of solvent molecules within the coordination shell was held at @xmath11 . further , solvent molecules within the coordination sphere are fully - coupled with the ion , but the ion interacts with the bulk medium at various fractional charge states @xmath72 , where @xmath73 is the charge on the ion and @xmath74 . ion - solvent pair correlation function was obtained from the @xmath75 simulation . the electrostatic contribution to @xmath62 is determined by integrating the average potential , @xmath76 , at the centre of the ion due to the bulk medium . @xmath72 is the charge seen by the bulk water molecules . ( we emphasize that the water molecules inside the coordination sphere always see a charge of @xmath73 . ) the electrostatic contribution to @xmath62 is approximated by the two point gauss - legendre quadrature @xmath77 which is exact to fourth order in perturbation theory@xcite . for obtaining the van der waals contribution to @xmath62 , we determine the distribution of ion - bulk interaction energies from the @xmath78 simulation . the van der waals contribution to @xmath62 is then obtained by approximating this distribution by a gaussian @xcite within the inverse form of the potential distribution theorem . ion - water clusters within the coordination sphere were studied at different fractional charge states to obtain the electrostatic contribution to the free energy of forming an ion - water cluster when there is no bulk medium outside the coordination sphere . ( ion - solvent pair correlation function was obtained from the @xmath75 simulation . ) to calculate the van der waals contribution to the free energy , we first obtain the distribution of ion - solvent interaction energies for the neutral ion . we obtain the uncoupled binding energy distribution by performing test particle insertions of the neutral ion in neat water clusters . the vdw contribution is then obtained by histogram overlap @xcite . for hydrated ions , an implicit assumption in using gas - phase clusters to calculate solution phase thermodynamic properties is that the average distribution of water molecules in the solution phase and gas phase clusters are similar . examining the ion - water pair correlation ( fig . [ fg : grkw6 ] ) does validate this assumption , but _ only _ for the low coordination states . , of water around k@xmath1 in the presence ( aq ) and absence ( g ) of the external medium for the @xmath80}_6 $ ] cluster . the radius of the coordination sphere @xmath4 = 3.7 . the pair correlation in presence of the medium shows enhanced structure relative to that for the cluster in the absence of the bulk . this suggests that the external medium promotes a better packing of solvent around k@xmath1 . * inset * : the distribution , @xmath81 , of the farthest water molecule from k@xmath1 . observe that in the absence of the bulk medium , this water molecule is closer to the boundary of the cluster . ] figure [ fg : grkw6 ] shows that the presence of the medium causes the cluster to be better packed , resulting in a somewhat sharper pair correlation for the cluster in the bulk than in its absence . as anticipated , the effect of the medium is most pronounced for the water molecule that is farthest from the ion ( fig . [ fg : grkw6 ] , inset ) . the distribution of the water molecules comprising the @xmath82 coordination states is insensitive to the presence of the external medium ( data not shown ) , a feature that is in accordance with our earlier @xcite finding that there is a core group of water molecules with which the ion interacts strongly enough that the effect of the bulk medium on these water molecules is small . the higher coordination states are however influenced by the bulk medium as well . for the cluster with no bulk medium outside , the 6@xmath83 water molecule is closer to the boundary of the cluster , a position that also allows this water molecule to associate with the remaining water molecules in the cluster rather than with the ion . similar incomplete shell effects , where the second hydration shell starts to form before the first shell is complete , have been inferred on the basis of experimental studies on gas phase clusters with water@xcite or ammonia@xcite as solvent ligands . figure [ fg : ions_ww ] shows that the bulk can stabilize configurations of the water molecules that would be unfavorable otherwise . the average excess internal energy @xmath84 of the @xmath11-water molecules in the coordination sphere for the higher coordination states , relative to clusters in the absence of the bulk medium , is higher for the cluster that is extracted from simulations including the bulk . further , @xmath84 is relatively insensitive to the presence of the bulk for the low coordination states . thus we find that the medium promotes better packing of water molecules around the ion by stabilizing configurations of the cluster which are otherwise unfavorable . , of ion - water clusters for different coordination states . for curves denoted by ` aq ' ( filled symbols ) , @xmath84 is obtained by extracting the cluster from simulations in the presence of the bulk medium . curves denoted by ` g ' ( open symbols ) correspond to clusters simulated in the absence of a bulk . the coordination radius @xmath4 = 3.7 . for higher coordination states , the average excess internal energy is lower in the case of a cluster without a bulk medium outside . ] @xmath38 , the free energy of forming an ion plus @xmath11-water molecule cluster comprises a local contribution , @xmath69 , and the response of the bulk medium , @xmath85 ( eq . [ eq : knbyknt1 ] ) . we next consider each of these contributions separately . figure [ fg : xi_ions ] shows that @xmath69 is insensitive to the presence of the medium for small clusters ( @xmath86 ) . however for larger clusters ( @xmath87 ) , not having a bulk medium to stabilize the cluster leads to a higher free energy . the difference , @xmath61 is always negative . thus ignoring this difference while implementing cluster variation will predict a less favorable contribution due to that coordination state : in effect , the probability of observing these higher coordination states will be predicted to be lower than when the difference is included . , to the free energy of forming an ion plus @xmath11-water molecule cluster . the filled symbols , @xmath69 , are results in the presence of the bulk ( aq ) and the unfilled symbols , @xmath88 , in its absence ( g ) . the radius of the coordination sphere is @xmath89 . for @xmath86 , @xmath69 is insensitive to the presence of the bulk medium outside the coordination shell , whereas for @xmath87 , @xmath90 is more favorable with the medium than without . ] , of forming na@xmath1 plus @xmath11-water clusters . * bottom * : the long - range contribution , @xmath91 , to the free energy of forming na@xmath1 plus @xmath11-water clusters . note that increasing the radius @xmath4 of the coordination sphere does not greatly affect @xmath69 for @xmath92 but increases it for @xmath87 . on the same scale , the response of the medium is more pronounced for the same change in @xmath4 and it tends to become insensitive to the presence of the coordinating solvents for large @xmath4.,title="fig : " ] , of forming na@xmath1 plus @xmath11-water clusters . * bottom * : the long - range contribution , @xmath91 , to the free energy of forming na@xmath1 plus @xmath11-water clusters . note that increasing the radius @xmath4 of the coordination sphere does not greatly affect @xmath69 for @xmath92 but increases it for @xmath87 . on the same scale , the response of the medium is more pronounced for the same change in @xmath4 and it tends to become insensitive to the presence of the coordinating solvents for large @xmath4.,title="fig : " ] our previous work showed that increasing the coordination radius @xmath4 decreases the chemical term @xmath38 . intuitively , we expect a more favorable clustering free energy with increasing coordination radius because the ion can be better accommodated by the coordination shell solvent molecules , that is the local contribution becomes favorable and we expect @xmath38 to decrease because @xmath69 is expected to decrease . however , we find that increasing the coordination radius has no effect on the local contribution , @xmath69 , for small ( @xmath92 ) clusters and _ becomes marginally unfavorable for larger @xmath87 clusters _ ( fig [ fg : na_cn ] , top ) . the observed decrease in @xmath38 with increasing @xmath4 is in fact found to arise from a increasingly favorable medium response : @xmath91 decreases as @xmath4 increases ( fig [ fg : na_cn ] , bottom ) . this feature emerges because the coordination shell solvent becomes insensitive to the presence of the bulk for large coordination radii . it is this feature that compensates for the increase in @xmath69 and leads to the observed decrease in @xmath38 . figure [ fg : clusv ] shows the local chemical contribution ( eq . [ eq : lnx0 ] ) to the hydration free energy ; a maximum term approximation has been used . ) , @xmath93 . for k@xmath1 , neglecting the bulk medium in the local ion water clustering reaction leads to the identification of the @xmath94 coordination state as the optimal ( or most probable ) coordination state . account for the medium indicates that @xmath95 is the most probable coordination state ( cf . fig . 7 in ref . ) . for na@xmath1 , in the presence ( absence ) of the medium @xmath95 ( @xmath96 ) states are indicated to be optimal ( cf . fig . 3 in ref . ) . ] note that eq . [ eq : lnx0 ] can be rewritten as @xcite @xmath97 where @xmath98 is the probability of observing @xmath11 solvent molecules in the coordination sphere in the absence of the solute . thus the maximum term approximation is simply @xmath99 . when the bulk medium is present , @xmath69 defines the local contribution to @xmath38 and when it is absent , @xmath88 defines the local contribution . in both cases , we use the same long - range contribution @xmath91 to @xmath38 . for k@xmath1 , fig . [ fg : clusv ] shows that neglecting the role of the bulk medium in the local ion - water clustering shifts the predicted most probable coordination state to @xmath94 , whereas the value obtained in simulations is @xmath95 for @xmath100 : within the maximum term approximation , @xmath101 is a minimum for @xmath94 ( @xmath95 ) in the absence ( presence ) of the medium . for na@xmath1 , neglecting the medium suggests an optimal coordination state of @xmath96 , whereas in the presence of the bulk , a value of @xmath102 is predicted . the value obtained in simulations@xcite is @xmath102 . for both na@xmath1 and k@xmath1 , any discrepancy between the optimal coordination predicted using cluster variation and those observed in simulations is well within the uncertainty in the calculations . earlier@xcite we had defined dominant hydration structures on the basis of how an increment in the coordination number contributes to @xmath71 , the local chemical contribution to hydration . if in going from @xmath103 , the contribution to @xmath71 ( eq . [ eq : lnx0 ] ) were only of the order of thermal energies ( and substantially smaller than the contribution obtained in going from @xmath104 ) , then the @xmath11-coordinate state was regarded as dominant . since this definition is closely tied to the local interactions , it is expected to be insensitive to the definition of any chemically reasonable coordination radii . [ fg : clusv ] shows that past @xmath96 for both na@xmath1 and k@xmath1 , the change in @xmath71 is only modest . on this basis we would conclude that for the potential model used here , @xmath96 is the dominant coordination state for na@xmath1 and k@xmath1 , as was found earlier @xcite . effort has focused on using thermochemical data obtained either experimentally or from quantum chemical calculations from gas phase clusters to estimate single ion hydration free energies . an assumption often implicit in these studies is that the configurations of the clusters in the gas phase are similar to their aqueous counterparts . our results show that this assumption is only true for small cluster sizes . for the na@xmath1 , k@xmath1 , and f@xmath3 ions considered here , clusters with less than three water molecules satisfy this requirement . for larger clusters , the external medium starts to influence the local ion - water interaction . the external medium stabilizes ion - water cluster configurations that are _ better packed _ around the ion , such that the local ion - solvent interaction contribution to the free energy of cluster formation is more favorable in the presence of the medium than without . for higher coordination states of the ion , and , more generally , for the coordination structure and thermodynamics of soft ions , our results show that the medium will play a sizable role in the coordination structure and thermodynamics of the hydrated ion . for these cases , accounting for the molecular characteristics of the bulk medium on the ion - water cluster is important in inferring the structure and thermodynamics of the hydrated ion . in ref . , the long - range contribution to @xmath56 was erroneously left out from the equations . but these contributions were all correctly considered in the numerical work and hence no result is affected . since presenting these corrections also provides a helpful alternative perspective on the equilibrium constants appearing in the quasichemical theory , we note those corrections here . where @xmath106 ( eq . [ eq : knbyknt2 ] ) and @xmath107 is the configurational partition function of a water molecules . @xmath108 is the potential energy of the ion plus @xmath11 solvent molecules within the coordination volume and @xmath109 is the potential energy of the solvent molecules in the absence of the ions . @xmath110 is the field of the bulk medium on the @xmath11 solvent ligands in the coordination volume in the absence of the ion . [ eq : error ] includes the factor @xmath111 that was missing in eq . 10a in ref . . based on the above definitions , we can show that @xmath112 where once again the factor @xmath111 was left out in ref . . @xmath113 is the interaction energy of the ion with the @xmath11 solvent molecules within the coordination sphere . note that @xmath114 ( eq . [ eq : knbyknt1 ] ) . [ eq : winsmall ] includes the factor @xmath111 that was missing in eq . 11 , ref .
|
thermochemistry of gas - phase ion - water clusters together with estimates of the hydration free energy of the clusters and the water ligands are used to calculate the hydration free energy of the ion .
often the hydration calculations use a continuum model of the solvent . the primitive quasichemical approximation to the quasichemical theory
provides a transparent framework to anchor such efforts .
here we evaluate the approximations inherent in the primitive quasichemical approach and elucidate the different roles of the bulk medium .
we find that the bulk medium can stabilize configurations of the cluster that are usually not observed in the gas phase , while also simultaneously lowering the excess chemical potential of the ion .
this effect is more pronounced for soft ions . since the coordination number that minimizes the excess chemical potential of the ion is identified as the optimal or most probable coordination number , for such soft ions , the optimum cluster size and the hydration thermodynamics obtained without account of the bulk medium on the ion - water clustering reaction can be different from those observed in simulations of the aqueous ion .
the ideas presented in this work are expected to be relevant to experimental studies that translate thermochemistry of ion - water clusters to the thermodynamics of the hydrated ion and to evolving theoretical approaches that combine high - level calculations on clusters with coarse - grained models of the medium .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the intrinsic , three - dimensional ( hereafter 3-d ) shape of clusters of galaxies is an important cosmological probe . the structure of galaxy clusters is sensitive to the mass density in the universe , so knowledge of this structure can help in discriminating between different cosmological models . it has long been clear that the formation epoch of galaxy clusters strongly depends on the matter density parameter of the universe @xcite . the growth of structure in a high - matter - density universe is expected to continue to the present day , whereas in a low density universe the fraction of recently formed clusters , which are more likely to have substructure , is lower . therefore , a sub - critical value of the density parameter @xmath1 favors clusters with steeper density profiles and rounder isodensity contours . less dramatically , a cosmological constant also delays the formation epoch of clusters , favoring the presence of structural irregularity @xcite . + an accurate knowledge of intrinsic cluster shape is also required to constrain structure formation models via observations of clusters . the asphericity of dark halos affects the inferred central mass density of clusters , the predicted frequency of gravitational arcs , nonlinear clustering ( especially high - order clustering statistics ) and dynamics of galactic satellites ( see @xcite and references therein ) . + asphericity in the gas density distribution of clusters of galaxies is crucial in modeling x - ray morphologies and in using clusters as cosmological tools . @xcite . assumed cluster shape strongly affects absolute distances obtained from x - ray / sunyaev - zeldovich ( sz ) measurements , as well as relative distances obtained from baryon fraction constraints @xcite . finally , all cluster mass measurements derived from x - ray and dynamical observations are sensitive to the assumptions about cluster symmetry . + of course , only the two - dimensional ( 2-d ) projected properties of clusters can be observed . the question of how to deproject observed images is a well - posed inversion problem that has been studied by many authors @xcite . since information is lost in the process of projection it is in general impossible to derive the intrinsic 3-d shape of an astronomical object from a single observation . to some extent , however , one can overcome this degeneracy by combining observations in different wavelengths . for example , @xcite introduced a model - independent method of image deprojection . this inversion method uses x - ray , radio and weak lensing maps to infer the underlying 3-d structure for an axially symmetric distribution . @xcite proposed a parameter - free algorithm for the deprojection of observed two dimensional cluster images , again using weak lensing , x - ray surface brightness and sz imaging . the 3-d gravitational potential was assumed to be axially symmetric and the inclination angle was required as an input parameter . strategies for determining the orientation have been also discussed . @xcite proposed a method that , with a perturbative approach and with the aid of sz and weak lensing data , could predict the cluster x - ray emissivity without resolving the full 3-d structure of the cluster . the degeneracy between the distance to galaxy clusters and the elongation of the cluster along the line of sight ( l.o.s . ) was thoroughly discussed by @xcite . they introduced a specific method for finding the intrinsic 3-d shape of triaxial cluster and , at the same time , measuring the distance to the cluster corrected for asphericity , so providing an unbiased estimate of the hubble constant @xmath2 . @xcite recently proposed a theoretical method to reconstruct the shape of triaxial dark matter halos using x - ray and sz data . the hubble constant and the projection angle of one principal axis of the cluster on the plane of the sky being independently known , they constructed a numerical algorithm to determine the halo eccentricities and orientation . however , neither @xcite nor @xcite apply their method to real data . + in this paper we focus on x - ray surface brightness observations and sz temperature decrement measurements . we show how the intrinsic 3-d shape of a cluster of galaxies can be determined through joint analyses of these data , given an assumed cosmology . we constrain the triaxial structure of a sample of observed clusters of galaxies with measured x - ray and sz maps . to break the degeneracy between shape and cosmology , we adopt cosmological parameters which have been relatively well - determined from measurements of the cosmic microwave background ( cmb ) anisotropy , type ia supernovae and the spatial distribution of galaxies . we also show how , if multiply - imaging gravitational lens systems are observed , a joint analysis of strong lensing , x - rays and sz data allows a determination of both the 3-d shape of a cluster and the geometrical properties of the universe . + the paper is organized as follows . the basic dependencies of cluster x - ray emission and the sze on geometry are reviewed in [ sec : multi_wave ] . in [ sec : combin_datasets ] , we show how to reconstruct the 3-d cluster structure from these data , presuming cosmological parameters to be known . in passing we note how the addition of suitable strong gravitational lensing data can constrain the cosmological parameters as well , although we do not impose lensing constraints in this paper . we then turn to face the data . our cluster sample is introduced in [ sec : data_samp ] , and in [ sec : morph_2d ] , we present 2-d x - ray surface brightness parameters for each sample member . the triaxial structure of the clusters is then estimated and analyzed in [ sec : tria ] . [ sec : disc ] is devoted to a summary and discussion of the results . in appendix [ sec : triaxial ] , we provide details on the triaxial ellipsoidal @xmath3-model , used to describe the intra - cluster gas distribution , while appendix [ sec : inclination ] is devoted to a discussion of the consequences of our assumption of clusters being triaxial ellipsoids aligned along the line of sight . in appendix [ sec : lensing ] the identifications of multiple sets of images of background galaxies in strong lensing events is discussed . throughout this paper , unless otherwise stated , we quote errors at the @xmath4 confidence level . in this section , we summarize the relationships between sz and x - ray observables , on the one hand , and cluster shape and distance on the other . the gravitational potential wells of galaxy clusters contain plasma at temperatures of about @xmath5-@xmath6 . cmb photons that pass through a cluster interact with the energetic electrons of its hot intra - cluster medium ( icm ) through inverse compton scattering , with a probability @xmath7 . this interaction causes a small distortion in the cmb spectrum , known as the sunyaev - zeldovich effect ( sze ) @xcite , which is proportional to the electron pressure integrated along the l.o.s . , i.e. to the first power of the plasma density . the measured temperature decrement @xmath8 of the cmb is given by : @xmath9 where @xmath10 is the temperature of the icm , @xmath11 the boltzmann constant , @xmath12k is the temperature of the cmb , @xmath13 the thompson cross section , @xmath14 the electron mass , @xmath15 the speed of light in vacuum and @xmath16 accounts for frequency shift and relativistic corrections . + if we assume that the icm is described by an isothermal triaxial @xmath3-model distribution , substituting eq . ( [ eq : tri5 ] ) into ( [ eq : sze1 ] ) with @xmath17 , we obtain : @xmath18 where @xmath19 is the central temperature decrement which includes all the physical constants and the terms resulting from the l.o.s . integration @xmath20 with : @xmath21}{\gamma \left[3 \alpha\right]}.\ ] ] @xmath22 is the angular diameter distance to the cluster , @xmath23 is the projected angular position ( on the plane of the sky ) of the intrinsic orthogonal coordinate @xmath24 , @xmath25 is a function of the cluster shape and orientation ( eq . [ eq : tri3 ] ) , @xmath26 is the axial ratio of the major to the minor axes of the observed projected isophotes and @xmath27 the projection on the plane of the sky ( p.o.s . ) of the intrinsic angular core radius ( eq . [ eq : tri6 ] ) . in a friedmann - lematre - robertson - walker universe filled with pressure - less matter and with a cosmological constant , the angular diameter distance between an observer at a redshift @xmath28 and a source at @xmath29 is : @xmath30 with @xmath31 where @xmath2 , @xmath32 and @xmath33 are the hubble parameter , the normalized energy density of pressure - less matter and the reduced cosmological constant at @xmath34 , respectively . @xmath35 is given by @xmath36 , and sinn is defined as being @xmath37 when @xmath38 , @xmath39 when @xmath40 , and as the identity when @xmath41 . a more general expression of the angular diameter distance , also accounting for dark energy and inhomogeneity in matter distribution , can be found in @xcite . cluster x - ray emission is due to bremsstrahlung and line radiation resulting from electron - ion collisions ; the x - ray surface brightness @xmath42 is proportional to the integral along the l.o.s . of the square of the electron density : @xmath43 where @xmath44 is the x - ray cooling function of the icm in the cluster rest frame . substituting eq . ( [ eq : tri5 ] ) into ( [ eq : sxb0 ] ) with @xmath45 , we get : @xmath46 where the central surface brightness @xmath47 reads : @xmath48 @xmath49 is the molecular weight given by : @xmath50 . here we discuss how 2-d sze and x - ray maps of a cluster can be used to constrain its 3-d shape . we follow a parametric approach . we model the cluster using an isothermal , triaxial @xmath3 profile , and adopt a concordance model for the cosmological distance - redshift relationships . details of the cluster model are given in appendix [ sec : triaxial ] . + this model has long been used to describe the electron distributions of galaxy clusters . it was originally introduced specifically for dynamically relaxed , isothermal clusters @xcite , but it was then observed to fit the x - ray emission of most galaxy clusters reasonably well . a serious drawback of this model is that electron density profiles with extreme axial ratios lead either to unlikely total mass density distributions , i.e. dumbbell shaped clusters , or to regions with unphysical ( negative ) density . nevertheless , we believe its extreme versatility makes it a useful tool for our purposes . + for an ellipsoidal distribution , the 3-d shape of a cluster is described by two axis ratios , @xmath51 and @xmath52 , and the orientation of the cluster is fixed by three euler angles , @xmath53 and @xmath54 . as shown in eqs . ( [ eq : tri0 ] , [ eq : gl2 ] ) , in our model the density profile of a cluster is characterized by three additional parameters : the central density @xmath55 , the slope @xmath3 and a core radius , @xmath56 . under the hypothesis of isothermal icm , a single value , @xmath10 , characterizes the temperature profile of the cluster . in all , nine parameters describe the cluster . + as discussed in the previous section , the cosmological dependence of the model enters through the luminosity - redshift relationship . for a flat model universe , this relationship is in turn determined by two parameters : the hubble constant , @xmath2 , and the matter density @xmath32 . + a projected axis ratio , @xmath26 , and an orientation angle , @xmath57 , characterize a family of ellipses in the p.o.s . derived from the 2-d projection of 3-d ellipsoids . by fitting an elliptical profile to the x - ray and/or sze data , these two parameters can ( in principle ) be measured . two other observables , the slope @xmath3 of the profile and the projected core radius @xmath27 can also be determined from data . two independent geometrical constraints relate 2-d and 3-d quantities ( eqs . [ eq : tri4e ] , [ eq : tri4f ] ) . + so far we have discussed only quantities derivable from spatial distributions . besides these , the cluster central electron density and the temperature of the icm can be inferred from x - ray observations with sufficient energy resolution . the observed values of the central temperature decrement , @xmath19 in eq . ( [ eq : sze3 ] ) and of the central surface brightness , @xmath47 in eq . ( [ eq : sxb2 ] ) , provide two further constraints . if some assumption is made on the orientation of the cluster , with eight independent equations and eight unknown physical parameters a full determination of the cluster shape can be obtained . if a rotational ellipsoidal morphology is chosen , a lower number of parameters is needed to describe the three - dimensional shape of clusters ; in this case no additional assumption on the inclination is required allowing a full determination of the cluster shape and orientation . this case is treated in details in a subsequent paper @xcite . we wish to point that , even though we do not do so in this work , strong lensing data can be combined with the x - ray and sze observations to break the degeneracy between the intrinsic shape of the lensing cluster and the cosmological parameters . if this were done , one could obtain simultaneous constraints on the cluster parameters and on the cosmology . + in particular , each set of strong gravitational images identified in a cluster strongly constrains the mass distribution of the lens . the convergence @xmath58 depends on the cosmology only through the ratio of distances @xmath59 . therefore @xmath58 depends only the cosmological density parameters @xmath60 , and not on the hubble constant @xmath2 . the value of @xmath61 changes according to the redshift @xmath62 of the lensed source . image systems produced by sources at different redshifts probe independent values of the ratio @xmath63 . + each image system provides a constraint on central value of the convergence @xmath64 ( eq . [ eq : gl6b ] ) . in turn , if both the positions and the redshift of a multiple image system are known , each measured value of @xmath61 provides , through eq . ( [ eq : gl6b ] ) , a further independent constraint on the cosmological energy densities . each multiply - imaged source provides an independent constraint which relates the cosmological parameters @xmath60 to the 3-d shape and orientation of the cluster ( @xmath65 ) . with a sufficient number of image systems , then , a measure of both the intrinsic shape and orientation of the cluster and a simultaneous estimate of all cosmological parameters involved can therefore be performed . it is of course well known that the angular diameter distance to a spherically symmetric cluster can be be inferred from microwave decrement and x - ray data . the angular diameter distance enters the sze and the x - ray emission through a characteristic length - scale of the cluster along the l.o.s . sze and x - ray emission depend differently on the density of icm , and therefore also on the assumed cosmology . a joint analysis of sze measurements and x - ray imaging observations , together with the assumption of spherical symmetry , thus can yield the distance to the cluster @xcite . specifically , one can solve eqs . ( [ eq : sze3 ] ) and ( [ eq : sxb2 ] ) for the angular diameter distance @xmath22 , by eliminating @xmath66 . + more generally , for a triaxial cluster the inferred angular diameter distance takes the form : @xmath67 where @xmath68 is an experimental quantity given by : @xmath69 under the assumption of spherical symmetry , the 3-d morphology of the cluster is completely known : @xmath70 , @xmath71 and the observed major core radius @xmath72 reduces to @xmath73 . hence eq . ( [ eq : dis2 ] ) becomes : @xmath74 and the cluster angular diameter distance can therefore be obtained directly from eq . ( [ eq : obl7 ] ) . the standard approach in the past decades has been to take advantage of this possibility to estimate @xmath22 under the assumption of spherical symmetry , in order to constrain the underlying cosmology . since in fact it is also true that @xmath75 , through eq . ( [ eq : crit3 ] ) an estimate of @xmath2 can be obtained if @xmath76 and @xmath33 are known from independent observations @xcite . + the same approach clearly can not be applied when the assumption of spherical symmetry is relaxed and clusters are considered as more general triaxial systems . in this case an estimate of the axis ratios , shape and orientation parameters is required before @xmath22 can be computed . conversely , if @xmath22 is known from the redshift and prior knowledge of the cosmology , then the x - ray and sze data can be used to constrain the 3-d morphology of the cluster . in this paper we will follow this latter approach . we assume the values of @xmath60 and of @xmath2 to be known ; @xmath77 can be then determined through eq . ( [ eq : crit3 ] ) . we will then use eq . ( [ eq : dis2 ] ) to infer the 3-d morphology of a sample of galaxy clusters . we believe the cosmological distance scale is now known with sufficient accuracy to warrant our approach . an impressive body of evidence from cmb anisotropy , type ia supernovae , galaxy clustering , large - scale structure , and the ly@xmath78 forest @xcite are consistent with a the picture of a universe with sub - critical cold dark matter energy density and with two - thirds of the critical density being in the form of dark energy . @xcite combine the three dimensional power spectrum from over @xmath79 galaxies in the sloan digital sky survey with the first - year wilkinson microwave anisotropy probe ( wmap ) data @xcite to measure cosmological parameters . their results are consistent with a flat ( @xmath80 ) cosmological model with @xmath81 , @xmath82 and with a non - zero cosmological constant . thanks to the high precision to which cosmological parameters are known , we are able to constrain measurements of @xmath77 for the sample objects within a @xmath83 error . for the remainder of this paper , we will assume that every cluster is triaxial , with one principal axis aligned along the l.o.s . ( see [ sec : combin_datasets ] ) . in appendix [ sec : inclination ] we show that the magnitude of the systematic error in inferred elongation parameters cause by such assumption is small compared to the uncertainties arising from the observational errors . such a straightforward assumption also leads to an extremely simple formalism to describe the resulting three - dimensional shape of clusters , reducing the errors caused by the uncertainties in the observational data . the assumption that the cluster is aligned along the l.o.s . implies : @xmath84 ; ( see appendix [ sec : triaxial ] ) . we label axes so that major axis is parallel to the @xmath85 ; then the projected axial ratio and core radius are : @xmath86 and @xmath87 . the angular diameter distance becomes : @xmath88 we now introduce the elongation @xmath89 , defined as the ratio of the radius of the cluster along the l.o.s . to its major axis in the p.o.s . , @xmath90 spherical clusters have the same radius along the l.o.s . and in the p.o.s . and for them @xmath91 . clusters which are instead more or less elongated along the l.o.s . than in the p.o.s . will have values of @xmath92 or @xmath93 , respectively . we now apply the formalism described in [ sec : application ] to a sample of galaxy clusters to infer new information about the extent of the clusters along the l.o.s . + we use two samples of clusters for which combined x - ray and sz analysis has already been reported . the sample discussed by @xcite consists of @xmath94 x - ray selected clusters with @xmath95 and @xmath96 and @xmath97 and for which high s / n detections of sze , high - s / n x - ray imaging and electron temperatures were available . to these we add the sample of @xcite , which contains @xmath98 clusters from x - ray - flux - limited sample of @xcite . details on the completeness of the latter subsample are given by @xcite . + basic data for our @xmath0 clusters , including previously published redshift , plasma temperature and microwave decrement information @xcite are presented in table [ tab : sample ] . lrrr ms 1137.5 + 6625 & @xmath99 & @xmath100 & @xmath101 + ms 0451.6 - 0305 & @xmath102 & @xmath103 & @xmath104 + cl 0016 + 1609 & @xmath105 & @xmath106 & @xmath107 + rxj1347.5 - 1145 & @xmath108 & @xmath109 & @xmath110 + a 370 & @xmath111 & @xmath112 & @xmath113 + ms 1358.4 + 6245 & @xmath114 & @xmath115 & @xmath116 + a 1995 & @xmath117 & @xmath118 & @xmath119 + a 611 & @xmath120 & @xmath121 & @xmath122 + a 697 & @xmath123 & @xmath124 & @xmath125 + a 1835 & @xmath126 & @xmath127 & @xmath128 + a 2261 & @xmath129 & @xmath130 & @xmath131 + a 773 & @xmath132 & @xmath133 & @xmath134 + a 2163 & @xmath135 & @xmath136 & @xmath137 + a 520 & @xmath135 & @xmath138 & @xmath139 + a 1689 & @xmath140 & @xmath141 & @xmath142 + a 665 & @xmath143 & @xmath144 & @xmath145 + a 2218 & @xmath146 & @xmath147 & @xmath148 + a 1413 & @xmath149 & @xmath150 & @xmath151 + a 2142 & @xmath152 & @xmath153 & @xmath154 + a 478 & @xmath155 & @xmath156 & @xmath157 + a 1651 & @xmath158 & @xmath159 & @xmath160 + a 401 & @xmath161 & @xmath162 & @xmath163 + a 399 & @xmath164 & @xmath165 & @xmath166 + a 2256 & @xmath167 & @xmath168 & @xmath169 + a 1656 & @xmath170 & @xmath171 & @xmath172 + [ tab : sample ] _ chandra _ and _ xmm _ observations of clusters in the past few years have shown that in general clusters exhibit elliptical surface brightness maps , and so can not be spherically symmetric . in order to obtain a uniform set of x - ray observables for our sample objects , we have re - analyzed archival x - ray data for each of them . we have used _ chandra _ and/or _ xmm _ data for all objects except a 520 , for which only _ rosat _ data are available . + we modeled the emission of all clusters in the @xmath173 band . pixel values of all detected point sources were replaced with values interpolated from the surrounding background regions ; the _ ciao _ tools _ wavdetect _ and _ dmfilth _ were used for this purpose . + using the _ sherpa _ software , we fitted the cluster surface brightness to elliptical 2-d @xmath3-models ( see eq . [ eq : sxb1 ] ) . results are listed in table [ tab:2d_fit ] . fitted models from _ chandra _ and _ xmm _ observations are roughly consistent . + the 25 clusters have a weighted median projected axis ratio of @xmath174 , in very good agreement with the value of @xmath175 obtained by @xcite from _ einstein _ data of a lower - redshift sample of 65 objects . only six of the @xmath0 clusters have a projection in the p.o.s . close to be circular ( @xmath176 ) . lrrrrrrc ms 1137.5 + 6625 & @xmath177 & @xmath178 & @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 + ms 0451.6 - 0305 & @xmath184 & @xmath185 & @xmath186 & @xmath187 & @xmath188 & @xmath189 & @xmath183 + cl 0016 + 1609 & @xmath190 & @xmath191 & @xmath192 & @xmath193 & @xmath194 & @xmath195 & @xmath183 + cl 0016 + 1609 & @xmath196 & @xmath197 & @xmath198 & @xmath199 & @xmath200 & @xmath201 & @xmath202 + rxj1347.5 - 1145 & @xmath203 & @xmath204 & @xmath205 & @xmath206 & @xmath207 & @xmath208 & @xmath183 + a 370 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath183 + ms 1358.4 + 6245 & @xmath215 & @xmath216 & @xmath217 & @xmath218 & @xmath219 & @xmath220 & @xmath183 + a 1995 & @xmath221 & @xmath222 & @xmath223 & @xmath224 & @xmath225 & @xmath226 & @xmath183 + a 611 & @xmath227 & @xmath228 & @xmath229 & @xmath230 & @xmath231 & @xmath232 & @xmath183 + a 697 & @xmath233 & @xmath234 & @xmath235 & @xmath236 & @xmath237 & @xmath238 & @xmath183 + a 1835 & @xmath239 & @xmath240 & @xmath241 & @xmath242 & @xmath243 & @xmath244 & @xmath183 + a 2261 & @xmath245 & @xmath246 & @xmath247 & @xmath248 & @xmath249 & @xmath250 & @xmath183 + a 773 & @xmath251 & @xmath252 & @xmath253 & @xmath254 & @xmath255 & @xmath256 & @xmath183 + a 773 & @xmath257 & @xmath258 & @xmath259 & @xmath260 & @xmath261 & @xmath262 & @xmath202 + a 2163 & @xmath263 & @xmath264 & @xmath265 & @xmath266 & @xmath267 & @xmath268 & @xmath183 + a 2163 & @xmath269 & @xmath270 & @xmath271 & @xmath272 & @xmath273 & @xmath274 & @xmath202 + a 520 & @xmath275 & @xmath276 & @xmath277 & @xmath278 & @xmath279 & @xmath280 & @xmath281 + a 1689 & @xmath282 & @xmath283 & @xmath284 & @xmath285 & @xmath286 & @xmath287 & @xmath183 + a 665 & @xmath288 & @xmath289 & @xmath290 & @xmath291 & @xmath292 & @xmath293 & @xmath183 + a 2218 & @xmath294 & @xmath295 & @xmath296 & @xmath297 & @xmath298 & @xmath299 & @xmath183 + a 2218 & @xmath300 & @xmath301 & @xmath302 & @xmath303 & @xmath304 & @xmath305 & @xmath202 + a 1413 & @xmath306 & @xmath307 & @xmath308 & @xmath309 & @xmath310 & @xmath311 & @xmath183 + a 2142 & @xmath312 & @xmath313 & @xmath314 & @xmath315 & @xmath316 & @xmath317 & @xmath183 + a 478 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & @xmath322 & @xmath323 & @xmath183 + a 1651 & @xmath324 & @xmath325 & @xmath326 & @xmath327 & @xmath328 & @xmath329 & @xmath183 + a 401 & @xmath330 & @xmath331 & @xmath332 & @xmath333 & @xmath334 & @xmath335 & @xmath183 + a 399 & @xmath336 & @xmath337 & @xmath338 & @xmath339 & @xmath340 & @xmath341 & @xmath202 + a 2256 & @xmath342 & @xmath343 & @xmath344 & @xmath345 & @xmath346 & @xmath347 & @xmath202 + a 1656 & @xmath348 & @xmath349 & @xmath350 & @xmath266 & @xmath351 & @xmath352 & @xmath202 + [ tab:2d_fit ] although clusters are rarely circular in projection , some previous joint analyses of x - ray and sze data have assumed spherical symmetry . in order to bound the effects of this simplification , we have also modeled the surface brightness profiles of each sample cluster with a circular @xmath3-model . the choice of circular rather than elliptical @xmath3 model does not affect the resulting of the central surface brightness , as shown in the top panel in fig . [ fig : correzioni ] . for a few clusters the fitted value of the slope @xmath3 differs slightly between circular and elliptical models ( middle panel of fig . [ fig : correzioni ] ) . as would be expected , however , significantly different values for the core radius are obtained with these two models ( bottom panel of fig . [ fig : correzioni ] ) . this behavior has already been noted by @xcite . + therefore , relaxing the assumption of circular projection on the p.o.s . when measuring the angular diameter distance ( eq . [ eq : obl7 ] ) , mainly affects the value of the projected core radius @xmath27 . the bottom panel in fig . [ fig : correzioni ] shows that the core radius obtained using a circular @xmath3-model ( @xmath353 ) is consistently lower ( black squares ) than that obtained from an elliptical model ( @xmath354 ) . @xmath353 can in fact be well approximated by the arithmetic mean of the two semi - axes of the elliptical isophotes in the p.o.s . + the angular diameter distance obtained assuming spherical symmetry ( table [ tab : dc ] ) can therefore , in first approximation , be corrected for the observed ellipticity of the cluster in the p.o.s . multiplying @xmath355 by the correction factor @xmath356 : @xmath357 as shown in the bottom panel of fig . [ fig : correzioni ] , the corrected values of the core radii @xmath358 @xmath359 provide a good approximation to @xmath360 ( gray squares ) . in order to estimate the l.o.s . extent of clusters , then , we need only to obtain values of @xmath68 from the x - ray and sze data ( via eq . [ eq : obl7 ] ) and compare them ( via eq . [ eq : simp3 ] ) to the angular size distance obtained from the measured redshift and our adopted cosmological model . since only the x - ray data are publically available , however , we are unable to jointly fit both sze and x - ray data . for this reason we must rely on published values of central cmb temperature decrement ( @xmath361 ) for our analysis . + a potential difficulty with this approach is that the available values of @xmath361 , from @xcite and @xcite , have been inferred assuming that clusters are circularly symmetric when seen in projection on the sky . while this assumption is quite reasonable given the limited spatial resolution of the data available to these authors , it is not , in general , consistent with the results of our analysis of higher - resolution x - ray data ( see table [ tab:2d_fit ] ) . in the limit of very high spatial resolution sze data , we would expect this inconsistency to have negligible effect on our results , just as we find that with high - resolution x - ray data , the same central x - ray surface brightness is inferred from fits of circular and elliptical models ( see the top panel in fig . [ fig : correzioni ] ) . + we have computed values of the angular diameter distances for all clusters in the sample both under the assumption of spherical symmetry , and relaxing the assumption to a more general triaxial morphology . for the spherical case results obtained modeling the cluster x - ray surface brightness profiles with circular @xmath3-models ( [ sec : sph_vs_ell ] ) were substituted into eq . ( [ eq : obl7 ] ) . for the triaxial case results from the elliptical @xmath3-models were instead used ( [ sec : morph_2d ] ) . the term @xmath362 , which accounts for the frequency shift and also includes relativistic corrections , was computed as described by @xcite . central values of the cmb temperature decrement ( @xmath19 ) were taken from @xcite and @xcite . the resulting values of @xmath363 are listed in table [ tab : dc ] . the largest source of error ( about @xmath364 of the total ) is the uncertainty on the sze measurement of @xmath19 . the second most significant error source is the uncertainty in the x - ray measurement of the intra - cluster plasma temperature ( about @xmath365 ) . both @xmath363 and @xmath355 are plotted in fig . [ fig : dc_vs_z ] . a comparison between the experimental quantities @xmath355 and @xmath363 and the values of @xmath366 , together with their relative errors , highlights the high precision to which the cosmological angular distance is known , compared to the two experimental estimates , in support of our approach of a fixed cosmological model . lrrr ms 1137.5 + 6625 & @xmath367&@xmath368&@xmath369 + ms 0451.6 - 0305 & @xmath370&@xmath371 & @xmath372 + cl 0016 + 1609 & @xmath373&@xmath374 & @xmath375 + rxj1347.5 - 1145 & @xmath376&@xmath377 & @xmath378 + a 370 & @xmath379&@xmath380&@xmath381 + ms 1358.4 + 6245 & @xmath382&@xmath383 & @xmath384 + a 1995 & @xmath385&@xmath386 & @xmath387 + a 611 & @xmath388&@xmath389 & @xmath390 + a 697 & @xmath391&@xmath392 & @xmath393 + a 1835 & @xmath394&@xmath395 & @xmath396 + a 2261 & @xmath397&@xmath398 & @xmath399 + a 773 & @xmath400&@xmath401 & @xmath402 + a 2163 & @xmath403&@xmath404 & @xmath405 + a 520 & @xmath403&@xmath406 & @xmath407 + a 1689 & @xmath408&@xmath409 & @xmath410 + a 665 & @xmath411&@xmath412 & @xmath413 + a 2218 & @xmath414&@xmath415 & @xmath416 + a 1413 & @xmath417&@xmath418 & @xmath419 + a 2142 & @xmath420&@xmath421 & @xmath422 + a 478 & @xmath423&@xmath424 & @xmath425 + a 1651 & @xmath426&@xmath427 & @xmath428 + a 401 & @xmath429&@xmath430 & @xmath431 + a 399 & @xmath432&@xmath433 & @xmath434 + a 2256 & @xmath435&@xmath436 & @xmath437 + a 1656 & @xmath438&@xmath439 & @xmath440 + [ tab : dc ] assuming a general triaxial morphology , the ratio between @xmath441 and @xmath77 provides an estimate of the ratio of the cluster axis along the l.o.s . and the cluster major axis in the p.o.s . we have computed values of @xmath441 for all clusters in the sample ( [ sec : dc_ell ] ) . for each cluster we have then also computed @xmath77 ( eq . [ eq : crit3 ] ) and have then estimated their @xmath89 . resulting values are listed in table [ tab:3d_morph ] . + since the observables in our analysis have asymmetric uncertainties , we apply corrections given by @xcite to obtain estimates of sample mean and standard deviation . + all clusters in our sample were x - ray selected ; x - ray surveys are surface brightness limited . clusters close to the detection limit which are elongated along the l.o.s . will be detected , while the ones which are more extended in the p.o.s . will be missed . if a surface brightness limit is fixed which is far above the detection limit of the survey , the problem should be eliminated . + in both the @xcite and the @xcite samples this `` correction '' limit was applied . our final sample shows in fact only mild signs of preferential elongation of the clusters along the l.o.s . ( see fig . [ fig : isto_e_los ] ) . of the @xmath0 clusters , @xmath442 clusters are in fact more elongated along the l.o.s ( @xmath443 ) , while the remaining @xmath444 clusters are compressed . the mean of the distribution of the elongations is @xmath445 . in presence of likely outliers , the median is a more stable estimator @xcite . the median of the @xmath89 s is @xmath446 . while on average we observe only a very slight preferential elongation of the clusters along the l.o.s . , residual of x - ray selection effects , clusters with extreme axes ratios are still preferentially selected if the elongation lies along the l.o.s . this is a clear example of how deeply x - ray selected cluster samples are affected by morphological and orientation issues . lrr ms 1137.5 + 6625 & @xmath447 & @xmath448 + ms 0451.6 - 0305 & @xmath449 & @xmath450 + cl 0016 + 1609 & @xmath451 & @xmath452 + rxj1347.5 - 1145 & @xmath453 & @xmath454 + a 370 & @xmath455 & @xmath456 + ms 1358.4 + 6245 & @xmath457 & @xmath458 + a 1995 & @xmath459 & @xmath460 + a 611 & @xmath461 & @xmath462 + a 697 & @xmath463 & @xmath464 + a 1835 & @xmath465 & @xmath466 + a 2261 & @xmath467 & @xmath468 + a 773 & @xmath469 & @xmath470 + a 2163 & @xmath471 & @xmath472 + a 520 & @xmath473 & @xmath474 + a 1689 & @xmath475 & @xmath476 + a 665 & @xmath477 & @xmath478 + a 2218 & @xmath479 & @xmath480 + a 1413 & @xmath481 & @xmath482 + a 2142 & @xmath483 & @xmath484 + a 478 & @xmath485 & @xmath486 + a 1651 & @xmath487 & @xmath488 + a 401 & @xmath489 & @xmath490 + a 399 & @xmath491 & @xmath492 + a 2256 & @xmath493 & @xmath494 + a 1656 & @xmath495 & @xmath496 + [ tab:3d_morph ] we can estimate the three ellipsoidal axis lengths ( @xmath497 , @xmath498 and @xmath499 ) from the measured values of @xmath89 and @xmath26 , and from these the ratio of the semi - major to the semi - minor axis , @xmath500 . @xmath500 is an extremely convenient tool to describe the intrinsic shape of a cluster since it allows , without the aid of further parameters , to quantify how far a cluster is from spherical symmetry . for most clusters in the sample , the confidence regions of @xmath497 , @xmath498 and @xmath499 are highly overlapping so that , for example , the upper bound of the 1-@xmath501 interval for the estimate of the ratio between the median and the minor axis , @xmath502 , may be larger than the upper limit of @xmath500 . + to obtain well defined estimates of the errors of the maximum , intermediate and minimum axis ratios for each cluster , assuming the @xmath503 s to be normally distributed , we have obtained @xmath504 random samples from each distribution . we have then selected the maximum , the intermediate and the minimum values of each set of three in order to build the distribution of the maximum , intermediate and minimum axis ratios . we have finally computed the standard deviations of such three distributions , that provide estimates for the errors for the axes ratios . the resulting values of @xmath505 are listed in table [ tab:3d_morph ] and their distribution is shown in fig . [ fig : isto_q_max ] . + @xmath500 has a mean value of @xmath506 , and a median of @xmath507 . this result is consistent with cosmological simulations in which the mean value of the maximum axial ratio ranges from @xmath508 @xcite to @xmath509 @xcite . the intermediate axis ratios , @xmath502 show a median of @xmath510 . at @xmath511 no cluster in the sample can be approximated as spherical ; @xmath512 clusters are spherical at the @xmath513 confidence level . + although our estimates are affected by large errors and the data sample is of modest size , we look for trends in the distribution of the maximum axial ratios . + no correlation is observed between the maximum axial ratio and redshift ( see fig . [ fig : qmax_vs_z ] where the solid and dashed lines represent the weighted and non weighted linear best fit to the data , respectively ) . + a poor correlation is observed also between the maximum axial ratio and the cluster gas temperature . we find at most a weak tendency for hotter clusters to exhibit smaller values of @xmath505 . the linear weighted best fit to the data is : @xmath514 . the trend is plotted in fig . [ fig : qmax_vs_t ] , where the solid and dashed lines represent the weighted and non weighted linear best fit to the data , respectively . the absence of such a correlation may indicate that , in our sample , high cluster temperatures are not predominantly the result of shocks associated with accretion of sub - clusters @xcite , since such accretion events seem likely also to produce departures from spherical morphology . + from the distribution of axial shapes of clusters in our sample , we can estimate the effect that the assumption of spherical symmetry has on the determination of the total cluster mass . if the mass is computed at large distances from the cluster center ( @xmath515 ) the difference between the two models is less than @xmath516 even for the most elongated clusters . if the mass is computed close to the cluster core the effect becomes larger , ranging from @xmath517 to @xmath518 , for less to more elongated clusters in our sample , respectively , when the mass is computed within a sphere of radius @xmath519 . triaxial cluster distributions could therefore at least partially account for the observed discrepancies in the total mass of clusters computed with lensing and x - rays measurements . + we then analyze a subsample of the @xmath444 clusters for which the presence of a cooling flow has been claimed ( i.e. for which the upper limit , @xmath520 confidence , to the central cooling time has been measured to be less than @xmath521 ) . cooling flow clusters are typically recognized as dynamically relaxed systems in which the icm is supported by thermal pressure which dominates over non - thermal processes . their x - ray emission is in most cases regular and symmetric and little or no substructures is visible at optical wavelengths . we find no indication , however , that cooling flow clusters are more likely to be spherical . [ fig : isto_q_max ] suggests that the distribution of maximum axial ratios for the cooling flow sample is indistinguishable from that of the sample as a whole ; a kolmogorov - smirnov test confirms this impression . + finally , we find no relationship between cluster elongation along the l.o.s . and 2-d ellipticity ( see fig . [ fig : qmax_vs_ehb ] ) . in particular , a circular ( projected ) surface brightness profile is not an indicator that a cluster is in fact spherical . triaxial ellipsoids can be represented in the ellipticity - prolateness plane @xmath522 @xcite . the ellipticity is defined as : @xmath523 and the prolateness as : @xmath524 where the axial ratios satisfy @xmath525 . the allowed region in the @xmath522 plane is a triangle delimited by the lines on which prolate and oblate clusters fall ( @xmath526 and @xmath527 , respectively ) and the line connecting their endpoints . [ ell - pro ] shows the distribution in ellipticity - prolateness for our sample . no cluster in our sample shows extreme values of the ellipticity parameter . as expected from some simulations @xcite , prolate shapes ( @xmath528 ) may be more likely ( @xmath94 clusters ) than oblate ones ( @xmath529 ) . once again , cooling flow clusters ( gray boxes ) are indistinguishable from the sample as a whole . in this paper we have discussed how observations of clusters in the microwave and x - ray spectral bands can be combined to constrain their intrinsic 3-d shapes , provided that the cosmological model is known . we have applied our method to a combined sample of @xmath0 clusters of galaxies . in doing so we make the simplifying assumption that the clusters are ellipsoids with one axis parallel to the l.o.s . + our sample clusters were originally selected on the basis of x - ray luminosity , with the selection threshold well above the detection limit , in order to avoid selection on the basis of x - ray surface brightness . even so , we observe that clusters with extreme axes ratios are still preferentially selected only if the elongation lies along the l.o.s . + the mean value of the axial ratio in low - density cosmological simulations ranges from @xmath508 @xcite to @xmath509 @xcite and @xmath530 @xcite . this is consistent with results we present here @xmath531 . the spherical hypothesis is generally rejected , with prolate - like shapes being slightly more likely than oblate - like ones . numerical investigations suggest there should be some tendency for axial ratios to be larger at higher redshift @xcite , though this effect is only marginal : the axial ratio increases only @xmath532 from @xmath34 to @xmath533 @xcite . our data are not sufficiently precise to test this prediction . a poor correlation is also observed between the maximum axial ratio and the cluster gas temperature . the absence of such a correlation may indicate that high cluster temperatures are not mainly the result of shocks associated with accretion of sub - clusters , since such events would likely produce departures from spherical morphology . + the uncertainties on our results are mainly due to the relatively large errors in the sze measurements of central temperature decrement , together with the quadratic dependence of the distance estimates on this parameter . more accurate sze measurements , with better effective angular resolution , are required to extend this analysis to a large sample spanning a greater redshift range . + a relevant number of cosmological tests are today based on the knowledge of the mass of galaxy clusters through x - ray measurements ; these masses are though usually computed assuming a spherical symmetry . it is therefore extremely important to assess the effect that such assumption has on the determination of the total cluster mass . we have estimated that while the effect is negligible when the mass is computed at large distances from the cluster center , the discrepancy between the two mass values becomes much more important as we get closer to the cluster core ( i.e. @xmath534 when the mass is computed within a radius of @xmath519 ) . triaxial cluster shapes may therefore at least partially account for the discrepancies between cluster mass computed with strong lensing and x - rays data . + although the presence of a cooling flow is often interpreted as a sign of dynamical relaxation , the properties of the subsample of @xmath444 cooling flow clusters do not differ from those of the whole sample : cooling flow clusters therefore do not show preferentially spherical morphologies . this work has been supported by nasa grants nas8 - 39073 and nas8 - 00128 . this paper is based on observations obtained from the _ chandra _ data archive , the _ xmm_-newton science archive and the _ rosat _ public data archive ; we gratefully thank the _ chandra _ x - ray observatory science center ( operated for nasa by the smithsonian astrophysical observatory ) , _ xmm_-newton space operation centre ( operated by esa ) and mpe for maintaining the archives active and running . we consider a cluster electron density distribution described by an ellipsoidal triaxial @xmath3-model . in a @xmath3-model , the electron density of the intra - cluster gas is assumed to be constant on a family of similar , concentric , coaxial ellipsoids . high resolution @xmath535-body simulations have shown the asphericity of density profiles of dark matter halos and how such profiles can be accurately described by concentric triaxial ellipsoids with aligned axis : ellipsoidal @xmath3-models therefore provide a more detailed description of relatively relaxed simulated halos , respect to the conventional spherically symmetric model @xcite . + in a coordinate system relative to the cluster , we then describe the cluster electron density as : @xmath536 where @xmath537 , with @xmath538 , define an intrinsic orthogonal coordinate system centered on the cluster s barycenter and whose coordinates are aligned with its principal axes ; @xmath539 is the characteristic length scale of the distribution , which in our case is defined as the core radius ; along each axis , @xmath503 is the inverse of the corresponding core radius in units of @xmath539 ; @xmath55 is the central electron density . the electron density distribution in eq . ( [ eq : tri0 ] ) is described by 5 parameters : @xmath55 , @xmath3 , the axial ratios @xmath540 and @xmath541 , and the core radius @xmath542 along @xmath543 : @xmath544 to write the electron density distribution given by eq . ( [ eq : tri1 ] ) in a coordinate system relative to the observer , three additional parameters are needed : the rotation angles -@xmath545 and @xmath54- of the three principal cluster axes respect to the observer . a rotation through the first two euler angles is sufficient to align the @xmath546-axis of the observer coordinates system @xmath547 with the l.o.s . of the observer , i.e. the direction connecting the observer to the cluster center . when viewed from an arbitrary direction , quantities constant on similar ellipsoids project themselves on similar ellipses @xcite . a third rotation -@xmath54- will align @xmath548 and @xmath549 with the symmetry axes of the ellipses projected on the p.o.s . of the observer ( p.o.s . ) . eight independent parameters are therefore required to uniquely geometrically characterize the electron density distribution of a triaxial galaxy cluster . + the axial ratio of the major to the minor axes of the observed projected isophotes , @xmath550 , is a function of the shape parameters and of the direction of the l.o.s . , defined by the first two euler angles , @xmath551 and @xmath552 . it is given by @xcite : @xmath553 where @xmath554 and @xmath555 are : @xmath556 the rotation angle between the principal axes of the observed ellipses and the projection onto the sky of the ellipsoid @xmath543-axis is @xcite : @xmath557.\ ] ] the apparent principal axis that lies furthest from the projection onto the sky of the @xmath543-ellipsoid axis is the apparent major axis if @xcite @xmath558 or the apparent minor axis otherwise . in what follows , we assume @xmath548 to lie along the major axis of the isophotes , so that : @xmath559.\ ] ] the projected axial ratio @xmath26 , the orientation angle of the projected ellipses , the slope @xmath3 and the projection in the p.o.s . of the core radius can be determined fitting observed images to the @xmath3-model ; further independent constraints are therefore still required to uniquely determine the gas distribution . + the x - ray surface brightness and the sz temperature decrement are given by projection along the l.o.s . of two different powers of the electron density @xmath560 . following @xcite , we calculate the projection along the l.o.s . of the electron density distribution , given by eq . ( [ eq : tri1 ] ) , to a generic power @xmath561 which , in the observer coordinate system , can be written as : @xmath562}{\gamma \left[3 m\beta/2 \right ] } \frac{d_{\rm c } \theta_{\rm c3}}{\sqrt{h } } \left ( 1 + \frac{\theta_{1}^2+e_{\rm proj}^2 \theta_{2}^2}{\theta_{c,\rm proj}^2 } \right)^{(1 - 3 m\beta)/2}\ ] ] where @xmath22 is the angular diameter distance to the cluster and @xmath23 is the projected angular position on the p.o.s . of @xmath24 . @xmath25 is a function of the cluster shape and orientation : @xmath563 the observed cluster angular core radius @xmath27 is the projection on the p.o.s . of the cluster angular intrinsic core radius @xmath564 where @xmath565 . the analysis presented above is based on the main assumption that one cluster principal axis is elongated along the line of sight . despite this assumption is quite strong , it does not affect significantly the results . to test the effect of inclination on the estimate of the axis ratios , we proceed in the following way . first , we generate a galaxy cluster , characterized by the maximum axis ratio @xmath566 and by a parameter related to the degree of triaxiality , @xmath567 ; oblate and prolate clusters correspond to @xmath568 and 1 , respectively . since we are only interested in inclination issues , we neglect other measurements errors . then , we generate a set of 25 viewing angles @xmath569 . we assume that orientations are completely random , i.e. the angle @xmath570 is between 0 and @xmath571 and follows the distribution @xmath572 , whereas @xmath573 follows a uniform distribution between 0 and @xmath574 . for each pair of orientation angles , we compute the projected ellipticity @xmath575 and the elongation @xmath576 and , then , we obtain an estimate of the axial ratios of the cluster in the hypothesis of one principal axis being aligned along the l.o.s . finally , we calculate mean and standard deviation of the set of axial ratios corresponding to different viewing angles . if the value of such a mean is near that of the simulated cluster , then inclination issues hardly affect our analysis . in fig . [ fig : f9 ] , we plot the effect on the estimate of @xmath566 for different values of t in the case of @xmath577 . error bars equal standard deviations . as we can see , the error is minimum for highly triaxial clusters ( @xmath578 ) , being @xmath579 . such an error should be added in quadrature to the value of @xmath580 estimated in the previous sections , but due to its smallness it does not contribute significantly . the error increases for oblate or prolate clusters , with values @xmath581 . in this paper we have been facing with the hypothesis of triaxial clusters , in fact the algorithm illustrated in the previous sections is optimized to describe triaxial spheroids . the capability of ellipsoids of revolution to reproduce the observed data set will be the subject of a forthcoming paper . this considerations are quite general and still holds for very different values of @xmath566 . the error due to inclination issue on the estimate of axial ratios can be usually neglected with respect to other uncertainties , in particular in the measurement of the sz temperature decrement . clusters of galaxies act as lenses deflecting light rays from background galaxies . in contrast to sze and x - ray emission , gravitational lensing does not probe directly the icm distribution but maps the cluster total mass . the icm distribution in clusters of galaxies traces the gravitational potential . since we are considering a triaxial @xmath3-model for the gas distribution , the gravitational potential turns out to be constant on a family of similar , concentric , coaxial ellipsoids . ellipsoidal potentials are widely used in gravitational lensing analyses to fit multiple image systems @xcite . + the distribution of the cluster total mass can be inferred from its gas distribution . if the intra - cluster gas is assumed to be isothermal and in hydrostatic equilibrium in the cluster gravitational potential , while non - thermal processes are assumed not to contribute significantly to the gas pressure , the total dynamical mass density reads : @xmath582 where @xmath583 is the gravitational constant and @xmath584 is the mean particle mass of the gas . if we assume that the electron density of the icm follows a @xmath3-model distribution given by eq . ( [ eq : tri1 ] ) , the total gravitating mass density , in the coordinate system relative to the cluster , is given by : @xmath585 \label{eq : gl2}\ ] ] where @xmath586 and @xmath587 is the ellipsoidal radius , @xmath588 . the projected surface mass density can subsequently be written , in the observer reference frame , as : @xmath589 where : @xmath590 although the hypotheses of hydrostatic equilibrium and isothermal gas are very strong , total mass densities obtained under such assumptions can yield accurate estimates even in dynamically active clusters with irregular x - ray morphologies . elliptical potentials motivated by x - ray observations were employed in the irregular cluster ac 114 to provide good fit to multiple image systems @xcite . + the lensing effect is determined by the convergence @xmath58 : @xmath591 which is the cluster surface mass density in units of the surface critical density @xmath592 : @xmath593 where @xmath594 is the angular diameter distance from the lens to the source and @xmath595 and @xmath22 are the angular diameter distances from the observer to the source and to the lens , respectively . fitting the observed surface mass density to a multiple image system , it is possible to determine the central value of the convergence : @xmath596 which , using eqs . ( [ eq : gl4 ] ) and ( [ eq : gl5 ] ) , can be written as : @xmath597 , d. n. , verde , l. , peiris , h. v. , komatsu , e. , nolta , m. r. , bennett , c. l. , halpern , m. , hinshaw , g. , jarosik , n. , kogut , a. , limon , m. , meyer , s. s. , page , l. , tucker , g. s. , weiland , j. l. , wollack , e. , & wright , e. l. 2003 , , 148 , 175 , m. , strauss , m. a. , blanton , m. r. , abazajian , k. , dodelson , s. , sandvik , h. , wang , x. , weinberg , d. h. , zehavi , i. , bahcall , n. a. , hoyle , f. , schlegel , d. , scoccimarro , r. , vogeley , m. s. , berlind , a. , budavari , t. , connolly , a. , eisenstein , d. j. , finkbeiner , d. , frieman , j. a. , gunn , j. e. , hui , l. , jain , b. , johnston , d. , kent , s. , lin , h. , nakajima , r. , nichol , r. c. , ostriker , j. p. , pope , a. , scranton , r. , seljak , u. , sheth , r. k. , stebbins , a. , szalay , a. s. , szapudi , i. , xu , y. , annis , j. , brinkmann , j. , burles , s. , castander , f. j. , csabai , i. , loveday , j. , doi , m. , fukugita , m. , gillespie , b. , hennessy , g. , hogg , d. w. , ivezi ' c , . , knapp , g. r. , lamb , d. q. , lee , b. c. , lupton , r. h. , mckay , t. a. , kunszt , p. , munn , j. a. , oconnell , l. , peoples , j. , pier , j. r. , richmond , m. , rockosi , c. , schneider , d. p. , stoughton , c. , tucker , d. l. , vanden berk , d. e. , yanny , b. , & york , d. g. 2004 , , 69 , 103501 , p. a. , colberg , j. m. , couchman , h. m. p. , efstathiou , g. p. , frenk , c. s. , jenkins , a. r. , nelson , a. h. , hutchings , r. m. , peacock , j. a. , pearce , f. r. , & white , s. d. m. 1998 , , 296 , 1061
|
we discuss a method to constrain the intrinsic shapes of galaxy clusters by combining x - ray and sunyaev - zeldovich observations . the method is applied to a sample of @xmath0 x - ray selected clusters , with measured sunyaev - zeldovich temperature decrements .
the sample turns out to be slightly biased , with strongly elongated clusters preferentially aligned along the line of sight .
this result demonstrates that x - ray selected cluster samples may be affected by morphological and orientation effects even if a relatively high threshold signal - to - noise ratio is used to select the sample .
a large majority of the clusters in our sample exhibit a marked triaxial structure ; the spherical hypothesis is strongly rejected for most sample members . cooling flow clusters do not show preferentially regular morphologies .
we also show that identification of multiple gravitationally - lensed images , together with measurements of the sunyaev - zeldovich effect and x - ray surface brightness , can provide a simultaneous determination of the three - dimensional structure of a cluster , of the hubble constant , and the cosmological energy density parameters .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
junctions consisting of a single organic molecule between two metallic leads hold great promise for future nanoscale devices , where their potential applications include switches , transistors , and sensors . experimentally , it has proved difficult to control their production in an atomistic manner , and so theoretical studies are crucial for a full understanding of their behaviour . it is known that inelastic effects play an important role in the behaviour of such devices @xcite , but as yet we lack a full understanding of the processes at play that will lead to a complete interpretation of experimental results . in this paper , we use a model - system nanojunction including electron - vibration coupling @xcite in order to determine , by investigating the whole parameter space , what level of diagrammatic expansion is appropriate to describe the electron - vibration interaction in such junctions . we cover the entire parameter space , thus accounting for all the physical analogues to our model . the parameters we can vary correspond to the lead - molecule - lead coupling , the electron - vibron coupling strength , and the resonance of the electronic level with the leads electronic states in general , an organic molecule - based nanojunction is unlikely to have its homo or lumo levels in alignment with the equilibrium fermi level of the leads , and so such a nanojunction will be dominated by what we term the off - resonant regime with strong tunneling at low bias . moreover , coupling between the leads and the central molecule is likely to be relatively weak , in the sense that the corresponding coupling to the leads is much smaller than the corresponding hopping integrals in the leads themselves . conversely , a system consisting of a nanoconstriction in a gold wire will have electronic levels in the constriction that are close to those of the leads , and a larger coupling between the central region and the leads , closer to the tight - binding hopping parameter of the leads . this work will distinguish between these regimes and discuss for which physical systems diagrams beyond the self - consistent born approximation ( scba ) become relevant for electron - phonon coupling . we study this using a full non - equilibrium greens - function ( negf ) technique @xcite which allows us to study all the different transport regimes in the presence of electron - vibron interaction . following the spirit of many - body perturbation theory and feynman diagrammatics , we include the commonly - used scba diagrams as well as second - order diagrams in terms of the electron - vibron interaction . a detailed description of the formalism and the numerical implementation of the negf code we have developed is given in ref . [ ] . in this work we studied the equilibrium and non - equilibrium electronic structures of the nanojunctions in the presence of electron - vibron coupling . in the present paper , we now give and analyse results for the full non - equilibrium transport properties , namely the non - linear @xmath0 characteristics , the conductance @xmath1 , and especially the iets signal @xmath2 calculated with our negf code . the paper is structured as follows . in section [ sec : theory ] we summarize the key aspect of our methodology detailed in ref . [ ] . results for the effects of the second order diagrams on the non - equilibrium non - linear transport properties are presented in section [ sec : results ] . they are separated into the features we observe for the purely inelastic effects in the iets signal at bias equal to an integer multiple of the vibron energy , and for the inelastic resonant features related to the vibron replica of the electronic resonances . our conclusions are given in section [ sec : conclusions ] . in addition , we explain in detail in appendix [ sec : appendix - vertex ] how one of the second - order diagrams acts as a vertex correction to the fock - like electron - vibron diagram . a fully atomistic description of the non - equilibrium inelastic transport properties we wish to study is , unfortunately , beyond the reach of current _ ab initio _ methods . instead we use a model system which retains the essential physics of the junction while reducing the calculations to a tractable size . a full description of both the model and our methodology is given in ref . [ ] , and so we review only the most salient features here . we use the single - site single - mode model ( sssm),in which the central molecule of the junction is represented by a single molecular level coupled to a single vibrational mode . we have already used this model and discussed its validity in our previous study on the equilibrium and non - equilibrium electronic structures of such a system coupled to two electron reservoirs @xcite . the total hamiltonian for the nanojunction is given by @xmath3 in this work , we represent the hamiltonian of the left ( @xmath4 ) and right ( @xmath5 ) leads @xmath6 with a non - interacting tight - binding model with semi - elliptic bands , although in principle it can take any valid form . the hopping between leads and the central region is given by @xmath7 , where @xmath8 is the hopping integral between the @xmath9 lead and the central region . the central region contains the electron - vibron interactions . we choose that an electron couples linearly , via its density , to the displacement of a single vibration mode . the hamiltonian for the central region in the sssm model is then given by @xmath10 where @xmath11 ( @xmath12 ) creates ( annihilates ) an electron in the molecular level @xmath13 , which is coupled to the vibration mode of energy @xmath14 via the coupling constant @xmath15 . a detailed analysis of the formalism of the non - equilibrium transport properties from negf and for interacting system is provided in ref . the current @xmath16 passing through each lead @xmath17 is expressed in terms of two time green s functions@xcite . it is transformed into frequency representation for the steady - state regime to give : @xmath18 we vary the applied bias by moving the chemical potentials of the left and right leads . with the equilibrium fermi energies @xmath19 , we introduce a quantity @xmath20 such that @xmath21 and @xmath22 following ref . [ ] . in this way we can create several forms for the potential drop across the junction . by setting @xmath23 , for example , we create an asymmetric drop whereby @xmath24 remains constant while @xmath25 is changed , whereas @xmath26 gives a symmetric potential drop where @xmath25 rises ( lowers ) as @xmath24 lowers ( rises ) by the same amount . it now remains to construct our non - equilibrium green s functions ( details given in ref . ) . the retarded and advanced green s functions are calculated using a dyson equation @xmath27 while the greater ( @xmath28 ) and lesser ( @xmath29 ) green s functions are obtained from a quantum kinetic equation with the form @xmath30 where @xmath31 is the _ non - interacting _ green s function of the isolated central region . in this work we consider both first- and second - order contributions to the electron - vibron coupling . the first - order diagrams are shown in fig . [ fig : fockhartree ] , and , if calculated self - consistently , equate to the commonly - used self - consistent born approximation ( scba ) . we also make use of the two second - order diagrams , those which involve two phonon excitations ( fig . [ fig : dph - dx ] ) . the first of these is similar in structure to the @xmath32 skeleton for electron - electron interactions , and consists of a fock - like diagram where the phonon is dressed by a single electron - hole bubble ( hence the appellation dph for dressed - phonon diagram , fig . [ fig : dph - dx ] left ) . the second , which we call the double - exchange ( dx ) diagram ( fig . [ fig : dph - dx ] right ) , includes two phonons simultaneously , with the second being emitted before the first is reabsorbed . the dx diagram is part of the skeleton diagrams corresponding to vertex corrections . we use these diagrams to construct expressions for the electron - vibron self - energy @xcite as the ( total or partial ) sum of each diagram : @xmath33 . note that in order to handle numerically sharply peaked functions or strongly discontinuous functions , we have found it necessary to include a very small but finite imaginary part in our expression for the bare vibron green s function @xcite . this also allows us to perform calculations with a smaller number of @xmath34-grid points , as long as our imaginary part @xmath35 in the bare vibron green s function is around two to three times the @xmath34-grid spacing . we have already discussed in detail the effects of the corresponding extra broadening on the lineshape of the spectral functions and on the values of the linear conductance in ref . in this section we present the effects of the second - order diagrams on the full non - equilibrium transport properties of the nanojunction in the presence of electron - vibron coupling . calculations of the green s functions are performed with different levels of approximation for the electron - vibron self - energies ( figs . [ fig : fockhartree ] and [ fig : dph - dx ] ) . fully self - consistent calculations using the first - order diagrams are annotated scba , those using one or both second - order diagrams are annotated sc(ba+dx ) , sc(ba+dph ) or sc(ba+dx+dph ) as appropriate . in addition we have performed non - self - consistent second - order corrections , i.e. by using the scba green s functions to calculate the second - order diagrams , we then determine the new green s functions without full self - consistency . these calculations are annotated scba+dx or scba+dph . the inelastic properties of the system are present in the current @xmath0 but are better represented by the second derivative of the current @xmath2 as it is the signal that is directly measured experimentally in the form of the inelastic electron tunneling spectrum ( iets)@xcite . the iets curves present features , peaks or dips @xcite at biases corresponding to the energy of a specific excitation , in our case to the energy of one or several excitations of the vibration mode . the peak feature is commonly associated with the opening of a new inelastic channel for the conductance of nanojunctions in the off - resonant regime , i.e. when the electronic level @xmath13 is sufficiently far from the equilibrium fermi level . in the case of the resonant transport regime ( when @xmath13 is close to @xmath36 ) , a dip feature is obtained in the iets . it is associated with electron - vibron backscattering effects and a decrease in the conductance at the threshold bias . furthermore , being the derivative of the conductance , the iets curves also present features at biases corresponding to peaks in the conductance . we have found @xcite that in order to get a better aspect ratio for the iets features corresponding to vibron excitations , it is more convenient to normalize the iets curves by the dynamical conductance , i.e. @xmath37/[di / dv]=d / dv \ln g(v)$ ] as in refs . . we divide our results into two sections ; the first for purely inelastic features , and the second for inelastic features associated with the electron resonance effects . the first category corresponds to features observed in the iets signal at bias equal to an integer multiple of the vibron energy @xmath38 , i.e. a tunneling electron excites @xmath39 vibrons . the second category corresponds to inelastic resonant tunneling via the vibron replica associated with the main electronic resonance at @xmath40 , and hence are observed in the iets signal for biases @xmath41 . we will see below that the second - order diagrams have different effects on the iets features depending upon their correspondence to one of these two categories . we consider first of all the off - resonant regime . in this limit the iets features associated with inelastic resonant tunneling are sufficiently far ( or sufficiently small for the higher vibron replica ) from the inelastic feature at @xmath42 . we can thus avoid a superposition of the two different kinds of features . for an asymmetric potential drop ( @xmath43 , left panel ) and a symmetric potential drop ( @xmath44 , right panel ) . results are shown for three different lead - molecule coupling parameters @xmath45 ( black , red , and blue lines ) , and for two different electron - vibron couplings ( medium @xmath46 , solid lines and triangles ; strong @xmath47 , dashed lines and circles ) . lines represent calculations at the scba level , symbols sc(ba+dph ) , from which we can see there is no distinguishable contribution to the inelastic peak from dph for any parameter set . the other parameters are @xmath48 . ] results are presented in fig . [ fig : w0peak ] for different electron - vibron couplings and for both symmetric and asymmetric potential drops . we note that , as would be expected intuitively , the size of the feature increases with the electron - phonon coupling @xmath15in fact the height with respect to the baseline is proportional to @xmath49 . as @xmath8 is increased , the iets signal decreases in overall magnitude , although the feature at @xmath14 itself remains clear ( fig . [ fig : w0peak ] inset ) . in the off - resonant regime , we do not find any difference to the curves when the second - order dressed - phonon ( dph ) diagram is included . possible reasons for this are discussed in section [ sec : electron - resonance ] . we also note that the results are dependent on the symmetry of the potential drops at the left and right contact . as shown in the separate panels of fig . [ fig : w0peak ] , the curves for symmetric potential drops have a much lower baseline than for the asymmetric potential drops , although the inelastic features have the same lineshape , position and magnitude ( see figs . [ fig : w0peak ] , [ fig : scba+dx ] and [ fig : iets_badx_andfit ] ) . this is because ( for positive bias ) in the asymmetric case , @xmath25 is rising bias while @xmath24 is kept fixed at its equilibrium value . thus @xmath25 is approaching the electron resonance twice as fast as in the symmetric case , where @xmath25 rises at the same rate as @xmath24 falls . the symmetric case therefore contains much less effect from the tail of the main electron resonance , and allows us to further isolate the purely inelastic part of the iets signal . we now consider the effect of the double - exchange diagram ( fig . [ fig : dph - dx ] right ) on the inelastic peak at @xmath14 . calculation of @xmath50 is extremely computationally intensive , as it can be reduced neither to a simple convolution product , nor even to a simple double - convolution product @xcite . calculations for the integration in the energy representation of the green s functions and the self - energies scale as the cube of the number @xmath51 of points in the energy - grid @xcite . moreover , as the finite imaginary part @xmath35 we have included in the vibron green s function causes extra broadening , if taken too large it washes out much of the effect of the dx diagram . in order to decrease @xmath35 we need to increase the number of grid points , and thus , even for our minimal model , a fully self - consistent calculation of the double - exchange diagram becomes intractable with @xmath52 . hence for the dx calculations , we usually work with @xmath53 , giving @xmath54 for the total spectral width considered in our calculations . . results for scba and sc(ba+dph ) are virtually identical . however , including the dx diagram , either self - consistently ( solid red line ) or non - self - consistently ( red diamonds ) raises the amplitude of the peak . this effect can be approximated by an scba calculation done with a larger @xmath15 ( black triangles ) , indicating that dx generates an effective renormalization of @xmath15 . the other parameters are @xmath55 . ] vertex . ] vertex . ] we work around this by calculating the effects of the dx diagram both in a self - consistent manner sc(ba+dx ) or non - self - consistenly as a second - order correction to the scba result . the results of this are plotted in fig . [ fig : scba+dx ] . we see that the effect of the dx diagram is to both increase the height of the feature and to raise its baseline . for the set of parameters shown the self - consistency in the dx diagram calculations is not crucial . we see that the effect of dx can also be reproduced with a scba calculation in which we increase the value of @xmath15 . therefore the dx diagram has the effect of renormalizing @xmath15 as it is part of the skeleton family of vertex correction , as shown in fig . [ fig : dx - feynman - mod ] . we can thus approximate the effect of the dx diagram in the iets with a fock - like diagram with one renormalized vertex @xmath56 . the amplitude of the peak at @xmath14 , instead of varying as @xmath49 , therefore now depends on @xmath57 , and so we can define an _ effective _ electron - vibron coupling constant @xmath58 , with @xmath59 . this allows us to make a more quantitative analysis of @xmath60 by fitting the scba+dx curve to an scba curve with electron - vibron coupling @xmath61 , as shown in fig . [ fig : scba+dx ] . furthermore , we can study how @xmath56 and @xmath15 are correlated by performing a series of calculations for different values of the parameters @xmath15 and @xmath14 , and then fitting the scba+dx results to those from scba calculations . the results of this are shown in fig . [ fig : gamma0correlation ] , from which we can see that , to a first approximation , the dx diagram consistently raises the effective electron - vibron interaction by approximately 3% within the range of parameters we used . as a function of the nominal coupling parameter @xmath15 , from a fit of several sets of data comparing scba+dx curves to scba ( as in fig . [ fig : scba+dx ] ) . the data for all three values of @xmath14 lie on a straight line with slope 1.03 ( the straight line with slope 1 is shown for comparison ) , implying that the vertex correction increases the effective electron - vibron coupling by 3% . ] however , we would like to point out that although the apparent effects of the dx diagram on the iets signal is to renormalize the coupling constant @xmath15 , the reality is much more subtle . in appendix [ sec : appendix - vertex ] we discuss in detail the renormalization effect of the dx diagram in terms of vertex corrections , and we show that such vertex corrections do not simply correspond to a mapping of the sssm hamiltonian onto a similar hamiltonian with a static renormalization of the electron - vibron coupling constant , i.e. @xmath62 . rather , the vertex correction actually generates a dynamical renormalization of the electron - vibron coupling constant , i.e. @xmath63 . this can be seen more clearly by considering scba calculations , for different values of the electron - vibron coupling parameter , and checking which of such calculations correspond the best to a sc(ba+dx ) calculation . the result is shown in fig . [ fig : iets_badx_andfit ] . although the difference between the best scba fits and the sc(ba+dx ) are not large , it is quite clear that a renormalized scba calculation does not provide exactly the same lineshape as a full sc(ba+dx ) calculation for all the range of biases @xmath64 around the inelastic peak at @xmath14 . showing different scba fits for the renormalized electron - vibron coupling . the scba ( dashed line ) and sc(ba+dx ) ( solid red line ) are shown for @xmath65 , together with scba calculations for @xmath66 and @xmath67 . although these provide a good approximation , neither gives an exact fit to the lineshape of the sc(ba+dx ) curve . the other parameters are @xmath68 . ] finally we expect to see a peak feature at @xmath69 in the iets for the off - resonant transport regime . this peak feature is the two - vibration excitation equivalent of the feature observed at @xmath70 . since this a higher - order process , the amplitude of the feature should be @xmath49 times smaller than the feature at @xmath71 . this feature has a rather small amplitude for all the electron - vibron coupling constants we have considered in this work since @xmath72 . an example of a close - up of the iets feature around @xmath69 is given in figure [ fig : offres_ietssidebandpeak ] . we find that the amplitude of the peak with respect to the linear baseline is indeed approximately @xmath73 ( one order of magnitude ) smaller than the corresponding amplitude of the peak at @xmath74 ( shown in fig . [ fig : scba+dx ] ) . once more we find that the effects of the dph diagram are negligible for this part of the iets signal . in fig . [ fig : offres_ietssidebandpeak ] we also include the results of a scba calculation for a larger coupling @xmath75 which mimics the renormalization effects of the dx diagram as discussion above . for different levels of approximation : the scba ( solid black line ) and sc(ba+dph ) ( red circles ) are virtually identical , as for the feature at @xmath14 . an scba calculation approximating the effect of dx is also included ( dashed line ) . a straight baseline is included for clarity . the other parameters are @xmath76 . ] the main electron - resonance peak occurs at the polaron - shifted value of @xmath77 , and consists of a peak - dip feature in the iets , as it corresponds to a resonant peak feature in the conductance . with no electron - phonon coupling ( @xmath78 ) , the iets curve has no features other than the one corresponding to the resonant transmission in the conductance at @xmath79 . once the electron - phonon coupling is turned on , phonon side - band peaks emerge in the spectral function at energies @xmath80 . the @xmath81 features correspond to phonon emission ( vibration excitation ) by an electron , while the @xmath82 features correspond to phonon - emission by a hole . in the iets , they appear as peak - dip features ( derivatives of an inelastic resonant peak in the conductance ) with amplitude decreasing as the bias is further increased . at lower biases however , there are no features at @xmath83 at the scba level except for very low values of the coupling to the leads ( @xmath84 , fig . [ fig : resonance - peaks ] top inset ) . including the dph diagram , however , introduces a small peak - dip features at @xmath85 just below the main peak - dip feature at @xmath86 in the iets , as shown in fig . [ fig : resonance - peaks ] ) . the dph diagram also has a strong influence on the lineshape of the other phonon side - band peaks above @xmath87 , increasingly so as @xmath13 is brought within range of the equilibrium chemical potentials ( i.e. @xmath88 in fig . [ fig : resonance - peaks ] bottom ) . . with no electron - vibron coupling ( solid black line ) , there is a single feature at @xmath89 . with electron - vibron coupling ( red and blue dashed lines ) , the main peak moves to @xmath77 and side - band peaks appear . the inset shows a tiny feature at @xmath90 that only occurs for @xmath91 in scba calculations but reappears for all @xmath92 once dph is included . bottom : the inelastic resonant features for various @xmath13 at @xmath93 for both scba ( lines ) and sc(ba+dph ) ( symbols ) calculations , showing the increasing influence of dph in the resonant transport regime @xmath94 . the other parameters are : @xmath95 . , title="fig : " ] + . with no electron - vibron coupling ( solid black line ) , there is a single feature at @xmath89 . with electron - vibron coupling ( red and blue dashed lines ) , the main peak moves to @xmath77 and side - band peaks appear . the inset shows a tiny feature at @xmath90 that only occurs for @xmath91 in scba calculations but reappears for all @xmath92 once dph is included . bottom : the inelastic resonant features for various @xmath13 at @xmath93 for both scba ( lines ) and sc(ba+dph ) ( symbols ) calculations , showing the increasing influence of dph in the resonant transport regime @xmath94 . the other parameters are : @xmath95 . , title="fig : " ] . calculations performed for different approximations for the electron - vibron self - energy : ( left panel ) scba ( black solid line ) , scba plus second order dph correction ( dashed red line ) , sc(ba+dph ) ( dot - dashed blue line ) ; ( right panel ) : scba ( solid black line ) , sc(ba+dx ) ( dashed red line ) and sc(ba+dx+dph ) ( dot - dashed blue line ) . the other parameters are @xmath96 . ] we now consider the combined effects of the dph diagram as well as of the dx diagram on the specific case of the iets feature at @xmath97 . this is shown in figure [ fig : electronres_sidebandpeak ] . the dph diagram increases strongly the peak - dip feature ( at @xmath98 ) obtained from scba calculations at medium / strong electron - vibron coupling . note here the importance of the self - consistency in the calculations : the second - order dph diagram calculated as a second - order correction to scba ( fig . [ fig : electronres_sidebandpeak ] left ) gives a completely wrong feature in the iets . interestingly , the self - consistent calculation with the dx diagram seems to give a similar feature to that observed in the scba calculations , but slightly shifted towards lower bias ( fig . [ fig : electronres_sidebandpeak ] right ) . this is completely consistent with the renormalization effects of the electron - vibron coupling by the dx diagram as discussed in the previous section . indeed , the dx diagram renormalizes the coupling @xmath15 towards a higher value @xmath99 . consequently the renormalization of the molecular level by @xmath100 is more important than for scba calculations , and thus the feature is moved towards lower bias . the calculations performed with both dx and dph diagrams ( fig . [ fig : electronres_sidebandpeak ] right ) generate a hybrid feature in the iets in comparison to individual calculations with the second - order diagrams . however the new iets is not simply obtained by a linear superposition of the individual effects of the dph and dx diagrams . it might at first seem strange that the dph self - energy is negligble at @xmath101 , where one might expect it to be influential , but that it has a significant effect at biases @xmath102 . while this is to some extent related to the strength of the electron - vibron coupling , there is another , more important , underlying cause . the dph diagram involves an electron - hole bubble , and so for this diagram to become relevant , simultaneous electron and hole states must be available . this is not the case when the spectral functions of the coupled electron - vibron system are mostly empty or mostly filled . when the bias is significantly low and both fermi levels @xmath103 are below the electron resonance level , these excitations are inaccessible and so there is no effects from the dph diagram . once the bias is increased to within range of @xmath104 , or @xmath105 however , these electron - hole states become accessible and the dph diagram becomes influential ( unless the spectral function is mostly filled ) . this is also borne out by the increasing contribution from dph to the lineshape of the phonon side - band peaks as the electron level @xmath13is decreased , moving dph s sphere of influence to lower and lower biases ( see lower panel in fig . [ fig : resonance - peaks ] ) . having examined the role of the second - order diagrams in detail for characteristic selected sets of parameters , we now show results across the entire parameter range . in order to present this in a concise manner , we have compiled maps of our iets results comparing scba calculations to those with sc(ba+dph ) indicating in which regimes the dph self - energy has a significant effect . for the off - resonant regime ( fig . [ fig : maps ] left ) , we can see that the greatest effect of dph is apparent at higher bias ( i.e. approaching the electron resonance @xmath13 ) and when the lead - molecule - lead coupling is small . for the resonant case , however , we can see that the dph self - energy only gives a non - negligble contribution in the region of parameter space where the coupling to the leads is small , and at low bias . this has potential implications for real molecular junctions . consider a junction which has its dominant molecular levels far from the equilibrium fermi levels of the leads . at sufficiently high bias , the dph self - energy will have a significant contribution to the inelastic spectra . this could occur in the case of a junction formed from an organic molecule if the electronic level of the molecule is within range of the intended operational bias of the junction . if however the dominant molecular electronic level is close to the leads fermi levels , and the coupling to the leads is large ( as would be the case , for example , in a gold nanoconstriction ) then dph will not give a significant contribution at any applied bias . . it gives a near - zero result ( blue and green areas ) when the two calculations give the same spectrum , and a positive number ( red areas , i.e. bottom right corner for scba and bottom left corner for sc(ba+dph ) ) where there is a substantial difference between the two . the other parameters are @xmath106 ( off - resonant / resonant ) , @xmath107.,scaledwidth=49.0% ] . it gives a near - zero result ( blue and green areas ) when the two calculations give the same spectrum , and a positive number ( red areas , i.e. bottom right corner for scba and bottom left corner for sc(ba+dph ) ) where there is a substantial difference between the two . the other parameters are @xmath106 ( off - resonant / resonant ) , @xmath107.,scaledwidth=85.0% ] by using the non - equilibrium green s functions technique , we have studied the effect of electron - vibron interaction on the inelastic transport properties of single - molecule nanojunctions for a model system . we have included not only the first - order diagrams ( ba ) but also the second - order diagrams ( double - exchange dx and dressed phonon dph diagrams ) for the electron - vibration interaction . we have calculated the inelastic electron tunneling spectrum ( iets ) across the full range of parameters available to our model . the effects of the second - order dx and dph diagrams are different and affect different features of the iets signal . the effects of these diagrams are generally less visible in the integrated quantities , such as the current or the derivated iets signal , than in the spectral functions @xcite . however their effects are non - negligible , and are important for the full understanding of the spectroscopic information conveyed by the iets signal . the effect of the dressed - phonon ( dph ) diagram is more important in the bias regions where one of the leads chemical potentials begins to impinge upon the electron resonance or one of its vibron replica ( i.e. for resonant inelastic features ) . its effect is reduced both by increasing the lead - molecule - lead coupling and/or reducing the electron - vibron interaction . the renormalization of the vibron propagator ( dph ) has been shown to be strongly dependent on the self - consistency of the calculations . it would be interesting now to study the effects of the full series of the electron - hole bubble on the renormalised vibron propagator ( i.e. full @xmath32-like diagram ) . the double - exchange diagram ( dx ) affects all the features in the iets signal ( i.e. resonant inelastic features and purely inelastic features at @xmath108 ) . the corrections are small in the weak - to - medium electron - vibron coupling because they are of the order of @xmath109 . however we have shown , numerically and analytically , that the effect of dx is similar to a dynamical renormalization of one vertex in the fock - like diagram . more interestingly , the complex form of the non - equilibrium dynamical renormalized electron - vibron coupling @xmath110 we have derived analytically can be adequately replaced in our iets calculations by a single static renormalized parameter @xmath61 . this important result leads us to believe that the second - order dx calculations , which are extremely costly in computing time even for our model system , can be incorporated in calculations for realistic systems by an appropriate renormalization of the vertex in a low - computing - cost scba calculation . in appendix a of ref . [ ] we have given all the details for the derivations of the first- and second - order electron - vibron self - energies . in this section we show how the second - order double - exchange ( dx ) diagram can be recast in an effective first - order fock - like diagram with a renormalized vertex . we recall that for the sssm model , the fock and dx self - energies defined on the keldysh contour @xmath111 are given by @xmath112 and @xmath113 the dx self - energy can be rewritten as an effective fock - like diagram after introducing a renormalized electron - vibron coupling parameter @xmath114 and the vertex function @xmath115 : @xmath116 with @xmath117 and where the vertex function is given by @xmath118 the above expression for the vertex function is compatible with the second - order expansion of the electron - vibron interaction . a generalization of the vertex function ( see fock - like diagram in figure [ fig : dx - feynman - mod ] ) to all orders of the interaction is possible , though beyond the scope of the present paper . note that at the lowest order , the renormalized electron - vibron coupling parameter would be given by @xmath119 with @xmath120 . hence eq.([eq : dxse_onck ] ) would simply be transformed eq.([eq : fockse_onck ] ) as expected . using the rules of analytical continuation on the real - time branches given in appendix a of paper ref . [ ] , we find the different components of the self - energies . then after taking the fourier transform of the different quantities in the steady - state limit , i.e. @xmath121 , we find the following expression for the different components of the energy - dependent self - energies : ( the index @xmath125 labels the branch of the keldysh time - loop contour @xmath111 and are related to the usual convention : time - ordered ( @xmath126 ) , anti time - ordered ( @xmath127 ) , greater ( @xmath128 ) and lesser ( @xmath129 ) components . ) because all the quantities are originally defined on @xmath111 and because the vertex function is a 3-point ( 3 times ) function , the non - equilibrium dynamical renormalized electron - vibron coupling is a complex function of three indices @xmath130 and of two energy variables . such a dynamical renormalization ( including non - equilibrium conditions ) is much more complicated than a simple static renormalization of the electron - vibron constant coupling @xmath131 . in our analysis of the iets signal in the off - resonant transport regime , we try to keep the interpretation of the results as simple as possible , and we show that the renormalization of the iets signal due to the dx diagram can be fairly well approximated by a simple static renormalization of the coupling constant @xmath15 for applied bias around the vibration frequency @xmath132 .
|
we study the effect of electron - vibron interactions on the inelastic transport properties of single - molecule nanojunctions .
we use the non - equilibrium green s functions technique and a model hamiltonian to calculate the effects of second - order diagrams ( double - exchange dx and dressed - phonon dph diagrams ) on the electron - vibration interaction and consider their effects across the full range of parameter space .
the dx diagram , corresponding to a vertex correction , introduces an effective dynamical renormalization of the electron - vibron coupling in both the purely inelastic and the inelastic - resonant features of the iets .
the purely inelastic features correspond to an applied bias around the energy of a vibron , while the inelastic - resonant features correspond to peaks ( resonance ) in the conductance .
the dph diagram affects only the inelastic resonant features .
we also discuss the circumstances in which the second - order diagrams may be approximated in the study of more complex model systems .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
@xcite found that the galactic cepheids follow a spectral type that is independent of their pulsational periods at maximum light and gets later as the periods increase at minimum light . * hereafter skm ) used radiative hydrodynamical models to explain these observational phenomena as being due to the location of the hydrogen ionization front ( hif ) relative to the photosphere . their results agreed very well with code s observation . skm further used the stefan - boltzmann law applied at the maximum and minimum light , together with the fact that radial variation is small in the optical @xcite , to derive : @xmath3 where @xmath4 are the effective temperature at the maximum / minimum light , respectively . if @xmath5 is independent of the pulsation period @xmath6 ( in days ) , then equation ( 1 ) predicts there is a relation between the @xmath7-band amplitude and the temperature ( or the colour ) at minimum light , and vice versa . in other words , if the period - colour ( pc ) relation at maximum ( or minimum ) light is flat , then there is an amplitude - colour ( ac ) relation at minimum ( or maximum ) light . equation ( 1 ) has shown to be valid theoretically and observationally for the classical cepheids and rr lyrae variables @xcite . for the rr lyrae variables , @xcite and @xcite used linear and non - linear hydrodynamic models of rrab stars in the galaxy to explain why rrab stars follow a flat pc relation at _ minimum _ light . later , @xcite used macho rrab stars in the lmc to prove that lmc rrab stars follow a relation such that higher amplitude stars are driven to cooler temperatures at maximum light . similar studies were also carried out for cepheid variables , as in skm , @xcite , ( * ? ? ? * hereafter paper i ) and ( * ? ? ? * hereafter paper ii ) . in contrast to the rr lyrae variables , cepheids show a flat pc relation at the _ maximum _ light , and there is a ac relation at the minimum light . therefore , the pc relation and the ac relation are intimately connected . all these studies are in accord with the predictions of equation ( 1 ) . in paper i , the galactic , large magellanic cloud ( lmc ) and small magellanic cloud ( smc ) cepheids were analyzed in terms of the pc and ac relations at the phase of maximum , mean and minimum light . one of the motivations for this paper originates from recent studies on the non - linear lmc pc relation ( as well as the period - luminosity , pl , relation . see paper i ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) : the optical data are more consistent with two lines of differing slopes which are continuous or almost continuous at a period close to 10 days . paper i also applied the the @xmath2-test @xcite to the pc and ac relations at maximum , mean and minimum @xmath7-band light for the galactic , lmc and smc cepheids . the @xmath2-test results implied that the lmc pc relations are broken or non - linear , in the sense described above , across a period of 10 days , at mean and minimum light , but only marginally so at maximum light . the results for the galactic and smc cepheids are similar , in a sense that at mean and minimum light the pc relations do not show any non - linearity and the pc(max ) relation exhibited marginal evidence of non - linearity . for the ac relation , cepheids in all three galaxies supported the existence of two ac relations at maximum , mean and minimum light . in addition , the cepheids in these three galaxies also exhibited evidence of the pc - ac connection , as implied by equation ( 1 ) , which give further evidence of the hif - photosphere interactions as outlined in skm . to further investigate the connection between equation ( 1 ) and the hif - photosphere interaction , and also to explain code s observations with modern stellar pulsation codes , galactic cepheid models were constructed in paper ii . in contrast to skm s purely radiative models , the stellar pulsation codes used in paper ii included the treatment of turbulent convection as outlined in @xcite . one of the results from paper ii was that the general forms of the theoretical pc and ac relation matched the observed relations well . the properties of the pc and ac relations for the galactic cepheids with @xmath8 can be explained with the hif - photosphere interaction . this interaction , to a large extent , is independent of the pulsation codes used , the adopted ml relations , and the detailed input physics . the aim of this paper is to extend the investigation of the connections between pc - ac relations and the hif - photosphere interactions in theoretical pulsation models of lmc cepheids , in addition to the galactic models presented in paper ii . in section 2 , we describe the basic physics of the hif - photosphere interaction . the updated observational data , after applying various selection criteria , that used in this paper are described in section 3 . in section 4 , the new empirical pc and ac relations based on the data used are presented . in section 5 , we outline our methods and model calculations , and the results are presented in section 6 . examples of the hif - photosphere interaction in astrophysical applications are given in section 7 . our conclusions & discussion are presented in section 8 . throughout the paper , short and long period cepheid are referred to cepheids with period less and greater than 10 days , respectively . the partial hydrogen ionization zone ( or the hif ) moves in and out in the mass distribution as the star pulsates . it is possible that the hif will interact with the photosphere , defined at optical depth ( @xmath9 ) of 2/3 , at certain phases of pulsation . for example , skm suggested that this happened at maximum light for the galactic cepheids , as the hif is so far out in the mass distribution that the photosphere occurs right at the base of the hif . the sharp rise of the opacity wall ( where the mean free path goes to zero ) due to the existence of hif prevents the photosphere moving further into the mass distribution and hence erases any `` memory '' of global stellar conditions , including the underlying pc relation . this lead to a flat relation between period & temperature , period & colour and period & spectral type at maximum light , as seen in skm and paper ii . at other phases , since the hif does not interact with the photosphere , the temperature of the star ( or the colour ) follows the underlying global pc relation . the hif - photosphere interaction also relies on the properties of the saha ionization equation and the structural properties of the outer envelopes of cepheids . it is well known that the partition functions in the saha ionization equation are formally divergent unless some atomic physics is used to truncate them . in the pulsation codes we used , we approximate the partition functions of various atoms by their ground state statistical weights . the properties of the saha ionization equation in cepheid envelopes are such that hydrogen starts to ionize at a temperature that is almost independent of density , for a certain range of low densities . outside of this range of density , the density dependence increases . thus , when the photosphere is very close to , or engaged with the hif and the density of these regions is reasonably low , the temperature of the photosphere is less dependent on the surrounding density and hence the global stellar parameters . at higher densities , the temperature at which hydrogen ionizes becomes more sensitive to density and hence more sensitive to global stellar parameters . if the photosphere is far from the hif , or disengaged , then the location of the photosphere and hence the temperature of the photosphere , is again strongly dependent on density and hence on global stellar parameters . that is why the photosphere needs to be close to , or engaged with the hif for this effect to take place . moreover , this dependence on density is not sharp so that for `` low '' and `` high '' densities the density dependence of the photospheric temperature is weak and strong respectively . an examination of figure 15.1 in @xcite demonstrates that this is plausible . thus as the star pulsates , the photospheric temperature has a density dependence that can be strong or weak depending on phase . an example where the density dependence is weak are the galactic long period cepheids at maximum light ( skm , paper ii ) : these cepheids display a flat pc relation at maximum light . these properties of the hif - photosphere interaction can , in turn , affect the temperature of the photosphere and hence the colour of the cepheid . here we investigate the idea that lmc cepheids with periods below 10 days are such that the hif and photosphere are engaged through most of the pulsation cycle . at periods greater than 10 days , the photosphere only engages with the hif at maximum light . the transition is sharp because the photosphere is either at the base of the hif or it is not . the transition occurs because as the period increases , the @xmath10 ratio increases and this implies the hif is located further inside in the mass distribution , changing the phase at which it can interact with the photosphere @xcite . the structure of galactic cepheids is such that this interaction only occurs at maximum light , even for cepheids with periods shorter than 10 days . in paper i , we constructed the light curves of fundamental mode cepheids in the lmc by using the extensive photometric dataset in the ogle ( optical gravitational lensing experiment ) database . however , the dataset used in paper i was downloaded in 2002 , prior to the updated version of the dataset that was available after april 24 , 2003 ( ogle website , udalski 2004 [ private communication ] ) . the updated version includes additional @xmath7- and @xmath11-band data for most of the cepheids . in addition , the periods have been refined by the ogle team using the complete set of photometric data . due to these reasons , we decided to repeat the light curve construction @xcite with the updated data and periods . since the cepheids in the ogle database are truncated at @xmath12 , due to the saturation of the ccd detector for the longer period ( hence brighter ) cepheids @xcite , we include some additional lmc cepheid data from @xcite , @xcite and @xcite to extend the period coverage to @xmath13 in our sample . the requirements that govern our choice of the published photometric data are : ( a ) latest observations that use the modern day ccd cameras ; ( b ) high quality data with large number of data points per light curve , which provide uniform phase coverage and small scatter of the light curve ; and ( c ) as homogeneous as possible ( i.e. , from a minimal number of sources ) to avoid any additional systematic errors . these requirements are essential to construct accurate light curves to allow the estimation of colours and magnitudes at maximum , mean and minimum light for our pc and ac study . hence we did not include some of the older photometric data in this study . the photometric data of all cepheids , comprising 771 from ogle database , 14 from @xcite+@xcite and 39 from @xcite , were mainly fit with @xmath14 to @xmath15 fourier expansions ( @xmath16 is the order of fourier expansion ) using the simulated annealing method described in @xcite to the @xmath7- and @xmath11-band photometric data . this is in contrast to paper i that only applied @xmath14 fourier fits . however , for some of the ogle long period cepheids ( @xmath17 days ) , it was found out that the quality of the fitted light curves could be improved by using a higher order fourier expansion , hence we extended the fit to @xmath18 for these long period cepheids . all the fitted light curves were visually inspected and the best - fit light curves from the different orders of the fourier expansions were selected . to the best of our knowledge , this analysis also represents a major improvement in the fourier analysis of the ogle data . the extinction is corrected with the standard procedure , i.e. @xmath19 with @xmath20 and @xmath21 @xcite . the values of @xmath22 for each ogle cepheids are taken from the ogle database @xcite , while for the cepheids in @xcite+@xcite and in @xcite , the values of @xmath22 are adopted from @xcite and/or @xcite . to guard against some `` bad '' cepheids or other contamination in our sample , and select only the good cepheids in both bands , we removed some cepheids in the sample according to the following criteria ( see also * ? ? ? * ) : = 7.5 cm 1 . cepheids without @xmath7- and/or @xmath11-band photometry , or the number of data per light curve ( in either bands or both ) is too low to fit a @xmath14 fourier expansion . cepheids with poorly fitted or unacceptable @xmath7- and/or @xmath11-band light curves in the sample , such as those with a large scatter of data points or with bad - phase coverage ( large gaps between the phased data points ) . most of the magnitudes , as well as the colours , at the maximum and/or minimum light from these fitted light curves are very uncertain . cepheids with possible duplicity in the ogle sample . some of the possible duplicated cepheids were removed in the ogle database by consulting table 4 of @xcite . 4 . cepheids with unusual colour . we first plot out ( as in figure [ figcut][a ] ) the extinction corrected pc relation at mean light . the plot shows that there are number of outliers in the period - colour plane , mostly with @xmath23 . the presence of these outliers is probably due to : ( a ) their extinction is either over- or under - estimated ; ( b ) they have blue or red companions that can not be resolved due to the problems of blending ; or ( c ) other unknown physical reasons . a detailed investigation of these outliers is beyond the scope of this paper , but it is clear that they should be removed from the sample . these outliers are removed with the adopted colour - cut of @xmath24 , a compromise between maximizing the number of cepheids in the sample and excluding the cepheids with unusual colour . cepheids with unusually low ( or high ) amplitude . some cepheids with unusually low @xmath7- and @xmath11-band amplitudes were found in the sample . their amplitudes are typically @xmath25 times smaller as compared to the amplitudes of other cepheids at given period . some examples of the light curves for these low amplitude cepheids are given in @xcite . in addition , most of the light curves for these low amplitude cepheids can be fitted with @xmath14 fourier expansion , while other cepheids with `` normal '' amplitude may require higher order fits . @xcite has briefly discussed some possible physical reasons for these cepheids to have such low amplitudes , e.g. they are just entering or leaving the fundamental mode instability strip @xcite or they have different chemical composition ( see , e.g. , * ? ? ? the detailed investigation of these low amplitudes cepheids is beyond the scope of this paper . here , we apply a conservative amplitude cut of @xmath26 mag . in the @xmath7-band to remove the low amplitude cepheids . besides that , we also remove ogle-286532 ( with unusually low amplitude ) and hv-2883 ( with unusually high amplitude ) as they are clear outliers in the @xmath27-amplitude plot ( not shown , but see * ? ? ? note that @xcite applied a cut of @xmath28 mag . to remove the low amplitude cepheids in ngc 6822 . other examples of removing the low - amplitude cepheids can also be found in @xcite . 6 . cepheids with @xmath29 and @xmath30 . in order to guard against possible contamination from the first overtone cepheids @xcite and to be consistent with the previous studies @xcite , we removed cepheids with @xmath29 ( see further justification in @xcite ) . regarding the removal of cepheids with @xmath30 , a preliminary analysis of the pc relation reveals that few of the longest cepheids should be removed from the sample , because they are clear outliers in the pc plot at maximum light ( see upper panel of figure [ figpcmax ] ) . without these longest period cepheids , the pc(max ) relation for the long period cepheids is flat , which is consistent with the results found in paper i. the hypothesis of the hif - photosphere interaction also suggests the flatness of the pc relation at maximum light for long period cepheids . however , as the period gets longer ( with @xmath31>1.5 $ ] ) , the photosphere disengages from the hif @xcite . these longest period cepheids have biased the slope of the pc(max ) relation by making the slope becomes steeper . these selection criteria are guided mainly by the philosophy that it is better to lose some `` bad '' but real cepheids rather than including those spurious and doubtful cepheids in the sample @xcite , or those with bad fitted light curves that will give inaccurate measurements of the maximum and minimum light . hence , the final sample consists of 641 lmc cepheids that will be considered further . the locations of the outliers from various selection criteria are shown in figure [ figcut ] for the pc(mean ) relation , @xmath7-band pl relation , @xmath32-@xmath27 relation , where @xmath33 are the fourier amplitudes . see @xcite and @xcite for details . ] and the colour - magnitude diagram ( cmd ) . note that some of the outliers are located within the `` good '' cepheids . however they can be eliminated due to various physical reasons as given above , especially those with poorly fit light curves that will give inaccurate measurements at maximum , mean and minimum light . a simple sigma - clipping algorithm ( e.g. , * ? ? ? * ) will not be able to remove these outliers @xcite . to construct the empirical pc & ac relations , we used the following quantities from the fourier fits to the cepheid data as obtained from previous section : * @xmath7-band amplitude : the difference of the numerical maximum and minimum from the fourier expansion , @xmath34 . * @xmath35 : defined as @xmath36 , where @xmath37 is the @xmath11-band magnitude at the same phase as @xmath38 . * @xmath39 : defined as @xmath40 , where @xmath41 is the mean value from the fourier expansion ( see * ? ? ? this is very similar to the conventional definition of the mean colour , @xmath42 , where @xmath43 denotes the intensity mean . * @xmath44 : defined as @xmath45 , where @xmath46 is the @xmath11-band magnitude at the same phase as @xmath47 . @xmath47 is the @xmath7-band magnitude closest to @xmath48 , the mean value from fourier expansion . * @xmath49 : defined as @xmath50 , where @xmath51 is the @xmath11-band magnitude at the same phase as @xmath52 . these quantities have been corrected for extinction as mentioned in previous section . the empirical lmc pc and ac relations at maximum , mean and minimum light for all , long and short period cepheids are summarized in table [ c9tabpc ] & [ c9tabac ] , and the corresponding plots are presented in figure [ c9figpc ] & [ c9figac ] , respectively . lccc phase & @xmath53 & @xmath54 & @xmath55 + + maximum & @xmath56 & @xmath57 & 0.099 + mean & @xmath58 & @xmath59 & 0.075 + phmean & @xmath60 & @xmath61 & 0.081 + minimum & @xmath62 & @xmath63 & 0.075 + + maximum & @xmath64 & @xmath65 & 0.098 + mean & @xmath66 & @xmath67 & 0.075 + phmean & @xmath68 & @xmath69 & 0.092 + minimum & @xmath70 & @xmath71 & 0.081 + + maximum & @xmath72 & @xmath73 & 0.097 + mean & @xmath74 & @xmath75 & 0.074 + phmean & @xmath76 & @xmath77 & 0.078 + minimum & @xmath78 & @xmath79 & 0.073 + lccc phase & @xmath53 & @xmath54 & @xmath55 + + maximum & @xmath80 & @xmath81 & 0.093 + mean & @xmath82 & @xmath83 & 0.092 + phmean & @xmath84 & @xmath85 & 0.099 + minimum & @xmath86 & @xmath87 & 0.098 + + maximum & @xmath88 & @xmath89 & 0.086 + mean & @xmath90 & @xmath91 & 0.085 + phmean & @xmath92 & @xmath93 & 0.103 + minimum & @xmath94 & @xmath95 & 0.082 + + maximum & @xmath96 & @xmath97 & 0.076 + mean & @xmath98 & @xmath99 & 0.074 + phmean & @xmath100 & @xmath101 & 0.078 + minimum & @xmath102 & @xmath103 & 0.079 + to test the non - linearity of the pc and ac relations , or the `` break '' at a period of 10 days , we apply the @xmath2-test as given in paper i and in @xcite . the null hypothesis in the @xmath2-test is single line regression is sufficient , while the alternate hypothesis is that two lines regressions with a discontinuity ( a break ) at 10 days is necessary to fit the data . the probability @xmath104 , under the null hypothesis , can be obtained with the corresponding @xmath2-values and the degrees of freedom . in general , the large value of @xmath2 ( equivalent to the small value of @xmath105 $ ] ) indicates that the null hypothesis can be rejected . for our sample , @xmath106 when @xmath107 ( the 95% confident level ) , therefore the null hypothesis can be rejected if the @xmath2-value is greater than @xmath108 with more than 95% confident level and the data is more consistent with the two - line regression . a glance of table [ c9tabpc ] and figure [ c9figpc ] suggests that the lmc pc relations are broken at maximum , mean and minimum light . these are confirmed with the @xmath2-test results with @xmath109 . similarly , the @xmath2-test results for the ac relation are : @xmath110 . hence , the lmc pc and ac relations are non - linear ( hence broken ) at maximum , means and minimum light . note that the flatness of the long period pc(max ) relation as given in table [ c9tabpc ] ( @xmath64 ) is in good agreement with the slope found in paper i ( @xmath111 ) . recall that equation ( 1 ) predicts that if the pc relation is flat at maximum light , then there is a correlation between the amplitude and the colour at minimum light . this is seen in table [ c9tabac ] ( and in figure [ c9figac ] ) for the long period ac(min ) relation , with a slope of @xmath112 . the stellar pulsation codes we used are both linear @xcite and non - linear @xcite . these codes , which include a 1-d turbulent convection recipe @xcite , are the same as in paper ii . briefly speaking , the codes take the mass ( @xmath113 ) , luminosity ( @xmath114 ) , effective temperature ( @xmath115 ) and chemical composition ( @xmath116 ) as input parameters . the chemical composition is set to be @xmath117 to represent the lmc hydrogen and metallicity abundance ( by mass ) . the mass and luminosity are obtained from the ml relations calculated from evolutionary models . the @xmath115 are chosen to ensure the models oscillate in the fundamental mode and located inside the cepheid instability strip . the pulsation periods for the models are obtained from a linear non - adiabatic analysis @xcite . all other parameters used in the pulsation codes had the same values for the lmc and galactic models ( paper ii ) . this included the @xmath118 parameters that are part of the turbulent convection recipe , though see section 8 . of course , one variable parameter was the metallicity . the only other difference between this study and paper ii , besides the metallicity , is the value set for the artificial viscosity parameter , @xmath119 . in this study , we set @xmath120 for the lmc models to improve the shape of the theoretical light curves , in contrast to the value of @xmath121 used for the galactic models . in paper ii , the ml relations are adopted from @xcite and @xcite . in order to be consistent with previous work , the ml relations used in this paper will also be adopted from these two sources . however , @xcite only provided two ml relations , one for @xmath122 which are used in paper ii , and another one for @xmath123 . hence we have to adopt the second ml relation for the lmc models . even though the lmc metallicity is higher than @xmath123 , the lmc is still considered as a low metallicity system in the literature . hence the @xcite ml relation can be approximately applied for the lmc models . an anonymous referee pointed out that an interpolation of the @xcite ml relations between @xmath122 and @xmath123 should also be used . we have included the interpolated ml relation in our model calculations . in the context of the hif - photosphere interaction , it is the ml relation which dictates at what period and at what phases this will occur . stellar evolutionary theory changes the ml relation as a function of metallicity . hence the coefficients of the ml relation are important in determining the nature of the hif - photosphere interaction ( paper ii ) . in short , the ml relations used are : ccccccc @xmath113 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath128 & @xmath129 + + 11.0 & 4.375 & 5050 & 46.4155 & 0.124 & 28.98 & -0.118 + 10.0 & 4.236 & 5100 & 35.6727 & 0.091 & 22.92 & -0.093 + 9.50 & 4.161 & 5250 & 28.2406 & 0.094 & 18.68 & -0.046 + 9.10 & 4.099 & 5260 & 25.3960 & 0.082 & 16.92 & -0.042 + 8.75 & 4.042 & 5310 & 22.3804 & 0.076 & 15.07 & -0.027 + 8.40 & 3.982 & 5380 & 19.3886 & 0.071 & 13.20 & -0.008 + 7.95 & 3.902 & 5330 & 17.7750 & 0.055 & 12.09 & -0.027 + 7.00 & 3.717 & 5410 & 12.6085 & 0.035 & 8.722 & -0.018 + 6.55 & 3.620 & 5490 & 10.2940 & 0.031 & 7.183 & -0.006 + 6.40 & 3.587 & 5485 & 9.81474 & 0.027 & 6.853 & -0.010 + 6.00 & 3.493 & 5510 & 8.37226 & 0.020 & 5.866 & -0.014 + 5.90 & 3.468 & 5500 & 8.12498 & 0.017 & 5.691 & -0.018 + 5.80 & 3.443 & 5525 & 7.69466 & 0.017 & 5.400 & -0.015 + 5.70 & 3.418 & 5560 & 7.23505 & 0.017 & 5.090 & -0.009 + 5.30 & 3.312 & 5600 & 6.01283 & 0.012 & 4.244 & -0.009 + + 7.20 & 4.272 & 5380 & 40.2561 & 0.275 & 24.51 & -0.162 + 6.80 & 4.192 & 5380 & 35.4374 & 0.264 & 21.91 & -0.122 + 6.20 & 4.063 & 5410 & 28.2629 & 0.225 & 17.94 & -0.076 + 5.95 & 4.005 & 5420 & 25.6378 & 0.211 & 16.43 & -0.060 + 5.40 & 3.869 & 5510 & 19.4314 & 0.170 & 12.82 & -0.015 + 5.15 & 3.803 & 5510 & 17.5523 & 0.160 & 11.66 & -0.007 + 4.65 & 3.660 & 5490 & 14.3143 & 0.131 & 9.611 & -0.010 + 4.20 & 3.518 & 5510 & 11.3659 & 0.101 & 7.729 & -0.011 + 4.00 & 3.450 & 5545 & 10.0011 & 0.089 & 6.854 & -0.005 + 3.95 & 3.432 & 5540 & 9.77393 & 0.085 & 6.701 & -0.008 + 3.80 & 3.378 & 5550 & 8.94637 & 0.075 & 6.157 & -0.009 + 3.70 & 3.341 & 5575 & 8.31297 & 0.070 & 5.745 & -0.005 + 3.65 & 3.322 & 5570 & 8.10751 & 0.066 & 5.605 & -0.008 + 3.60 & 3.302 & 5530 & 8.09994 & 0.058 & 5.583 & -0.022 + 3.60 & 3.302 & 5600 & 7.71463 & 0.065 & 5.352 & -0.001 + + 6.80 & 4.092 & 5280 & 30.9661 & 0.207 & 19.51 & -0.091 + 5.20 & 3.701 & 5340 & 15.9744 & 0.094 & 10.63 & -0.050 + 4.40 & 3.457 & 5460 & 10.0562 & 0.056 & 6.898 & -0.028 + 4.20 & 3.389 & 5550 & 8.51317 & 0.054 & 5.906 & -0.007 + 3.80 & 3.243 & 5630 & 6.46303 & 0.039 & 4.532 & -0.002 + 1 . ml relation given in @xcite : @xmath130 2 . ml relation given in @xcite : @xmath131 3 . ml relation interpolated between two @xcite relations at @xmath132 and @xmath123 to yield a relation at @xmath133 : @xmath134 the units for both @xmath113 and @xmath114 are in solar units . note that these ml relations cover reasonably broad @xmath10 ratios given in the literature . the input parameters for the lmc models with these ml relations and the periods calculated from linear non - adiabatic analysis are given in table [ tabinput ] . after the full amplitude models are constructed from the pulsation codes , the temperature and the opacity profile can be plotted in terms of the internal mass distribution ( @xmath135 $ ] , where @xmath136 is mass within radius @xmath137 and @xmath113 is the total mass ) at a given phase of pulsation . as in paper ii , the locations of the hif ( sharp rise in the temperature profile ) and photosphere ( at optical depth @xmath138 ) can be identified in the temperature profile . to quantify the hif - photosphere interaction ( if the photosphere is next to the base of the hif or not , see also paper ii ) , we calculate the `` distance '' , @xmath139 , in @xmath140 between the hif and the photosphere from the temperature profile . the definition of @xmath139 can be found in paper ii . a small @xmath139 means there is a hif - photosphere interaction , and vice versa . the theoretical quantities from the models can be compared to the observed quantities using the following prescriptions : ccccc @xmath6 & @xmath141 & @xmath5 & @xmath142 & @xmath143 + + 46.4155 & 27481.28 & 5445.00 & 18329.74 & 4826.00 + 35.6727 & 19742.83 & 5434.01 & 14321.20 & 4962.65 + 28.2406 & 16894.15 & 5502.35 & 12093.99 & 4978.72 + 25.3960 & 14460.62 & 5562.28 & 10570.70 & 5010.85 + 22.3804 & 12772.75 & 5615.34 & 9283.257 & 5069.43 + 19.3886 & 11263.84 & 5687.98 & 8143.460 & 5145.20 + 17.7750 & 9034.392 & 5561.39 & 7044.743 & 5147.04 + 12.6085 & 5637.467 & 5528.15 & 4800.840 & 5278.11 + 10.2940 & 4371.170 & 5537.42 & 3839.001 & 5377.80 + 9.81474 & 4004.569 & 5518.39 & 3580.396 & 5383.61 + 8.37226 & 3217.688 & 5596.34 & 2929.354 & 5437.17 + 8.12498 & 3032.639 & 5581.53 & 2793.441 & 5438.96 + 7.69466 & 2864.745 & 5602.37 & 2640.439 & 5467.54 + 7.23505 & 2705.430 & 5639.03 & 2490.271 & 5504.98 + 6.01283 & 2105.589 & 5669.57 & 1986.586 & 5567.91 + + 40.2561 & 23500.98 & 5821.99 & 11853.38 & 4947.11 + 35.4374 & 19604.60 & 5830.18 & 9898.124 & 4962.26 + 28.2629 & 14661.29 & 5875.98 & 7659.896 & 5057.96 + 25.6378 & 12836.04 & 5884.01 & 6916.299 & 5108.02 + 19.4314 & 9538.226 & 5985.21 & 5555.179 & 5303.60 + 17.5523 & 8035.700 & 5921.07 & 4947.967 & 5142.58 + 14.3143 & 5490.345 & 5783.01 & 3639.964 & 5164.11 + 11.3659 & 3850.971 & 5814.49 & 2722.473 & 5242.47 + 10.0011 & 3286.030 & 5817.00 & 2377.950 & 5300.52 + 9.77393 & 3136.139 & 5801.77 & 2301.751 & 5296.73 + 8.94637 & 2738.003 & 5798.30 & 2071.106 & 5331.19 + 8.31297 & 2499.416 & 5794.49 & 1917.696 & 5365.34 + 8.10751 & 2374.713 & 5780.35 & 1850.048 & 5369.40 + 8.09994 & 2235.292 & 5723.18 & 1796.586 & 5355.41 + 7.71463 & 2271.393 & 5805.49 & 1772.270 & 5408.34 + + 30.9661 & 14958.03 & 5611.77 & 8486.049 & 4963.81 + 15.9744 & 5697.042 & 5665.97 & 4209.487 & 5079.81 + 10.0562 & 3197.451 & 5670.64 & 2560.871 & 5283.85 + 8.51317 & 2723.566 & 5716.48 & 2201.628 & 5382.21 + 6.46303 & 1852.537 & 5684.63 & 1593.714 & 5507.38 + = 7.5 cm 1 . as in paper ii , we use the basel atmosphere database @xcite to construct a fit giving temperature and effective gravity as a function of @xmath144 colour . the effective gravity is obtained at the appropriate phase from the models ( see paper ii ) . these prescriptions are used to convert the temperatures to the @xmath144 colours . the bolometric corrections ( @xmath145 ) are obtained in a similar manner . the anonymous referee has suggested that @xmath144 may not be a good way to convert between temperature and colour unless both of the micro - turbulence and surface gravity are included . as indicated above this is the case , and in any case our results and those of paper ii for galactic models , show good agreement between theory and observations . a number of previous authors have used this method and some authors commented that this colour can be used as an indicator of temperature ( e.g. * ? ? ? * ; * ? ? ? the empirical relations we studied in this series were also mainly in the @xmath144 colour . 2 . in addition to the basel atmosphere , we also use the atmosphere fit from @xcite , referring this as the sbt atmosphere in our paper . the sbt atmosphere does include both of the effective gravity and the micro - turbulence in their table 6 for the temperature and colour conversion . these conversions are tabulated for two micro - turbulence velocities of @xmath146 and @xmath147 , as well as for various metallicities . to apply these conversions to our lmc models , we first interpolated the conversions between @xmath148=0.0 $ ] and @xmath148=-0.5 $ ] to @xmath148=-0.3 $ ] , which is appropriate for the lmc metallicity . the @xmath144 colours at the maximum , mean and minimum light are then obtained from the given effective temperature and the effective gravity for both of the micro - turbulence velocities . we use the prescriptions given in @xcite to convert the observed colours to the temperatures appropriate for the lmc data as follows : @xmath149 + note that these functions are also obtained from the basel atmosphere database . we can compare the colours obtained from the basel and sbt atmosphere for our models constructed in this paper . the results are presented in figure [ figatmos ] . from this figure it can be seen that the colours obtained from both of the atmosphere fits agree within @xmath150mag . the difference is even smaller if the micro - turbulence velocity of @xmath147 is used . this indicates that the @xmath144 colours can be used to indicate the temperature . since the results of our models are qualitatively compared to the observations ( see next section ) and not used to quantitatively derive any theoretical pc and/or ac relations , an accuracy of @xmath151mag . , independent of period , from the atmosphere fit is acceptable and would not cause problems for our results . note that the sbt atmosphere are only defined for @xmath152 and @xmath153 , few of our models either the @xmath115 or @xmath154 or both are beyond these ranges at certain phases , hence no colours can be obtained from the sbt atmosphere ( for example some points are missing at minimum light for few of the long period models , as shown in figure [ figatmos ] ) . due to these reasons , we continue adopt the basel atmosphere fits to convert the temperatures and @xmath144 colours , after taking account of the effective gravity in the fits , as a function of phase . cccccccc @xmath6 & @xmath155 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 ( asc ) & @xmath160 ( des ) + + 46.4155 & 24249.2 & 24423.695 & 5330.06 & 24282.129 & 4882.55 & 5319.42 & 4880.66 + 35.6727 & 17207.7 & 17078.471 & 5293.90 & 17188.051 & 4922.99 & 5305.14 & 4924.47 + 28.2406 & 14484.7 & 14466.869 & 5457.79 & 14529.348 & 5073.65 & 5459.55 & 5069.64 + 25.3960 & 12540.8 & 12461.087 & 5443.66 & 12544.735 & 5096.58 & 5452.77 & 5096.18 + 22.3804 & 10997.1 & 10996.587 & 5493.44 & 11039.123 & 5157.95 & 5493.51 & 5152.87 + 19.3886 & 9587.02 & 9560.1922 & 5545.87 & 9618.0583 & 5232.85 & 5549.58 & 5228.51 + 17.7750 & 7983.35 & 7988.5457 & 5473.94 & 7994.9159 & 5213.83 & 5473.06 & 5211.77 + 12.6085 & 5208.60 & 5208.1564 & 5493.34 & 5202.6440 & 5349.00 & 5493.36 & 5350.84 + 10.2940 & 4170.31 & 4162.1135 & 5560.81 & 4174.0993 & 5450.34 & 5564.43 & 5449.06 + 9.81474 & 3860.04 & 3861.4313 & 5556.01 & 3852.8715 & 5446.51 & 5555.34 & 5449.11 + 8.37226 & 3109.66 & 3106.7337 & 5570.85 & 3110.1806 & 5481.59 & 5572.55 & 5481.37 + 8.12498 & 2939.41 & 2935.0599 & 5553.67 & 2939.6504 & 5476.34 & 5556.38 & 5476.23 + 7.69466 & 2775.73 & 2775.1245 & 5581.66 & 2777.4057 & 5501.67 & 5582.05 & 5500.77 + 7.23505 & 2618.53 & 2620.1965 & 5620.29 & 2618.8674 & 5534.39 & 5619.15 & 5534.20 + 6.01283 & 2051.97 & 2050.8394 & 5646.29 & 2051.3277 & 5583.97 & 5647.27 & 5584.50 + + 40.2561 & 18754.3 & 18580.019 & 5704.60 & 18824.542 & 5144.86 & 5718.84 & 5141.42 + 35.4374 & 15896.6 & 15885.248 & 5749.07 & 15861.179 & 5147.12 & 5750.16 & 5158.63 + 28.2627 & 11157.8 & 11188.594 & 5692.54 & 11143.155 & 5111.26 & 5688.21 & 5112.18 + 25.6378 & 10388.8 & 10366.128 & 5782.33 & 10358.696 & 5184.81 & 5785.59 & 5188.79 + 19.4314 & 7743.89 & 7741.3087 & 5871.14 & 7750.7288 & 5332.20 & 5871.60 & 5330.76 + 17.5523 & 6538.19 & 6537.3819 & 5833.02 & 6532.0823 & 5310.84 & 5833.20 & 5312.28 + 14.3143 & 4564.83 & 4621.4163 & 5750.64 & 4565.7876 & 5280.59 & 5732.80 & 5280.34 + 11.3659 & 3289.17 & 3288.4584 & 5711.02 & 3281.3799 & 5328.36 & 5711.31 & 5331.32 + 10.0011 & 2811.35 & 2823.6619 & 5726.94 & 2800.7972 & 5376.66 & 5721.20 & 5381.50 + 9.77393 & 2702.54 & 2712.0209 & 5715.81 & 2692.4263 & 5378.63 & 5711.29 & 5383.51 + 8.94637 & 2385.34 & 2395.4576 & 5704.28 & 2388.3903 & 5412.32 & 5699.14 & 5410.62 + 8.31297 & 2189.29 & 2193.3351 & 5708.03 & 2188.7842 & 5448.47 & 5705.90 & 5448.80 + 8.10751 & 2094.38 & 2103.6034 & 5698.27 & 2093.0120 & 5449.90 & 5693.36 & 5450.83 + 8.09994 & 2004.90 & 1998.5077 & 5648.26 & 1999.8838 & 5419.02 & 5651.76 & 5422.70 + 7.71463 & 2004.07 & 2009.2157 & 5713.42 & 2004.9007 & 5490.38 & 5710.72 & 5489.80 + + 30.9661 & 12359.6 & 12202.760 & 5566.66 & 12354.904 & 5029.65 & 5585.98 & 5030.07 + 15.9744 & 5011.60 & 5031.4017 & 5550.08 & 5024.6207 & 5173.09 & 5544.24 & 5169.84 + 10.0562 & 2860.25 & 2866.2012 & 5593.28 & 2859.5002 & 5346.96 & 5590.81 & 5347.35 + 8.51317 & 2447.87 & 2441.2040 & 5643.58 & 2453.1647 & 5463.35 & 5646.11 & 5460.12 + 6.46303 & 1749.72 & 1749.6046 & 5719.11 & 1744.6691 & 5577.03 & 5719.24 & 5581.12 + the effective temperatures for the full amplitude models in table [ tabinput ] at the corresponding maximum and minimum light ( or luminosity ) are given in table [ c9tabmaxmin ] . for the effective temperatures at mean light , the temperatures for the mean light at ascending and descending branch of the light ( or luminosity ) curve are not the same ( e.g. , in paper ii ) , hence table [ c9tabmean ] gives the effective temperature at these phases for our lmc models . the layout of table [ c9tabmean ] is the same as table 3 from paper ii . following paper ii , the locations of the photosphere can be identified in the temperature and opacity profiles . these are displayed in figure [ c9bono4]-[c9chiosi13 ] with a @xmath161 , a @xmath162 and a @xmath23 model , respectively . the left and right panels of figure [ c9bono4]-[c9chiosi13 ] are the temperature and opacity profiles respectively . the photospheres are marked as filled circles in these figures . finally , the plots of the @xmath139 , the `` distance '' between the photosphere and the hif from the temperature profiles , as a function of pulsating period for the lmc models are presented in figure [ c9deltalmc ] with the three ml relations used . in paper ii , it is found that the distribution of @xmath139 as a function of period is almost independent of the adopted ml relation . this is also seen in the lmc models as depicted in figure [ c9deltalmc ] . figure [ c9bono4]-[c9chiosi13 ] and figure [ c9deltalmc ] bear witness to the fact that at maximum light , the photosphere lies at the base of the hif for all of the models . although there is a slight deviation for some longer period models , the location of the photosphere is close to the hif within the error bars ( which are defined as the coarseness of the grid points around the location of the hif ) . as in paper ii for the galactic models , the closeness of the photosphere to the base of the hif , for reasonably low densities , results in a flat or almost flat pc relation for the long period lmc cepheids . in the case of minimum light , even though figure [ c9deltalmc ] implies that @xmath163 is nearly constant across the period range and the photosphere is near the base of the hif , as in the case of maximum light , @xmath163 does follow a shallow correlation with period after 10 days . judging from the error bars of @xmath163 and from figure [ c9bono4]-[c9chiosi13 ] , there is tentative evidence that the photosphere is disengaged from the hif for @xmath161 at minimum light . hence the temperatures or the colours at minimum light are more dependent on period for @xmath164-@xmath27 relation may not be correlated with the slopes of the pc relation ] and the global properties . theoretical quantities that can be computed from the models and compared with data include the pulsation periods , the @xmath7-band amplitudes and the fourier parameters , the temperatures and colours at the maximum , mean and minimum light . these are the pc plots , the ac plots , the period - temperature plots and the fourier parameters plots portrayed in figures [ c9modelpc]-[c9lmcfourier ] . the temperatures in table [ c9tabmaxmin ] & [ c9tabmean ] , after conversion to the @xmath144 colours as mentioned in previous section , are superimposed along with the observed lmc pc relations as plotted in figure [ c9modelpc ] . similarly , figure [ c9modelpt ] graphs the same quantities but on the @xmath165-@xmath27 plane with the observed @xmath144 colours converted to temperatures using the prescriptions given in section 5 . the theoretical bolometric light curves are converted to the @xmath7-band light curves with the bolometric corrections obtained from the basel database mentioned previously . from the theoretical @xmath7-band light curves , the amplitudes can be estimated and these are displayed in figure [ c9modelac ] along with the colours from models to compare with the empirical ac relations . the fourier parameters of the theoretical @xmath7-band light curves can also be obtained with ( @xmath166 ) fourier expansion . these fourier parameters are compared with the observational data in figure [ c9lmcfourier ] . several features are noticed from figure [ c9modelpc]-[c9lmcfourier ] : 1 . the general trends of the models qualitatively match the observational data . there are greater discrepancies between the data and short period models , particularly in matching the observed light curve amplitudes . 2 . the models with the ml relation from @xcite , with lower @xmath10 ratio , do better in matching the observations . these models also tend to lie near the envelopes of the pc , ac , @xmath165-@xmath27 and @xmath167-@xmath27 relations defined by the observational data . 3 . the slopes of the period - colour ( or period - temperature ) relations at maximum and minimum light from the models roughly match the observational data , i.e. , the theoretical pc(max ) relation is approximately flat and there is a relation at minimum light . 4 . the temperatures from the models with the @xcite ml relation is cooler ( hence redder ) than the models with the @xcite ml relation and the observed data at maximum light . in contrast , the temperatures ( or the colours ) at minimum light from the models with these two ml relations are consistent with each other and are located near the blue edge of the observed data . the means at the descending branches are in better agreement with the observed data than the means at the ascending branches . this is because the observed means , @xmath168 , are obtained mostly from the descending branches . though previous researchers have noted that temperatures on the ascending and descending branches are not the same at mean light ( as cepheids exhibit loops in cmd ) , what is new here is the way the nature of the hif changes during the pulsation . the behaviors of the models from the interpolated @xcite ml are closer to the models from @xcite ml relation because their slopes are very similar . the amplitudes of the theoretical light curves ( in both of the bolometric and @xmath7-band light curves ) are smaller than the observations at given period , especially for the models with the @xcite ml relation . these can be seen from the ac relations as given in figure [ c9modelac ] and the left panels of figure [ c9lmcfourier ] . overall , some agreements and disagreements are found between the theoretical quantities and the observational data . it is also found out that there are some problems associated with the pulsation codes when the lmc models are constructed : these include the smaller amplitude of the model light curves and the cooler temperatures at the maximum light ( especially with @xcite ml relation ) . note that from equation ( 1 ) , cooler temperatures at maximum light imply that the amplitudes will be lower at given period . varying other parameters in the pulsation codes , including the @xmath118 parameters , does not improve the situation , though perhaps a more detailed and systematic study of the dependence of lmc cepheid pulsation models on the @xmath169 parameters could resolve this situation . however , we believe that the qualitative nature of the photosphere - hif interactions as given in figure [ c9deltalmc ] will still hold even in models which fare better in mimicking observed amplitudes . this is in part because figure [ c9deltalmc ] suggest that the behaviors of @xmath139 as a function of period are nearly , though not completely , independent of amplitudes , as the models with @xcite ml relation have higher amplitudes ( although still smaller than the observations ) than the models with the @xcite ml relation . however , better codes that fix these problems or the 3-d convection codes are needed in the future studies . the temperature profiles from the galactic models given in paper ii and the lmc models are compared in figure [ c9pt ] at maximum and minimum light . the upper panels of figure [ c9pt ] suggest that at maximum light , the photosphere is not far from the base of the hif in both of the galactic and the lmc models . in contrast , the photosphere is further away from the hif in the galactic models than the lmc models at minimum light . the hif is located further out in the mass distribution for the galactic models . the plots of the @xmath139-@xmath27 relation from the galactic and lmc models at maximum and minimum light are also compared in figure [ c9delta ] . it can be seen from the figure that at maximum light , the behavior of both galactic and lmc models is similar , where the photosphere is near the base of the hif . at minimum light , the long period models show that the photosphere is disengaged from the hif , while the behavior of the short period models is different between the galactic and lmc models . the photosphere of the short period lmc models seems to be located closer to the hif at minimum light , but it is not the case for the short period galactic models . this could lead to shallower slopes of the pc(min ) relation seen in the lmc cepheids as compared to the galactic counterparts . in terms of the hif - photosphere interaction , there is some tentative evidence from the models that the lmc long period cepheids behave like the galactic cepheids , while the short period lmc cepheids behave like the rr lyrae stars at minimum light . figure [ c9density ] graphs the density ( defined as @xmath170 , where @xmath7 is the specific volume ) at the photosphere as a function of the period of the model at minimum , maximum and ascending and descending mean light . galactic models generally tend to have the lowest density and , in particular , have significantly lower densities at minimum light than the lmc models . we note that the galactic models always have a photospheric density lower than about @xmath171 whereas the photospheric density for the lmc models only falls below this figure after a period of 10 days . at maximum light , all long period models have a low photospheric density . what we get from this figure is that it provides some evidence that there is a difference in photospheric density between the lmc and galactic models . moreover , this difference appears to be consistent with what is required by our theoretical scenario : short period lmc models have a higher photospheric density than their galactic counterparts . however , for a discussion of some caveats , see section 8 . we now discuss two important applications of the photosphere - hif interaction : reddening corrections and the explanation of the observed non - linear lmc pl ( and pc ) relations . @xcite original interest in the spectral properties of cepheids at maximum light was to estimate reddening . skm used this to correct a number of reddening for galactic cepheids . @xcite used equation ( 1 ) and the theoretical explanation provided in skm to derive a relation linking the colour excess to the colour at maximum light , the @xmath7-band amplitude and the period . such a relation is predicted from equation ( 1 ) . @xcite estimates the error with this method to be comparable to other multi - colour methods . a more interesting application of the hif - photosphere interaction is to explain the recent detected non - linear lmc pl relation as presented in @xcite , paper i , @xcite and @xcite . paper i used the @xmath2-test to provide strong statistical evidence that the optical cepheid pl relation at mean light in the lmc is non - linear around a period close to 10 days . @xcite used the macho and 2mass datasets together with additional long period cepheids from the literature to further support the existence of non - linear lmc pl relation in the optical and near infra - red wave - bands . in contrast , current data indicate that the galactic pl relation is linear at mean light @xcite . non - linearity of the lmc pl relations can be tested using the @xmath2-test with the data given in section 3 . the empirical results of the fitted lmc pl relations at maximum , mean and minimum light using the updated data are presented in table [ c9tabpl ] . the plots of the pl relations at maximum / minimum light and at mean light are shown in figure [ c9plmaxmin ] & [ c9plmean ] , respectively . the @xmath2-test results for these pl relations are : @xmath172 , and @xmath173 . the large @xmath2-values for both @xmath7- and @xmath11-band pl relations at mean and minimum light strongly indicate that the pl relations at these two phases are not linear , and the data is better described with the broken ( i.e , two regressions ) pl relation . however , the small @xmath2-values at maximum light , with corresponding @xmath174-values of @xmath175 and @xmath176 for the @xmath7- and @xmath11-band pl(max ) relations respectively , show that the null hypothesis of the @xmath2-test can not be rejected ( a value of @xmath177 and/or @xmath178 is required for doing this ) . hence there is no observed break seen in the pl(max ) relation and the data is consistent with single line regression . note that the same slopes of the pl(max ) relations for long period cepheids in both bands are consistent of the finding that the pc(max ) relation is flat for these cepheids . lcccccc phase & @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 & @xmath184 + + maximum & @xmath185 & @xmath186 & 0.260 & @xmath187 & @xmath188 & 0.170 + mean & @xmath189 & @xmath190 & 0.208 & @xmath191 & @xmath192 & 0.141 + minimum & @xmath193 & @xmath194 & 0.204 & @xmath195 & @xmath196 & 0.140 + + maximum & @xmath197 & @xmath198 & 0.257 & @xmath199 & @xmath200 & 0.174 + mean & @xmath201 & @xmath202 & 0.228 & @xmath203 & @xmath204 & 0.158 + minimum & @xmath205 & @xmath206 & 0.243 & @xmath207 & @xmath208 & 0.174 + + maximum & @xmath209 & @xmath210 & 0.260 & @xmath211 & @xmath212 & 0.169 + mean & @xmath213 & @xmath214 & 0.203 & @xmath215 & @xmath216 & 0.138 + minimum & @xmath217 & @xmath218 & 0.194 & @xmath219 & @xmath220 & 0.131 + our tentative theoretical explanation for the non - linear nature of the lmc pl relations across a period of 10 days replies on the hif - photosphere interaction . @xcite and @xcite have established the connection between the pc and pl relations : both these relations arise from the more general plc relation . these relations refer to quantities evaluated at mean light . the existence of such a connection relies on the period - mean density theorem , the instability strip and the stefan - boltzmann law . if we assume the stefan - boltzmann law can be applied at every phase , then it is straightforward to show that a plc relation ( though possibly with different coefficients ) exists at every phase point . thus the standard plc relation and indeed the pc and pl relation expresses at mean light are just the averages of the same relations at different phases points . consequently one way to understand the behavior of plc / pl / pc relations at mean light is to understand their behavior at different phase points . what we try to do in this paper is point out some evidence from our models that shows how the changing behavior of the pc relations at different phases can , in principle , arise from a consideration of the photosphere - hif interaction at these phases . since the mean light pc and pl relation are the average of those at all phases , these properties can affect the pc and , as a consequence , the pl relation ( via the plc relation ) . in fact , the new data with superb phase resolution from such micro - lensing projects such as ogle and macho demands a multiphase analysis . this approach can potentially lead to a deeper understanding of the pulsation and evolution of cepheid variables . for example , @xcite looked at pc relations in the galaxy and lmc as a function of phase . they found that short and long period lmc cepheids have a shallower and steeper slope at most pulsation phases than galactic cepheids respectively . in this paper , we have confronted updated pc and ac relations at maximum , mean and minimum light for lmc cepheids observed by the ogle team , and additional cepheids from the literature , with theoretical , full amplitude pulsation models of lmc cepheids . the observed pc and ac relations provide compelling evidence of a non - linearity or break at a period of 10 days . we also constructed theoretical cepheid pulsation models appropriate for the lmc using the florida pulsation codes @xcite to study the hif - photosphere interaction . the empirical results presented in this paper , as well as in other papers such as @xcite and @xcite , provide strong empirical evidence that the pc and pl relations for the lmc cepheids are non - linear , in the sense described in previous sections . issues such as extinction and a lack of long period cepheids that may cause the non - linear lmc pl and pc relations have been addressed and argued against in paper i , @xcite and @xcite , and will not be repeated here . other arguments against the non - linear lmc pl relation include the results presented in @xcite , as the authors found no evidence for a non - linear pl relation in the lmc at @xmath221-bands . however , @xcite treated the data of @xcite extensively and found , in a statistically rigorous way , that the reason why @xcite found linear @xmath221 pl relations , is due to the small number of short period cepheids ( @xmath222 ) in their sample . @xcite also reduce the number of ogle / macho lmc cepheids and show how the @xmath2-test can produce a non - significant result when the number of short / long period cepheids become small . instead , using the 2mass data that are cross - correlated with macho cepheids , @xcite have found that the lmc @xmath223-band pl relations are non - linear - band than in @xmath224-band , as shown in @xcite . ] and the @xmath225-band pl relation starts to become linear . @xcite also discussed why this is the case . another argument against the non - linear pl relation is that the pl relation should be universal , as found in @xcite . we argue that their results are based on a handful of cepheids ( @xmath226 ) and on short periods cepheids in a cluster whose membership to the lmc is in question . their shallower galactic pl relation based on the revised infra - red surface brightness method also contradicts the steeper galactic pl relation based on independent methods from open cluster main - sequence fitting @xcite . it is worthwhile to point out that our sample selection does not affect the detection on non - linear lmc pl relation at mean light . since the mean magnitudes of a cepheid light curve is less affected by our constrains on selecting the cepheids with good light curves , we can use the published ( reddening corrected ) mean magnitudes to test the non - linear lmc pl relation . the anonymous referee kindly provided a large sample of lmc cepheids that combined the published mean @xmath7-band magnitudes from the ogle , @xcite and @xcite datasets . there are a total of 115 long period cepheids in this sample and the @xmath2-test still return a significant detection of the non - linear lmc pl relation . the ogle+@xcite combined data also give very similar results . similar tests have also been done in @xcite by using the macho data alone and the macho+@xcite combined data . the non - linear lmc pl relation is still present from the @xmath2-test results on these two datasets . therefore we believe our sample selection does not affect the detection of the non - linear lmc pl relation . the detection of non - linear lmc pl relation from totally independent ogle and macho data , using totally independent reddening estimates , suggested that this non - linearity is real and our paper is the first attempt to theoretically explain this non - linearity in terms of the hif - photosphere interaction . due to small number of lmc models , it is impossible to derive the theoretical pc and ac relations with a small error on the slope and compare directly to the empirical relations . however , these lmc models can be qualitatively compared to the observations by converting some physical quantities to the observable quantities and vice versa , such as the temperature - colour conversion . hence we compared our model light curves to the observations in terms of theoretical pc and ac relations at the phases of maximum , mean and minimum light and also in terms of the fourier parameters from theoretical light curves with observations . the theoretical quantities from the models generally agree with the observations , but it was found out that these models tend to have smaller amplitudes and ( hence ) the temperature is cooler at maximum light than the real cepheids . though our models have some drawbacks in this comparison , our main interest is in comparing the interaction of the photosphere and hif as a function of phase with similar results presented in paper ii for galactic cepheid pulsation models . the aim is _ not _ to compare our models rigorously with observations but rather to study models which match observations reasonably well in the context of the theoretical framework described in previous sections and in paper i & ii . nevertheless we argued that the qualitative nature of the photosphere - hif interaction is not seriously affected by these problems . our postulate is that at certain phases , this interaction can affect the pc relation due to the properties of the saha ionization equation : specifically for reasonably low densities in cepheid envelopes , hydrogen ionizes at a temperature that is almost independent of period . consequently , when the photosphere is located at the base of the hif , the photospheric temperature and hence the colour is almost independent of period . however , when this engagement occurs , but the density is greater , then the temperature at which hydrogen ionizes again becomes sensitive to global surroundings and hence on period . when the photosphere is not engaged with the hif in this way , its temperature is again dependent on period and global stellar parameters . for galactic cepheids , this hif - photosphere interaction occurs mainly at maximum light for cepheids with @xmath227 ( paper ii ) . at minimum light , there is a strong correlation between the hif - photosphere distance and period leading to a definite ac relation at minimum light for galactic cepheids ( skm , paper i & ii ) . in this paper , we have found tentative evidence that , for short period lmc models which match observations in the period - color plane , the hif - photosphere interaction occurs at most phases but at densities which are too high to produce a flat pc relation . why would these short period lmc cepheids be different in this regard to short period galactic cepheids ? one possibility could be that this is partly because these lmc cepheids are hotter than their galactic counterparts @xcite . the hif - photosphere are disengaged for most of the pulsation cycle for long period lmc cepheids . this happens because as the period increases , so does the @xmath10 ratio which pushes the hif further inside the mass distribution . when the hif - photosphere are disengaged in this way , the photospheric temperature is more dependent on density and hence on period . the change is sudden because the hif - photosphere are either engaged or they are not . this can lead to a sudden change in the pc relation at 10 days as shown by the observations @xcite . however , at maximum light the hif - photosphere are engaged at low densities for long period lmc cepheids leading to the observed flat pc relation for these stars . taken together with equation ( 1 ) , this theoretical scenario is consistent with the observed pc - ac behavior described in paper i and in this study . the anonymous referee has noted that these suggestions about photospheric density can be tested by spectroscopic means . we now enumerate some caveats to our argument that could be addressed in future papers . 1 . since the smc pc relation at mean light is linear ( e.g. , paper i ) , how do smc ( i.e. , metal - poor ) models fit into the theoretical scenario outlined in this paper and paper ii , if at all ? this is a difficult question and its full answer is beyond the scope of this paper however , as the metallicity decreases , we do note that the smc has a different ml relation to the lmc and galaxy and so does the temperatures associated with the instability strip . these will change the relative location of the hif and photosphere @xcite and possibly alter the phase at which they interact . further the amplitudes for smc cepheids are smaller due to the lower metallicity @xcite . this will also affect the hif - photosphere interaction . one difference which can be consistent with this is the fact that the pc relation at maximum light in the smc is not flat ( see paper i ) but it is the case for the galaxy and lmc pc relations . this indicates that at maximum light , there is less interaction between the hif and photosphere at low densities . this leads to an observed linear pc relation at mean light for the smc cepheids . these will be investigated further in a future paper in this series . 2 . could the well - known hertzsprung progression play any part in causing the observed changes in the galactic and lmc pc relations ? it may also be that higher order overtones becoming unstable or stable , though with the fundamental mode still being dominant , may also have an impact on the pc relation in some as yet unknown way ( paper ii ) . 4 . the behavior of short period lmc cepheids still needs to be understood , for example , what causes the difference between the bottom left panels of figures [ c9deltalmc ] and [ c9delta ] ? that is , why is it that for short period galactic / lmc cepheids , the hif - photosphere are disengaged / engaged ? our experience suggests that constructing short period full amplitude fundamental mode cepheids requires more care than the long period case because the first overtone has a non - negligible growth rate . because of this we feel a thorough study of these short period cepheids merits a separate paper . 5 . would more advanced pulsation codes which , for example , can match the observed amplitudes and which contain a more accurate model of time dependent turbulent convection , yield similar results , especially for figure [ c9delta ] ? could such codes fare better in modeling short period lmc cepheids ? smk acknowledges support from hst - ar-10673.04-a . we thank an anonymous referee for several useful suggestions and providing the data for our testing . we would also like to thank e. antonello , r. buchler & j. kwan for useful discussions , and r. bell & m. marengo for the discussion regarding the atmosphere fits .
|
period - colour ( pc ) and amplitude - colour ( ac ) relations are studied for the large magellanic cloud ( lmc ) cepheids under the theoretical framework of the hydrogen ionization front ( hif ) - photosphere interaction .
lmc models are constructed with pulsation codes that include turbulent convection , and the properties of these models are studied at maximum , mean and minimum light . as with galactic models , at maximum light the photosphere is located next to the hif for the lmc models . however very different behavior is found at minimum light . the long period ( @xmath0days )
lmc models imply that the photosphere is disengaged from the hif at minimum light , similar to the galactic models , but there are some indications that the photosphere is located near the hif for the short period ( @xmath1 days ) lmc models .
we also use the updated lmc data to derive empirical pc and ac relations at these phases .
our numerical models are broadly consistent with our theory and the observed data , though we discuss some caveats in the paper .
we apply the idea of the hif - photosphere interaction to explain recent suggestions that the lmc period - luminosity ( pl ) and pc relations are non - linear with a break at a period close to 10 days .
our empirical lmc pc and pl relations are also found to be non - linear with the @xmath2-test .
our explanation relies on the properties of the saha ionization equation , the hif - photosphere interaction and the way this interaction changes with the phase of pulsation and metallicity to produce the observed changes in the lmc pc and pl relations .
cepheids stars : fundamental parameters
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the first 500 myr after the big bang mark the current frontier in our exploration of cosmic history . understanding when and how the first galaxies started to form , how they grew their stellar mass and eventually turned into the diverse population of galaxies we see today is one of the most intriguing and challenging questions of modern observational astronomy . this is the main science driver for the director s discretionary time @xmath7 frontier field program ( hff ; e.g. * ? ? ? the hff will make use of lensing magnification of 4 - 6 foreground clusters to probe the ultra - faint galaxy population as early as 400 - 500 myr after the big bang . furthermore , the hff additionally creates six deep parallel blank field pointings in order to mitigate the uncertainties of lensing magnification and cosmic variance . while great progress has been made recently in probing galaxy build - up out to @xmath8 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , beyond @xmath1 , our understanding of galaxies is still very limited due to small number statistics . consequently the evolution of the cosmic star - formation rate density from @xmath1 to @xmath2 is still uncertain . the analysis of the full hubble ultra - deep field 09/12 ( hudf09/12 ) data and of two fields from the cosmic assembly near - infrared deep extragalactic legacy survey ( candels ) revealed a rapid decline of the sfrd by @xmath9 in only 170 myr from @xmath1 to @xmath2 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * but see also ellis et al . the two detections of @xmath10 galaxies in the cluster lensing and supernova survey with hubble ( clash ; * ? ? ? * ; * ? ? ? * ) have not changed this broad picture of a steeper decline compared to lower redshift trends . by adding up to twelve additional very deep sightlines , the hff program will be the prime dataset to clarify the sfrd evolution at @xmath11 before the advent of the james webb space telescope _ ( jwst)_. furthermore , given the power of lensing clusters ( see * ? ? ? * ) , the hff program will also provide a unique dataset to study resolved morphologies of very high - redshift , multiply imaged galaxies ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and will likely result in galaxy samples bright enough for spectroscopy ( e.g. * ? ? ? * ; * ? ? ? * ) . it may even be possible to probe the faint - end cutoff of the high - redshift ultra - violet ( uv ) luminosity functions with the hff dataset once all observational uncertainties and biases are under control @xcite . results on @xmath12 galaxies have been reported using partial hff data from the first observing epochs ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and very recently also from the full dataset of a2744 @xcite . the majority of these analyses to date have been limited , however , to the presentation of possible candidates only . the recent analysis of the complete dataset over abell 2744 by @xcite provided the first multiply imaged @xmath2 galaxy candidate identified from the hff program ( see also * ? ? ? the candidate jd1 is found to be a triply imaged source with an intrinsic apparent magnitude of only @xmath1329.9 mag , comparably faint as the previous @xmath2 galaxies identified in the deepest data over the hudf @xcite . the locations of all three multiple images of jd1 are consistent with the prediction of the cluster lensing maps for a @xmath14 source , which significantly decreases the likelihood of this source being a lower redshift contaminant . in this paper we make use of the complete hff dataset of the first cluster , abell 2744 , and its parallel field in order to search for additional @xmath2 galaxy candidates and to derive the first constraints on the star - formation rate density of @xmath2 galaxies based on hff data . in particular , we will discuss the effect of shear- and position - dependent completeness for high - redshift galaxy catalogs . this proves to be very important , yet has been largely overlooked so far . this paper is organized as follows : in section [ sec : data ] , we describe the dataset and sample selection . a detailed description of our completeness simulations and how shear affects the selection volume of galaxies is given in section [ sec : completeness ] . our results on the @xmath2 star - formation rate densities are presented in section [ sec : results ] , before summarizing in section [ sec : summary ] . throughout this paper , we adopt @xmath15 kms@xmath16mpc@xmath16 , i.e. @xmath17 , consistent with the most recent measurements from planck @xcite . magnitudes are given in the ab system @xcite , and we will refer to the hst filters f435w , f606w , f814w , f105w , f125w , f140w , f160w as @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath23 , @xmath24 , respectively . the hff program images each cluster / blank field for 140 orbits split over seven filters with the acs and wfc3/ir cameras . these filters are @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath23 , and @xmath24 . in this paper , we use the fully reduced version 1 hff dataset of abell 2744 and its parallel field provided by stsci . ] these images were calibrated , cosmic - ray cleaned , background corrected , astrometrically aligned , and drizzled to the same output frames . in particular , we use the images drizzled at 60 mas pixel scale . the final mosaics provided by stsci also include all ancillary data available over these fields in the same filters from additional programs . of particular importance is the frontier field uv imaging program ( go13389 , pi : siana ) which adds 16 orbits of acs data over the parallel field ( split over @xmath18 and @xmath19 ) . for the cluster field , we create a weighted combination of the individually provided epoch 1 and 2 acs images using the weightmaps , which adds the pre - existing data over this cluster ( go11689 , pi : dupke ) . the final @xmath25 depth of the images in empty regions of sky is @xmath26 mag as measured in circular apertures of 04 diameter . for more detailed information on these data , see koekemoer et al . ( 2014 , in preparation ) and visit the frontier field webpage at stsci . galactic extinction is accounted for by adjusting zeropoints for each hst filter using a milky way extinction curve @xcite and @xmath27 @xcite . this only results in minor corrections of @xmath28 mag in the wfc3/ir filters and up to 0.05 mag in the @xmath18 filter . gravitational lens models for all hff clusters were produced by five teams using different methods . these are made available through the frontier field webpage on mast . it is important to note that all these models are only based on ancillary data taken before the hff campaign , and they are expected to improve and converge with the additional constraints from the many faint multiple images found in the hff data @xcite . for details on the models see , e.g. , @xcite . here we use the five models that also released both components of the shear tensor which allow us to compute the radial and tangential magnification factors in order to be able to properly estimate the selection volume of high redshift galaxies ( see section [ sec : shear ] for more details ) . this includes the models of bradac et al . ( e.g. , * ? ? ? * ) , merten et al . ( e.g. , * ? ? ? * ) , zitrin et al . ( e.g. , * ? ? ? * ) , and williams et al . ( e.g. , * ? ? ? * ) . the results shown in the remainder of this paper are based on the lensing map provided by zitrin et al . ( zitrin - nfw ) for a2744 . however , our results on the overall number densities of @xmath2 galaxies do not change significantly when considering other magnification maps , consistent with the findings of @xcite . one significant concern with the data obtained over the cluster fields is the intra - cluster light ( icl ) which significantly increases the background and limits the detectability of faint galaxies . the brightness of the icl lies @xmath29 mag above the surface brightness limit of the hff data over a large part of the cluster field ( see * ? ? ? * ) and thus significantly limits the direct detectability of faint sources . furthermore , for abell 2744 the critical curve for lensing @xmath2 galaxies runs partially through the icl , which may significantly reduce the chance of finding highly magnified galaxies in standard sextractor catalogs due to blending with the icl and spurious detections . in order to mitigate some of this effect , we subtract the icl using a 25 wide median filter . when filtering , we exclude the cores of bright sources in order to minimize over - subtraction around bright galaxies or stars . the median subtracted images are then fed to sextractor @xcite to produce source catalogs using standard parameters . we found this procedure to result in somewhat more reliable catalogs and flux estimates for faint , small sources compared to running the standard sextractor background subtraction on the original images . however , in future analyses , it may be possible to improve upon our treatment of the icl using more sophisticated modeling and subtraction accounting for the icl and bright cluster galaxies simultaneously . this may likely result in even more complete catalogs of high - redshift sources toward the cluster center . whatever method is used , however , both the cluster galaxies themselves and the increased background due to the icl result in reduced search volumes of high redshift galaxies in cluster images ( see later section [ sec : completeness ] and figure [ fig : posdepcompleteness ] ) . similar to previous selections , we identified galaxies at @xmath30 by exploiting the spectral break shortward of ly@xmath31 due to inter - galactic hydrogen . red @xmath32 colors and non - detection in shorter wavelength filters are the key features used in the selection . in order to directly compare the hff sample with previous analyses ( e.g. * ? ? ? * ) , we restrict the search here to galaxies with @xmath4 , which selects sources at @xmath33 . we identified sources in a @xmath34 image constructed from the @xmath24 and @xmath23 images and measured photometry with sextractor run in dual image mode . all images were psf - matched to the @xmath24 point - spread function . colors were measured in small kron apertures ( kron factor 1.2 ) , typically 02 radius and total magnitudes were derived from larger elliptical apertures using the standard kron factor of 2.5 , with an additional correction to total fluxes based on the encircled flux measurements of stars in the @xmath24 band . based on these catalogs , we applied the same selections as we used previously in @xcite : @xmath35 @xmath36 furthermore , sources were required to be detected in @xmath24 and @xmath23 with @xmath37 in each and at least @xmath38 in one of the bands . the @xmath39 for each candidate source was computed following @xcite as @xmath40 , with @xmath41 the flux in band @xmath42 and @xmath43 the associated uncertainty . sgn(@xmath41 ) is equal to 1 if @xmath44 and @xmath45 if @xmath46 , and the summation is over the @xmath18 , @xmath19 , @xmath20 , and @xmath21 bands . the limit of @xmath47 efficiently excludes lower redshift contaminants while only reducing the selection volume by a small amount ( 20% ; see also * ? ? ? * ; * ? ? ? * ) . , @xmath22 , @xmath23 , and @xmath24 . as can be clearly seen , both source positions are affected by diffraction spikes from a nearby star and a bright cluster galaxy . to provide the most reliable estimates , we perform manual flux measurements . . ] applying the above selection criteria to the publicly released hff data of a2744 , we identify two candidates , a2744-jd1a and a2744-jd1b . these are two images of a single , triply imaged source , independently discovered earlier by ( * ? ? ? * see also ishigaki et al . the third image ( jd1c in zitrin et al . ) lies very close to a bright foreground source and is not present in our catalogs despite aggressive deblending parameters used in our sextractor runs . visual inspection indicates that it is a viable source . however , it is not included in the rest of this paper , as our effective volume simulations account for sources lost due to photometric scatter or blending with neighbors , as is the case for jd1c . the two detected images of jd1 lie on either side of the @xmath2 critical curve at ( ra , dec)@xmath48 ( 00:14:22.20 , @xmath4930:24:05.3 ) and ( 00:14:22.80 , @xmath4930:24:02.8 ) and are shown in figure [ fig : stamps ] ( see also fig 1 in * ? ? ? * ) . for both sources the photometry is heavily affected by diffraction spikes , in the first case caused by a nearby bright star and for the second source by a bright galaxy . nevertheless , it is clear that both sources are real . while both images satisfy the color selection criteria in our standard catalog , the sextractor photometry is likely unreliable due to the diffraction spikes . we therefore performed manual aperture photometry to confirm the color measurements and total magnitudes . in particular , in our manual measurement , we estimate the sky value in a small annulus around the source , excluding pixels which are obviously affected by the nearby diffraction spikes . photometry is then measured in small , circular apertures of increasing size up to 06 diameter . finally , we use the encircled energy of stars to correct the fluxes to total as given in the wfc3 handbook @xcite . using this approach , we find total magnitudes of @xmath50 and @xmath51 for the two images jd1a and jd1b , respectively , and colors @xmath52 and @xmath53 . these measurements are consistent with the photometry from @xcite where these images are discussed in detail and the source s photometric redshift is determined to be @xmath54 . this is confirmed by the lensing geometry which satisfactorily predicts the location of the three images only if the source lies at @xmath11 . the magnification of the two images is @xmath9 as predicted by @xcite . however , the full range of allowed magnification factors also predicted by other frontier field lensing models is @xmath55 . while uncertain , this source is likely to be of comparable brightness to the faintest @xmath2 candidates found in the xdf / hudf12 dataset @xcite , i.e. , the @xmath3 source xdfj-38126243 @xcite or the @xmath56 candidate udf12 - 4265 - 7049 @xcite . the same search for galaxies with @xmath4 in the parallel field of the hff cluster a2744 did not result in any candidate @xmath2 galaxy . while we do find a number of high - quality sources with colors within @xmath57 of this cut , these galaxies most likely lie at slightly lower redshift @xmath58 and will be discussed in a future paper . our selection function simulations which we discuss in the next sections do statistically account for sources lost due to photometric scatter , and we therefore proceed with zero @xmath2 galaxy candidates from the parallel field of a2744 . for @xmath2 galaxies at fixed apparent magnitude behind abell 2744 ( blue lines ) . the critical curve for lensing @xmath2 galaxies based on the zitrin - nfw model is shown in orange . the magnitude distribution of simulated galaxies is assumed to be flat for @xmath59 mag and the completeness is normalized to areas of @xmath60 ( 15% of the image ) , where the absolute completeness is @xmath61 . the relative completeness over much of the cluster center is significantly reduced due to the increased background . however , lower completeness is also found around the critical curve even in the absence of bright foreground sources . this is due to the sheared morphologies of galaxies . this effect has been largely ignored in lf analyses behind lensing clusters so far . it may be possible to increase the source completeness with the use of more sophisticated modeling and subtraction of the intra - cluster light and bright foreground galaxies as well as with the adoption of a detection smoothing kernel adapted to the expected shear at a given location within the image . nonetheless , it will not result in the same completeness levels expected in ultra - deep blank fields . ] in the next sections we show the importance of position - dependent source blending and shear on the completeness and selection efficiency of highly - magnified , high - redshift candidates . while faint @xmath2 galaxies are only marginally resolved with @xmath62 , they do have a finite size of @xmath63 kpc and are not point sources ( e.g. * ? ? ? * ; * ? ? ? this has important implications for the completeness of galaxy selections around the critical curves of lensing clusters . the limited surface brightness sensitivity of hst leads to a significant reduction of the selection efficiency for the most highly magnified and sheared sources ( see e.g. , * ? ? ? this effect has so far been largely ignored in previous determinations of selection volumes and uv lfs behind lensing clusters due to the computational challenges involved ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , with few exceptions . for instance , @xcite discuss how the signal - to - noise boost from lensing depends on a galaxy s two dimensional profile due to shear . @xcite used completeness simulations which account for the mapping of galaxy images from the source plane to the image plane using the lens models . this has recently also been incorporated in new determinations of the uv lfs behind the hff clusters ( e.g. * ? ? ? * ; * ? ? ? as we show below it is crucial to include lensing shear and magnification - dependent completeness when deriving lensed number densities of galaxies . we do this by taking shear into account to first order using the shear tensor to compute the tangential and radial magnification as well as the direction of the shear angle ( see e.g. * ? ? ? we therefore rely on hff lensing models which provide the two components @xmath64 and @xmath65 of the shear tensor . these are second order derivatives of the lensing potential @xmath66 , such that @xmath67 , @xmath68 ( see e.g. * ? ? ? * for a proper derivation ) . the results in the remainder of this paper are all based on the zitrin - nfw model . using different models has no significant impact on the overall number densities of @xmath2 sources , even though the predicted magnification for individual sources can show a wide range ( see also * ? ? ? based on the shear tensor , we derive the shear angle @xmath69 at each location in the image , as well as the tangential ( @xmath70 ) and radial shear factors ( @xmath71 ) , which can be computed based on the convergence @xmath72 and shear @xmath73 maps provided by the lens models : @xmath74 using these three quantities , we can estimate the effect of high magnifications on the selection function and completeness of galaxies . we follow standard procedures for blank fields and insert artificial galaxies with different light profiles , sizes , luminosities , and redshifts into the original science images . after re - running our detection algorithm with the same parameters as for the original images , the completeness @xmath75 is simply given by the fraction of sources that are detected and observed at magnitude @xmath76 . the only difference compared to non - cluster field completeness simulations is that we apply the position - dependent shear to the artificial galaxies before inserting these into the images . for computational efficiency we limit the tangential and radial shear factors to @xmath77 . our estimates therefore become unreliable above @xmath78 ( where the completeness is overestimated ; see section 3.3 ) . this only affects a small fraction of the image plane , however ( @xmath135% ; see also * ? ? ? the resulting completeness and effective selection volumes depend on the assumed properties of the simulated galaxy population . in particular , the galaxy size and morphological profile distributions as well as the intrinsic color distributions are important parameters of such simulations ( see e.g. * ? ? ? * ) . here , the color distributions are set according to the luminosity dependent distribution of uv continuum slopes as measured by bouwens et al . ( 2013 ; see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in order to account for non - regular morphologies observed in star - forming galaxies at high redshift , lbg completeness simulations often rely on scaling actual observed lbgs ( e.g. , at @xmath79 ) and redshifting them accounting for the difference in cosmological angular diameter distance and for intrinsic size evolution . in the case of simulations where lensing magnification is taken into account such redshifting is not possible due to the insufficient resolution of actual @xmath79 galaxy observations with hst to reliably reproduce higher redshift , highly magnified sources . we therefore use another common approach adopting idealized galaxy light profiles . in practice , we use a 50% mix of exponential disks and devaucouleur profiles ( corresponding to sersic profiles with @xmath80 or @xmath81 respectively ; sersic 1968 ) . as shown later , this assumption has a small , but noticeable effect on the resulting completeness . the intrinsic size distribution of galaxies is chosen according to a log - normal distribution with mean evolving as @xmath82 as is consistent with most studies of lbg size evolution at @xmath83 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in particular , the size distributions are normalized at @xmath84 where we set the peak of the distribution at @xmath85 kpc and we assume a constant width of @xmath86 ( consistent with the distribution of halo spin parameters ) . the simulated light profiles are then sheared according to the lens model at the position where they are inserted , before convolving them with the wfc3/ir psf . since we are simulating the light profiles of magnified sources , it is important to also account for any trend in size with mass or luminosity . smaller galaxy sizes at lower luminosities are sometimes used to argue that lensing shear has no effect on galaxy completeness ( e.g. * ? ? ? however , both the mass - size and the luminosity - size relations at high redshift are found to be very shallow following @xmath87 as measured for @xmath88 lbgs by @xcite similar to the luminosity scaling @xmath89 found by @xcite . these measurements are completely consistent with the surprisingly constant size scaling for late type galaxies at all redshifts @xmath90 seen in the candels dataset ( @xmath91 ; see * ? ? ? * ) and there is no convincing evidence for a change in these scaling relations at higher redshifts ( but see * ? ? ? this suggests that a galaxy magnified by a factor @xmath92 is _ intrinsically _ only @xmath93 smaller than a non - lensed galaxy observed in the field . in order to account for this size scaling in our shear simulations over the cluster field , we scale our assumed size distribution from the blank field by @xmath94 before inserting galaxies in the image . this thus corresponds to an assumed scaling of the size distribution of @xmath95 . ) corresponding to just 15% of the area of the image in this cluster , where the completeness is about 80% . the values are computed for galaxies at _ fixed apparent _ magnitude @xmath59 under the assumption that galaxy sizes scale as @xmath96 ( dark blue dots and line ) . calculations are only shown up to @xmath97 , above which our estimates start to become unreliable ( overestimated ) due to the use of 3wide bins to compute the position - dependent completeness and due to our limiting the shear factors for computational efficiency . a representative gray errorbar on the left shows the 1@xmath98 dispersion in the relation across the image . while lensing preserves surface brightness , highly sheared sources are spread out over many pixels resulting in a lower detection probability . the assumed size distribution has thus a significant impact on the expected completeness as shown by the gray dashed line where no size scaling with luminosity was assumed . even in the absence of shear , however , magnified sources have reduced completeness due to blending with foreground galaxies and intra - cluster light as shown with the dashed red line . the faint blue lines show the impact of different galaxy profiles . the upper line shows the result using only @xmath81 sersic profiles , while the lower curve is for exponential discs with @xmath80 . ] for @xmath2 galaxies as a function of magnification factor @xmath99 and @xmath24-band magnitude behind abell 2744 . the different curves show different observed @xmath24 magnitude bins with decreasing luminosity ( bright to faint ) . the colors represent different light profiles ( @xmath80 blue , @xmath81 red ) . as seen in the previous figure , the impact of the assumption on the light profile is noticeable , but relatively small . this figure demonstrates that the reduced relative completeness for highly magnified galaxies with @xmath100 mag is mostly dominated by blending with foreground galaxies , while shear impacts sources at @xmath101 mag , i.e. about 1.5 mag from the nominal completeness limit of the data . ] figure [ fig : posdepcompleteness ] shows the relative detection completeness for galaxies in the observed magnitude range @xmath59 mag as a function of position in the cluster field . the completeness is normalized to the median found in areas of the image with @xmath60 ( 15% of the image ) , where the absolute completeness is @xmath61 . while the relative completeness decreases significantly around brighter sources in the field due to blending , it is clearly apparent that the completeness is also reduced around the critical curve of the cluster where no bright foreground sources are present . in those areas of the image , the main reason for the reduced completeness is shear and magnification . even though lensing conserves surface brightness , a source which is highly magnified above the survey detection limit is spread over many more pixels than a non - sheared source at the same observed magnitude , reducing its s / n and detection probability . the relative completeness as a function of magnification averaged over the whole cluster field is shown in figure [ fig : magdepcompleteness ] . as expected , we find a significant decrease in completeness toward higher magnification . even though the scatter is significant in this relation , a source magnified by @xmath102 has on average a @xmath103 lower chance of being detected and included in a high - redshift catalog compared to a source which is only magnified by @xmath60 . however , as already pointed out , magnification - dependent completeness is present even when ignoring shear and magnification , simply due to blending with bright cluster galaxies closer to the critical curves ( red dashed line in fig . [ fig : magdepcompleteness ] ) . the shear adds to the incompleteness on top of this by a factor @xmath104 . figure [ fig : magdepcompleteness ] also shows that the size scaling does have a significant impact on the derived completeness relation . using our default scaling of @xmath95 , we find that very highly magnified sources are up to a factor @xmath105 more complete than assuming no size scaling at all . the large discrepancy between these two estimates , however , shows that accurate size scaling relations are necessary to accurately compute the selection volumes of high - redshift galaxies , adding to the uncertainties in lfs estimated from cluster fields . one possible strategy for mitigating these uncertainties is to use a differential technique to derive the relative normalization of the uv lf in various redshift bins from their relative surface densities ( see e.g. , * ? ? ? this is based on the assumption that the sizes and surface brightness profiles of galaxies in different redshift intervals are largely self - similar vs. luminosity . however , these assumptions in deriving the lf evolution need to be properly tested and calibrated . figure [ fig : magdepcompleteness ] also shows the impact of our assumption on the distribution of galaxy light profiles . if we simulate only sersic @xmath80 exponential disk profiles , the resulting completeness is slightly reduced ( by @xmath106 at @xmath92 ) compared to the steeper @xmath81 light profile . clearly , the impact of a size - luminosity relation is thus significantly more important , and it will be crucial to accurately determine this with future observations . finally , figure [ fig : completenessfmag ] shows the magnification - dependent completeness as a function of observed magnitude and galaxy profile . this demonstrates that the drop in completeness at high magnification seen in figure [ fig : magdepcompleteness ] is mostly driven by sources with @xmath107 mag , i.e. @xmath131.5 mag brighter than our 5@xmath98 detection limit . for sources with observed magnitudes brighter than this , the reduced completeness at higher magnification is mostly dominated by blending with the bright foreground cluster galaxies . we stress that the completeness estimates derived here only apply to galaxy catalogs using standard source detection algorithms . it may be possible to increase the source completeness around the critical curves with the use of a smoothing kernel adapted to the expected shear . furthermore , our calculations assume idealized light profiles . clumpy substructure in galaxies may further increase their detection probability . quantifying these effects is beyond the scope of this paper , however . nevertheless , whatever detection algorithm is used , it is clear that accounting for a positional dependence of the completeness is crucial for any luminosity function or star - formation rate density analysis behind lensing clusters , which has so far been largely overlooked in the literature . candidate images in the a2744 hff data assuming the @xmath2 uv lf from @xcite . _ top _ differential number counts per wfc3/ir field - of - view and per unit magnitude _ bottom _ cumulative number counts . dark red lines correspond to the cluster field ( assuming the magnification map zitrin - nfw ) and the dark gray lines show the parallel blank field predictions . the thick solid lines are the expected numbers based on full simulations of the selection efficiency and completeness , which also include shear for the cluster field . the dot - dashed lines are idealized predictions assuming a selection efficiency of @xmath108 and a selection volume of @xmath109 for comparison . the dashed line for the cluster field shows the prediction when ignoring the position - dependent completeness and assuming no shear in the simulations ( as is often done in the literature ) . clearly , the magnification - dependent completeness results in a significant reduction of expected images at faint magnitudes in the cluster field . the total number of expected images is comparable in both fields , in contrast to that predicted with idealized assumptions . the absence of @xmath2 galaxy candidates in the parallel field is indicated by the 1@xmath98 upper limit in dark gray in the top panel , while the detection of two images ( of the same galaxy ) in the cluster field is shown as dark red square . ] galaxies as a function of _ intrinsic _ , de - lensed magnitude . as in the previous figure , dark red lines show the cluster field and dark gray lines correspond to the blank field . the thick solid lines are the expected numbers based on full simulations of the selection efficiency and completeness , which also include shear for the cluster field . by comparison to the dashed line ( corresponding to a simulation without shear ) it is clear that the reduced completeness due to lensing shear and foreground blending at high magnification significantly reduces the search volume at @xmath110 mag , which somewhat limits the power of clusters to probe the ultra - faint galaxy population . ] clearly , the reduced completeness at high magnification also somewhat reduces the power of lensing clusters to probe deeper down the lf than ultra - deep blank fields . in this section , we estimate how this affects the expected number of @xmath2 galaxy candidates in the a2744 hff data set . using the previous simulations we compute the magnification - dependent selection efficiency , @xmath111 . this is given by the fraction of inserted galaxies at magnification @xmath99 with redshift @xmath112 and observed magnitude @xmath76 , which are both detected and satisfy our color selection criteria . this therefore accounts both for completeness at a given observed magnitude , as well as for photometric scatter which statistically removes galaxies from our lbg color selection box . using this selection function , we can compute the number of expected galaxy images ( double - counting multiple images ) in bins of magnitude for a given uv lf @xmath69 from @xmath113 where @xmath114 is the observed , magnified luminosity of a source , @xmath99 is the magnification and @xmath115 is the image solid angle ( i.e. observed pixels ) which is magnified by @xmath99 . @xmath116 is the cosmological volume per unit solid angle and redshift . the same equation also holds for blank fields , where @xmath117 everywhere . from the above equation it is clear that ignoring the position - dependent completeness and the reduction of selection efficiency due to shear in cluster fields typically results in higher expected numbers than may actually be present for a given lf this is demonstrated in figure [ fig : nexpz10 ] , where we show the number of expected galaxy images for both the cluster field and the parallel blank field of a2744 . the lf for this figure is taken from the analysis of the full candels - deep and xdf / hudf09 dataset . this is still uncertain due to the small number of candidates and @xcite therefore derive two possible @xmath2 uv lfs based on the previous data , one in which they assume evolution in @xmath118 relative to the @xmath1 lf and one where only the normalization @xmath119 is evolving . both derivations of this lf have an extremely steep faint - end slope @xmath120 . for fig . [ fig : nexpz10 ] , we show the results for the lf which evolves only in @xmath121 , which fits the candels and xdf / hudf09 datasets better . in addition to the curve resulting from the full selection function simulation , we also show an idealized prediction assuming @xmath108 and integrating the volume over @xmath109 . this overpredicts the expected surface density distribution of candidates by a factor @xmath105 for bright galaxies for both pointings . for the cluster field , we further show the expected number of sources if shear- and magnification - dependent completeness due to blending are ignored ( as is often done in the literature ) . including shear reduces the total expected number of sources in the cluster field by a factor 1.6@xmath122 for this assumed lf . figure [ fig : nexpz10mint ] shows the differential number counts expected for @xmath2 galaxy candidates as a function of _ intrinsic _ , unlensed magnitude . due to the average magnification in the cluster field , the number counts peaks at about 0.5 - 1 magnitude fainter than in the parallel field . however , our simulations show that accounting for magnification - dependent completeness and shear significantly limits the power of lensing clusters to probe galaxies fainter than @xmath123 mag , i.e. intrinsically much fainter than the hubble - ultra deep field . idealized calculations show that the cumulative galaxy number counts are expected to be larger behind a lensing cluster than in the field if the effective slope of the lf is steeper than @xmath124 in which case the reduced observed solid angle due to lensing is more than compensated for by the large abundance of faint , lensed galaxies ( e.g. * ? ? ? given our assumed uv lf with a faint - end slope of @xmath120 the cluster field would thus be expected to show a significantly larger number of high - redshift sources at all magnitudes . however , once we include magnification - dependent completeness , the cluster field in fact shows a very similar total number of expected images of @xmath2 galaxy candidates as the blank field ( within @xmath12515% ) . for the best - fit lf evolving in @xmath119 from @xcite shown in figure [ fig : nexpz10 ] , we predict to detect 0.46 images in the cluster field and 0.49 @xmath2 sources in the parallel . for an lf evolving in @xmath126 instead , we predict 1.3 images in the cluster and 1.1 sources in the parallel field . if these numbers are similar for all the other five hff clusters , the frontier field program is thus expected to find between 6 to 14 new @xmath2 galaxy candidates assuming the two different @xmath2 uv lfs of @xcite are representative . we stress that these numbers depend strongly on the exact evolution of the uv lf at @xmath11 ( see also * ? ? ? nevertheless , at @xmath2 alone , the hffs are likely to more than double the number of reliable lbg candidates known to date . we now combine the first hff cluster and blank field around a2744 to derive a new , independent estimate of the cosmic sfrd . from figure [ fig : nexpz10 ] it is clear that the two images of jd1 behind a2744 which satisfy our selection criteria will result in a higher cosmic sfrd at @xmath2 than we previously determined in the candels - deep and xdf / hudf12 data . we estimate the hff constraint on the @xmath2 cosmic sfrd from the total expected number of galaxy images per wfc3/ir field relative to the earlier @xmath2 uv lf estimate by @xcite . in particular , we use their parametrization for @xmath119-only evolution and search for the normalization , which reproduces two predicted images in the cluster field . with the assumed schechter function parameters of @xmath127 mpc@xmath128mag@xmath16 , @xmath129 mag , and @xmath120 , we predict a total of only 0.46 galaxy images in the cluster field . considering the cluster field alone , finding two images therefore requires a higher normalization , @xmath119 , by a factor @xmath130 compared to the xdf / hudf12 lf . such an increase would , however , result in a total of 2.2 predicted galaxies in the hff parallel blank field , which is marginally inconsistent with not finding any candidate with @xmath4 . we combine the two constraints from the hff cluster and parallel field by multiplying the poissonian probabilities of finding 2 or 0 sources in the two fields , respectively , for a given uv lf normalization @xmath119 . this results in a combined best - fit of @xmath131 mpc@xmath128mag@xmath16 , which is completely consistent , but @xmath132 dex higher than found in the ultra - deep fields . using this lf normalization , we estimate a cosmic sfrd from the a2744 fields . this is done by integrating the lf to obtain the luminosity density , which we then convert to a sfrd using the standard conversion between uv luminosity and sfr @xcite . no correction for dust extinction was applied , consistent with the expectation for very little dust in these galaxies at @xmath11 . while we adopt the widely used approach for deriving the sfr , we note that the conversion of uv luminosity to sfr assumes both an initial mass function ( imf ) as well as a star - formation history . if galaxies are significantly younger than 100 myr or if their star - formation is very bursty ( as predicted by some models ; e.g. * ? ? * ; * ? ? ? * ) , the standard conversion factor may need to be corrected ( see e.g. * ? ? ? * ) . the investigation of these alternatives goes beyond the scope of this paper , however . when integrating the uv lf down to @xmath133 , which corresponds to a sfr limit of 0.7 @xmath134 , we obtain a sfrd of @xmath135 m@xmath136yr@xmath16mpc@xmath128 . this is shown in figure [ fig : sfrdevol ] , where we also plot the previous estimates for comparison . our approach to estimate the sfrd , which is an integrated quantity , is not sensitive to the exact magnification of the candidate source . i.e. even if the magnification at the exact source position presented in @xcite were underestimated and the galaxy intrinsically had a sfr of less than our limit of 0.7 @xmath134 , the sfrd value derived here is still valid . as long as the magnification map produces an accurate differential number count distribution ( which is marginalized over the image plane ) our approach of seeking the best - fitting normalization to the uv lf and integrating this to a fixed sfr limit produces accurate results . while the new constraint from the a2744 hff fields is somewhat higher than the previous ultra - deep field constraints , it is consistent with the rapid decline across @xmath1 to @xmath2 that is predicted by theoretical models . in particular fig [ fig : sfrdevol ] also shows the average sfrd evolution of a series of semi - analytical / empirical models @xcite and from sph simulations @xcite as well as the sfrd from the illustris simulation @xcite . where necessary , we shifted the theoretical models to account for our use of a salpeter imf when converting the uv luminosity to sfr . all theoretical models agree on a very rapid decline in the cosmic sfrd by @xmath137 from @xmath1 to @xmath2 when limited at @xmath138 @xmath134 , indicating that a rapid build - up of galaxies above this limit is a generic prediction of any model of galaxy formation ( see also * ? ? ? nevertheless , given the still large errorbars , a more gradual decline in the sfrd as empirically estimated based on the uv lf evolution across @xmath139 ( see gray line in fig [ fig : sfrdevol ] ) can still not be completely ruled out . if the faint - end slope of the uv lf does not steepen further at higher redshift @xcite or if the escape fraction stays constant , this more gradual decline may be necessary for galaxies to complete reionization in agreement with the high optical depth measurement by wmap ( e.g. * ? ? ? * ; * ? ? ? note , however , that the rapid decline in the observed sfrd may simply be a consequence of our fixed detection limit in luminosity and is likely still compatible with a more gradual evolution of the _ total _ sfrd . this is supported both by the higher sfrd estimates of gamma ray burst counts ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which are sensitive to the total sfrd , and by simulations ( e.g. , compare with * ? ? ? this paper presented a first estimate of the cosmic sfrd at @xmath2 based on frontier field data . in particular , we show that extensive completeness simulations including source blending and lensing shear close to the critical curve are crucial for any analysis of cluster data . we find a significantly lower completeness at higher magnification than for comparable blank field searches at a fixed observed magnitude ( figures [ fig : posdepcompleteness ] and [ fig : magdepcompleteness ] ) . this can be ascribed to several effects : blending with bright foreground cluster galaxies , higher background due to intra - cluster light , but also due to shear spreading out highly magnified sources over many pixels . sources at high magnification are on average only 70% complete in the a2744 image compared to a blank field even when the effect of shear is ignored ( due to blending with foreground sources and the icl ) . shear further reduces the completeness at @xmath102 by @xmath104 . however , the exact completeness at high magnification sensitively depends on the assumed size distribution for very faint sources below the detection limit of current blank field data ( see fig [ fig : magdepcompleteness ] ) . this effect therefore adds to the overall uncertainty of lf and sfrd estimates from cluster lensing fields . this position - dependent completeness has often been overlooked in the literature ( but see * ? ? ? * and recently ishigaki et al . 2014 , atek et al . however , it has important consequences on the expected number of high - redshift candidates seen behind lensing clusters compared to blank fields . in figure [ fig : nexpz10 ] , we show that the reduced completeness results in a similar number of source images predicted for the a2744 cluster and parallel field , very different from what is commonly assumed . following previous blank field studies , we search the hff a2744 cluster and parallel field data for @xmath2 galaxy candidates using a criterion @xmath4 and non - detections at shorter wavelength . while no candidates are found in the parallel field , we find two images of the same source lensed by the cluster ( previously identified in * ? ? ? * ) which both satisfy our selection criteria . combining the one multiply imaged candidate over the cluster field with the null detection in the parallel field , we derive a cosmic sfrd at @xmath2 which is consistent , but @xmath132 dex higher than found earlier in the ultra - deep blank fields ( see figure [ fig : sfrdevol ] ) . not surprisingly , this independent measurement based on the first completed hff cluster does not allow us to significantly rule out different possible scenarios for the sfrd evolution between @xmath1 and @xmath2 . the combination of these new results with all other estimates from the literature remain consistent with a rapidly declining sfrd as is predicted by cosmological simulations and dark - matter halo evolution in @xmath5cdm . the completed multi - cluster hff dataset will allow to further increase the sample size of galaxies at @xmath2 and to significantly tighten this first estimate of the cosmic sfrd . once biases due to magnification - dependent incompleteness are taken into account , the hff survey will be a key dataset to study the galaxy population at @xmath11 before the advent of the @xmath140 .
|
we search the complete hubble frontier field dataset of abell 2744 and its parallel field for @xmath0 sources to further refine the evolution of the cosmic star - formation rate density ( sfrd ) between @xmath1 and @xmath2 .
we independently confirm two images of the recently discovered triply - imaged @xmath3 source by zitrin et al .
( 2014 ) and set an upper limit for similar @xmath0 galaxies with red colors of @xmath4 in the parallel field of abell 2744 .
we utilize extensive simulations to derive the effective selection volume of lyman - break galaxies at @xmath2 , both in the lensed cluster field and in the adjacent parallel field .
particular care is taken to include position - dependent lensing shear to accurately account for the expected sizes and morphologies of highly - magnified sources .
we show that both source blending and shear reduce the completeness at a given observed magnitude in the cluster , particularly near the critical curves .
these effects have a significant , but largely overlooked , impact on the detectability of high - redshift sources behind clusters , and substantially reduce the expected number of highly - magnified sources .
the detections and limits from both pointings result in a sfrd which is consistent within the uncertainties with previous estimates at @xmath2 from blank fields .
the combination of these new results with all other estimates are also consistent with a rapidly declining sfrd in the 170 myr from @xmath1 to @xmath2 as predicted by cosmological simulations and dark - matter halo evolution in @xmath5cdm .
once biases introduced by magnification - dependent completeness are accounted for , the full six cluster and parallel frontier field program will be an extremely powerful new dataset to probe the evolution of the galaxy population at @xmath6 before the advent of the jwst .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it was observed by douglas and hull @xcite that d - branes on @xmath1 with a constant neveu - schwarz ( ns ) two - form potential @xmath2 give rise to an effective world - volume theory on a non - commutative torus . even though this initial observation was re - considered and generalized by many authors @xcite , all the subsequent work is restricted to flat backgrounds . a perturbative analysis along the lines of @xcite , on the other hand , shows that the quantization of world - volume geometries should be a much more general phenomenon which persists in the case of curved backgrounds . in this work we shall present the first non - perturbative ( in @xmath3 ) investigation of world - volume geometries in a curved string background with non - vanishing ns 3-form field @xmath0 . by the string s equation of motion . ] an exact treatment of d - branes in curved backgrounds is possible within the framework of boundary conformal field theory . here we illustrate the basic techniques and some general features of the resulting world - volume geometries in a particular example , namely the su(2 ) wzw theory , and study d - branes in the wzw model associated with the gluing condition @xmath4 . we shall argue that their world - volumes may be regarded as fuzzy two - spheres when the level @xmath5 is sent to infinity , i.e. when the background becomes flat . for finite level , @xmath6 is non - zero and we shall find non - associative deformations of these fuzzy spheres , which are closely linked to the theory of quantum groups . while the infinite level result can be predicted from the semi - classical analysis in @xcite together with the general phenomenon of world - volume quantization in flat backgrounds @xcite , our results on the finite level provide a non - trivial extension of the standard rules . apparently , many features of the world - volume geometry are not captured by the perturbative treatment of d - branes on group manifolds that was suggested recently in @xcite . we shall follow a general procedure which allows us to extract world - volume geometry from the world - sheet description of any ( generalized ) d - brane , even when it is given in purely algebraic terms . the essential input data are the operator product expansions ( ope ) of boundary fields ( open string vertex operators ) . since they depend on the ordering of the operators , it is not surprising that the brane world - volume obtained in this way is a non - commutative space , in general . we shall see that non - associativity may show up as well . our approach is inspired by a project initiated by j. frhlich and k. in @xcite ( see also @xcite for earlier ideas in the same direction ) , where it was proposed to construct non - commutative target space geometries from opes of closed string vertex operators . this was developed further in @xcite . it appears , however , that non - commutative geometry emerges in a more natural way and on a more fundamental level in the open string case , cf . the picture below . our findings add to the growing evidence that brane physics surpasses classical geometry even though the emergence of a non - commutative world - volume need not necessarily mean that a d - brane behaves non - geometrically in the sense of the criterion formulated in @xcite . this criterion rests on a comparison of low - energy effective field theories in the stringy and in the large - volume regime , and we do not attempt to test it in the present paper . but we would like to point out that the structures contained in the non - commutative world - volume also form the main ingredient of the effective action of the brane . while we have chosen the su(2)@xmath7 example mainly because of its simplicity and because there exists a semi - classical curved background picture , it is also an important ingredient of the cft formulation of the neveu - schwarz 5-brane , see e.g. @xcite . given that questions like stability of the configuration can be clarified , our findings should be relevant for the geometry of d - branes in the presence of a stack of 5-branes . similarly , our su(2 ) wzw results could be applicable in the study of branes on an ads@xmath8 string background , see e.g. @xcite . before we show how one can read off fuzzy geometry from branes in the wzw model , let us briefly review the emergence of non - commutative spaces in the more standard case of branes in flat @xmath9-dimensional euclidean space @xmath10 , or on a flat torus @xmath11 . consider a d - brane which is localized along a @xmath12-dimensional hyper - plane @xmath13 in the target , with tangent space @xmath14 . the conformal field theory associated with such a euclidean d - brane is defined on the upper half of the complex plane . it contains an @xmath9-component free bosonic field @xmath15 subject to neumann boundary conditions in the directions along @xmath14 and dirichlet boundary conditions for components perpendicular to the world - volume of the brane . from the free bosons , one may obtain various new fields , in particular the open string vertex operators @xmath16 which can be inserted at any point @xmath17 on the real line . when there is no magnetic field on the brane , the ope of these u(1)-primaries reads ( with @xmath18 and for @xmath19 ) v_k_1(x_1 ) v_k_2 ( x_2 ) = ( x_1-x_2)^k_1k_2/2 v_k_1 + k_2(x_2 ) + , [ opeb0 ] where the dots indicate less singular non - primary contributions . we can rewrite this relation by introducing the objects @xmath20(x ) \ : = \ \frac{1}{(2\pi)^{p/2 } } \int_{tv_p } d^pk \ \hat f(k ) \ v_k(x)\ ] ] for each function @xmath21 with fourier transform @xmath22 . then the boundary ope ( [ opeb0 ] ) translates into a `` definition '' of pointwise multiplication of functions , v[f](1 ) v[g](0 ) = v[fg ] ( 0 ) + . [ pointw]we have specialized to coordinates @xmath23 and @xmath24 for convenience , arbitrary insertion points can be recovered via conformal covariance . the effect of switching on a @xmath2-field is described by adding the term s_b = dz d|z b _ x^(z,|z ) |x^(z,|z ) [ bterm]to the action of the original theory without @xmath2-field . one can easily see that this is a pure boundary term with no influence on the bulk properties of the theory . it only changes the boundary conditions . if we assume for definiteness that @xmath13 is spanned by the first @xmath12 coordinates @xmath25 , the new boundary conditions read ( with @xmath26 ) _ y x^(z,|z ) = b^ _ _ x x^(z , |z ) z = |z , = 1 , , p . [ bglue ] this means that the ( exact ) free boson propagator becomes ( @xmath27 ) x^(x_1)x^(x_2)_b = - ( ^ + _ s^ ) |x_1-x_2| - i _ a^(x_1-x_2 ) [ xcorr]where @xmath28 and @xmath29 denote the symmetric resp . anti - symmetric part of the matrix @xmath30 . explicitly , _ [ magic]in particular , when @xmath2 is large we obtain @xmath31 , which means that @xmath29 is the poisson bi - vector corresponding to the symplectic form @xmath2 . ( [ xcorr ] ) immediately yields the boundary ope for a non - vanishing @xmath2-field , @xmath32 as before , this can be used to define a ( deformed ) product @xmath33 for functions through @xmath34(1 ) v[g](0 ) = v[f \star g ] ( 0 ) + \ldots\ ; , $ ] where now ( f g ) ( x ) : = e^i ^_a ^x_^y _ f(x ) g(y ) _ y = x . [ moyal]this is the associative , non - commutative moyal - weyl product of functions @xmath35 on the world - volume @xmath13 of the brane . in the context of the derivation we have given , non - commutativity of @xmath33 arises because the ordering of boundary fields in general does matter , cf . the sign - term in eq . ( [ xcorr ] ) . the algebra of functions with product ( [ moyal ] ) is , of course , the non - commutative brane world - volume uncovered by douglas and hull using a different approach . it is a deformation of the ordinary algebra of functions , with deformation parameter(s ) given by ( the matrix ) @xmath29 . in @xcite , the term ( [ bterm ] ) was viewed as a bulk perturbation of the @xmath36 theory , i.e. techniques of conformal perturbation theory were applied to the operator @xmath37 being inserted into arbitrary correlation functions of the @xmath36 theory . this perturbative analysis , which can be extended to arbitrary @xmath38-models ( at least in the case @xmath39 ) , leads to a string theoretic picture of kontsevich s quantization of poisson manifolds @xcite , see also the work of cattaneo and felder @xcite . it clearly displays that the quantization of world - volume geometries should be expected beyond the case of constant @xmath2-fields . this will be confirmed through our exact analysis of the wzw model ( see discussion of the limit @xmath40 below ) . as we remarked in the introduction , new phenomena are bound to occur when @xmath41 does not vanish . in such cases , the classical world - volume of a brane comes equipped with some generalization of an ordinary poisson - structure , and there exists no general notion of `` quantization '' for such geometries . hence , the investigation of branes in a non - vanishing ns 3-form field strength @xmath0 can teach us new lessons on how to quantize certain non - poisson geometries . in our example of branes on su(2 ) we shall recover some variants of well - known quantum group algebras . our formulation of the simple example of flat branes in a constant @xmath2-field motivates the following _ general procedure _ : when we want to associate non - commutative spaces to branes which are given as boundary conditions on the world - sheet , we take the ope of boundary fields ( open string vertex operators corresponding to internal excitations of the brane ) as a basic input . then we choose a suitable subset of boundary fields ( e.g. primaries as above ) and use them as abstract generators of an algebra of `` functions '' on the ( non - commutative ) world - volume of the brane , with multiplication table given by the boundary ope ( projected onto the subset , and evaluated at @xmath23 and @xmath24 , say ) . further comments on this general prescription will be given later , but now we would like to test it in the case of su(2 ) wzw models , where the semi - classical picture provides certain expectations as to how the `` quantized world - volume '' of branes should look like . the su(2 ) wzw model at level @xmath5 describes strings moving on a three - sphere @xmath42 of radius @xmath43 , which is equipped with a constant ns 3-form field strength @xmath44 where @xmath45 denotes the usual volume form on the unit sphere , and @xmath46 are components of the 1-form @xmath47 . in superstring theory , this geometry appears in the space transverse to a stack of @xmath5 ns 5-branes . these branes act as sources for @xmath5 units of ns 3-form flux through a three - sphere surrounding their ( 5 + 1)-dimensional world - volume . the world - sheet swept out by an open string in @xmath42 is parametrized by a map @xmath48 from the upper half - plane h into the group manifold su(2)@xmath49 . from this field @xmath50 one obtains lie algebra valued chiral currents @xmath51 as usual . we shall be interested in maximally symmetric d - branes on su(2 ) , which are characterized by the gluing condition @xmath52 along the boundary @xmath53 . they were analyzed from a semi - classical point of view in @xcite , and we shall briefly recall the findings of this approach . ( for a detailed path integral description of branes in su(2 ) , see @xcite . ) we first decompose the tangent space @xmath54 at each point @xmath55su(2 ) into a part @xmath56 tangential to the conjugacy class through @xmath57 and its orthogonal complement @xmath58 ( with respect to the killing form ) . in @xcite , the following two basic observations were made : 1 . with gluing conditions of the type @xmath59 , the endpoints of open strings on su(2 ) are confined to conjugacy classes , i.e. @xmath60 2 . along the individual branes , i.e. along the conjugacy classes of su(2 ) , the gluing condition becomes @xmath61 except for two degenerate cases , namely the points @xmath62 and @xmath63 on the group manifold , the conjugacy classes are two - spheres in su(2 ) . taking into account the usual correspondence between @xmath64 and the flat space coordinate @xmath65 , recalling that the metric on the three - sphere scales with @xmath5 , and comparing with the gluing conditions ( [ bglue ] ) , we infer that the d - branes associated with @xmath59 carry a non - vanishing 2-form potential ( b - field ) @xmath66 in the limit @xmath67 the three - sphere grows and approaches flat 3-space . one can parameterize it by a parameter @xmath68 taking values in the lie algebra su(2 ) , such that @xmath69 . then , the formula for the @xmath2-field reads @xmath70 this is the kirillov 2-form on the spheres in the algebra su(2)@xmath71 . extrapolating formula ( [ magic ] ) to our curved background , we can construct a bi - vector @xmath72 introducing an orthonormal basis @xmath73 in su(2 ) , and the left- and right - invariant vector fields @xmath74 on the group manifold , one can give an elegant formula for the bi - vector @xmath29 , @xmath75 the schouten bracket of @xmath29 ( which generally characterizes the deviation from the jacobi identity ) is of the form @xmath76\ = \ \frac{1}{6}\ ; f_{abc}\ ; ( e^a_l- e^a_r)(e^b_l - e^a_r ) ( e^c_l - e^c_r)\ .\ ] ] here @xmath77 are the lie algebra structure constants , the same as those in the expression for the field strength @xmath6 . this calculation makes sense for an arbitrary simple lie group . in general , the right hand side does not vanish and gives the obstruction for the jacobi identity . in the case of @xmath78su(2 ) , @xmath79 vanishes for dimensional reasons : it is a 3-vector tangent to the 2-dimensional conjugacy classes . in the infinite volume limit @xmath67 , the bi - vector @xmath29 becomes @xmath80 which is the kirillov - kostant poisson bi - vector . consequently , the geometry of the limiting theory @xmath81 is very close to the well - known situation of flat branes in a flat background with constant @xmath2-field , and we expect that the world - volume algebras of our branes in the wzw model will be quantizations of two - spheres . for finite @xmath5 , however , the background is curved and carries a non - vanishing ns 3-form @xmath6 . this will result in a non - associative deformation of the @xmath82 theory . since the three indices of the new object @xmath6 can relate three - fold products with different positions of brackets , the violation of associativity will turn out to be rather mild . the semi - classical extension of the above analysis shows that , for fixed gluing conditions , only a finite number of su(2 ) conjugacy classes satisfy a dirac - type flux quantization condition @xcite . these `` integer '' conjugacy classes are the two points @xmath62 and @xmath63 along with @xmath83 of the spherical conjugacy classes ( those passing through the points @xmath84 for @xmath85 ) . the wzw model on the upper half - plane is known in enough detail to support and specify the rather crude arguments of the previous subsection by an exact cft analysis . in fact , for the situation we are dealing with ( gluing conditions @xmath59 in a `` parent '' cft on the full complex plane with diagonal modular invariant partition function ) , cardy @xcite was able to list all @xcite possible boundary conditions . there exist @xmath86 of them , differing in the bulk field one - point functions ( brane charges ) and labeled by an index @xmath87 . without entering a detailed description of these boundary theories @xcite , we recall that their state spaces have the form [ partdec ] _ = _ j n_^j ^j where @xmath88 , @xmath89 , denote irreducible highest weight representations of the affine lie algebra @xmath90 , and where @xmath91 are the associated fusion rules . note that only integer spins @xmath92 appear on the right hand side of ( [ partdec ] ) . there exists a variant of the state - field correspondence which assigns a boundary field @xmath93 to each element @xmath94 ( see e.g. @xcite ) . in particular , the su(2 ) wzw boundary theory labeled by @xmath95 contains su(2)-multiplets associated to primary boundary fields , namely @xmath96 and @xmath97 . all these boundary fields are defined for arguments @xmath17 on the real line and their correlators have , in general , no unique analytic continuation into the upper half - plane . in the flat target case , we chose u(1)-primaries as generating elements of the world - volume algebra . now , it is more appropriate not to break the group symmetry by hand and , therefore , to keep the full su(2)-multiplets @xmath98 . for a fixed order @xmath99 of arguments on the real line , the ope of two such boundary fields reads [ boundope ] ^i_i(x ) ^j_j(y ) ~ _ k , k ( x - y)^h_i+h_j - h_k lll i & j & k + i & j & k c^,_ijk ^k_k(y ) , where @xmath100 is the conformal dimension of @xmath101 and @xmath102 $ ] denote the clebsch - gordan coefficients of the group su(2 ) . the latter simply compensate for the different transformation behavior of the fields on the left and right hand side under the action of the zero - mode subalgebra of @xmath103 . hence , the non - trivial information in ( [ boundope ] ) is contained in the new structure constants @xmath104 . in a consistent theory , these must obey sewing constraints , which were first analyzed by lewellen in @xcite ; see also @xcite . recently , these constraints were reconsidered by runkel @xcite for the a - series of virasoro minimal models . his findings carry over to su(2 ) wzw models on the upper half - plane and show that the only possible solution to the sewing constraints is given by the fusing matrix @xmath105 of the wzw theory , [ cval1 ] c^,_ijk = f_k ll & + i & j _ . it is one of the fundamental results on the relation between quantum groups and conformal field theory ( see e.g. @xcite ) that the fusing matrix of the wzw model is obtained from the @xmath106 symbols of the quantum group algebra @xmath107 according to [ cval2 ] f_k ll & + i & j _ = lll i & j & k + & & _ q q = e^ . in the limit @xmath108 , the @xmath106 symbols of the quantum group algebra approach those of the classical algebra @xmath109 , thus the structure constants @xmath110 of the boundary ope become @xmath106 symbols of the group su(2 ) when the level @xmath5 is sent to infinity . note that in this limit , the conformal dimensions @xmath111 tend to zero so that the opes ( [ boundope ] ) of boundary fields become regular as in a topological theory . we are now prepared to follow the procedure sketched at the end of section 2 and to read off the world - volume geometry of branes in the su(2)-wzw model . so let us think of the boundary fields @xmath112 as being assigned to elements @xmath113 of some vector space , and let us use the operator product expansion ( [ boundope],[cval1],[cval2 ] ) to define a multiplication by the prescription [ fsopek ] y^i_i y^j_j = _ k , k lll i & j & k + i & j & k c^,_ijk y^k_k . as in ( [ boundope ] ) , the summation on the right hand side runs from @xmath114 to a maximal spin @xmath115 . first , we shall investigate this product in the limiting case @xmath116 , where it produces a familiar algebraic structure . passing to finite levels leads to the following two changes : there is a @xmath5-dependent deformation of structure constants @xmath117 , cf . ( [ cval2 ] ) , and the range of the summation in ( [ fsopek ] ) becomes a function of the level , @xmath118 . we shall separate these two phenomena by looking at an intermediate case where @xmath5 is non - rational and where we omit the @xmath5-dependent restriction on the @xmath119-summation . _ infinite level @xmath81_:recall that , in the case of infinite level , the structure constants @xmath117 in eq . ( [ fsopek ] ) are given by the @xmath106 symbols of the group su(2 ) . the semi - classical analysis showed that @xmath120 , so we expect the world - volume algebra to be associative . indeed this can be confirmed using the biedenharn - elliot ( or pentagon ) relation for the @xmath106 symbols , along with the fact that @xmath106 symbols of the form ( [ cval2 ] ) vanish whenever @xmath121 . hence , for infinite level our relations define an infinite set of associative algebras @xmath122 with finite linear bases consisting of dim@xmath123 elements . since the dimension of each of these algebras is a perfect square , one may already suspect that they are full matrix algebras , i.e. that @xmath124 with @xmath125 . to describe the isomorphism , we first note that @xmath126 admits an action of the group su(2 ) by conjugation with group elements evaluated in the @xmath127-dimensional representation of su(2 ) . under this action , the su(2)-module @xmath126 decomposes into a direct sum of irreducible representations @xmath128 , [ fsdec ] m_n ( ) _ j=0^n-1 v^j . only integer @xmath92 appear , so this agrees with the decomposition of the state space @xmath129 in eq . ( [ partdec ] ) for boundary wzw models at sufficiently large ( or infinite ) level @xmath5 . thus , we can identify our elements @xmath130 with a basis of the spaces @xmath128 . the isomorphism ( [ fsdec ] ) allows to work out multiplication rules for any two such basis elements from the multiplication of @xmath131-matrices . the result @xcite turns out to coincide with our formula ( [ fsopek ] ) , which shows that @xmath132 and @xmath133 are indeed isomorphic as associative algebras . the non - commutative spaces @xmath132 are known as _ fuzzy spheres _ and are obtained when one quantizes functions on a two - sphere with the usual poisson structure ( see e.g. @xcite and references therein ) . the two - spheres may also be identified with co - adjoint orbits of su(2 ) . according to kirillov , their quantization gives all representations of the lie algebra su(2 ) or of its universal enveloping algebra @xmath134 . note that the size @xmath135 of our matrices agrees with the number of components for an su(2)-multiplet of spin @xmath95 . hence , through the investigation of maximally symmetric branes on su(2 ) at @xmath81 , we have recovered kirillov s theory of co - adjoint orbits . _ finite non - rational level @xmath5_:let us stress that this case does not appear among the exact boundary theories above ( for non - compact wzw models , it is the generic situation ) . we include it here merely as an intermediate step before presenting the structure for finite integer level @xmath5 . to be more precise , we consider the algebras spanned by @xmath130 with relations ( [ fsopek ] ) in which the structure constants @xmath117 are given by the @xmath106 symbols ( [ cval2 ] ) of the quantum group algebra @xmath136 , but with summation over the same range as in the case @xmath81 . the resulting algebras @xmath137 with @xmath138 not a root of unity cease to be associative . but they are still quasi - associative in the sense that y^i_i ( y^j_j y^k_k ) ( ^i_in ^j_jm ^k_kl ) ( ) = ( y^i_n y^j_m ) y^k_l [ qassoc ] where the @xmath139 denote representations of @xmath140 and where @xmath141 is drinfeld s `` re - associator '' @xcite . the proof of this statement is sketched in the appendix . when we perform a standard quasi - classical limit , commutators are replaced by the brackets corresponding to the bi - vector @xmath29 . for a general compact simple lie group @xmath29 fails to satisfy the jacobi identity . this corresponds to the leading non - vanishing term in the @xmath142-expansion of the re - associator @xmath143 , @xmath144 where @xmath73 is , as above , an orthonormal basis in the lie algebra , and @xmath77 are the corresponding structure constants . when applied to the relation ( [ qassoc ] ) , the lie algebra generators @xmath73 act by the adjoint vector fields @xmath145 . in the case of @xmath78su(2 ) this leads to vanishing of the first order correction to the associativity law . this is in accordance with vanishing of @xmath146 $ ] in this case . note that even in the su(2 ) case higher order corrections to the associativity law do not vanish . let us briefly mention that our quasi - associative algebras @xmath137 are closely connected to associative deformations of the fuzzy sphere which employ the clebsch - gordan coefficients of the deformed @xmath136 instead of their classical analogs . some details on these algebras and their associativity can be found in the appendix . for now , let us only remark that they are factors of the quantum spheres introduced by podle in @xcite . their relation to our algebras @xmath137 is based on the fact that one can obtain the clebsch - gordon maps of classical lie algebras from their @xmath147-deformed counterparts with the help of drinfeld s `` twist element '' @xmath148 . the latter provides the following factorization formula for the re - associator : @xmath149 where @xmath150 denotes the co - product of @xmath140 . combining these two roles of the twist element @xmath105 , one can show that our algebras @xmath137 are `` twist equivalent '' to associative factors of a podle sphere or , more explicitly , to the same matrix algebras @xmath151 as in the case of infinite level . hence , we simply recover the representations for the usual @xmath147-deformation of @xmath140 at generic values of the deformation parameter . _ finite integer level @xmath5_:the associated algebras @xmath152 are spanned by the generators @xmath153 with the label @xmath92 chosen from the set @xmath154 . multiplication of these elements is defined through eq . ( [ fsopek ] ) with structure constants @xmath117 now given by the @xmath106 symbols of @xmath136 at the root of unity @xmath155 . in addition , the summation on the right hand side is now restricted to run from @xmath114 to @xmath156 . viewed as su(2)-modules , the linear spaces @xmath152 decompose as follows : @xmath157 s^2_{\ik/2 - \a } & \ \mbox { for } \ \ \frac{\ik}{4 } \leq \a \leq \frac{\ik}{2 } \end{array } \right . \ \ .\ ] ] again , the algebras @xmath152 are only quasi - associative , and they provide examples of the geometries considered in @xcite . using the concept of representations introduced in @xcite , it is not difficult to show that each of the quasi - associative algebras @xmath158 possesses precisely one indecomposable representation on a vector space @xmath159 of dimension @xmath160 \ik - 2\a + 1 & \ \mbox { for } \ \ \frac{\ik}{4 } \leq \a \leq \frac{\ik}{2 } \end{array } \right . \ \ .\ ] ] according to our previous discussion , the algebras @xmath158 and their representations on @xmath161 , generalize kirillov s theory of co - adjoint orbits to quantum groups at roots of unity . in other words , the algebras @xmath158 we obtain are `` quantizations '' of integer conjugacy classes on su(2 ) . summing over all possible brane sectors , i.e. over the index @xmath162 , we construct a deformed universal enveloping algebra . of course , quantum group algebras were constructed within the framework of chiral conformal field theory before , see e.g. @xcite . as long as we avoid roots of unity , our new derivation from boundary conformal field theory reproduces well - known algebraic structures . differences between the two approaches occur only when @xmath147 is a root of unity . in that case , boundary conformal field theory improves upon the old constructions in two respects . first of all , the theory gives `` physical '' representations exclusively so that there is no need for additional truncations . furthermore , the dimensions @xmath163 of the representation spaces are invariant under the simple current symmetry which interchanges @xmath95 and @xmath164 . when we increase the level @xmath5 , the radius of the three - sphere grows and we can fit more and more branes into the background . at the same time , the 3-form field strength decreases and the world - volume algebras become `` more associative '' while their non - commutativity survives . this is to be compared to the non - commutative targets obtained in @xcite from closed strings : the @xmath67 limit of these targets is simply the classical group su(2 ) . the different behavior of closed and open string geometry may be explained as follows : both closed and open strings feel the presence of the ns 3-form field @xmath6 at finite level . open strings are also sensitive to the concrete choice of a 2-form potential @xmath2 , while closed strings `` see '' only its cohomology class . in the flat space limit @xmath81 , the cohomology becomes trivial while @xmath2 itself stays non - zero and is responsible for non - commutativity on the brane . we have derived non - commutative world - volume algebras for d - branes in the su(2 ) wzw model , using a general scheme that can be applied to arbitrary branes given as conformal boundary conditions , including supersymmetric cases . in the process , we have seen how abstract objects from the cft description , like cardy s boundary states and runkel s ope coefficients , acquire a geometrical meaning if in terms of non - commutative ( and sometimes non - associative ) spaces . the su(2 ) wzw model provides just the simplest example of a string background with a non - vanishing 3-form field strength @xmath6 , but we think that it illustrates quite nicely much of the behavior one should expect from more complicated backgrounds . in particular , the discussion of su(2 ) branes carries over to boundary wzw models with other structure groups g ( at least in the compact case ) and leads to a quantization of integer conjugacy classes in g. it might be interesting to investigate also branes that are not maximally symmetric , i.e. where the gluing conditions respect only a subalgebra of the maximal chiral symmetry algebra @xcite . boundary cft yields world - volumes independently of whether limiting classical pictures are available or not , and it actually provides more structure than a mere set of non - commutative algebras . connes program @xcite shows that , in order to talk about the geometry of a non - commutative space , it is necessary to fix further `` spectral data '' , including a hilbert space on which the ( associative ) world - volume algebra and a generalized dirac or laplace operator act . how these data can be extracted from a cft has been discussed , for the bulk case , in @xcite . the importance of the laplace operator , which is related to the conformal hamiltonian @xmath165 , can also be seen in the context of our definition of non - commutative world - volumes : in order to re - derive the ope of boundary operators from the algebraic structure of the world - volume , the spectrum of conformal dimensions must be known , cf . the remark after eq . ( [ pointw ] ) . in a cft on the upper half - plane , additional structure is available , e.g. in the form of boundary condition changing operators which induce transitions between two different boundary conditions @xmath166 . the ope of the boundary fields @xmath167 with boundary condition changing operators gives rise to bi - modules @xmath168 over the world - volume algebras of the two associated branes . in the case of d - branes on a group manifold , these bi - modules allow to construct tensor products for representations of the associated quantum group . opes involving two boundary condition changing operators provide even more data , namely a full braided tensor category . some comments on our general scheme to extract a world - volume algebras from the boundary cft description of branes are in order . it involves a choice of `` generating elements '' among the boundary fields . from a pure cft perspective , one could restrict to primary operators only , or one could work with all boundary operators and thus with an infinite - dimensional world - volume . in a sense , the latter algebra would include all internal excitations of the `` static '' space defined using primary fields . the wzw case , where it proved natural to keep the full group multiplets associated with primary boundary fields , suggests that there are distinguished `` intermediate '' choices . for a large class of cfts , the appropriate generalization of the lowest - dimension spaces of wzw models is likely to be given by the special subspaces introduced in @xcite ; see also @xcite . placing the cft into a string theory context can remove the arbitrariness and provide clear guidelines as to which world - volume generators to select from the boundary fields : string theory contains additional parameters like @xmath3 , and the relevant generators of the world - volume algebra are those surviving in some limiting regime . e.g. in the flat background case , one can remove all higher excitations by sending @xmath3 to zero while keeping the @xmath2-field finite ; see @xcite and also @xcite . it may be possible that a number of interesting limits exists ; then one expects that the world - volume of a brane can look very different in different regimes , and that full string theory can `` interpolate '' between those geometries . the next task would be to calculate the effective action on the in general non - commutative world - volume of the brane . the lowest - order terms are , of course , already given by our `` multiplication table '' ( the ope coefficients ) . in principle , higher - order contributions can be computed from the same data , but in practice one still needs to integrate over world - sheet moduli . in the context of the douglas - hull model , the effective field theories were found to be non - commutative supersymmetric gauge theories with some amount of non - locality @xcite . seiberg and witten could show that these models are equivalent to ordinary gauge theories on a flat brane @xcite . it remains to be seen whether classical structures are stretched further when more general cft backgrounds are taken as a starting point . perhaps it is worthwhile to compare the induced field theories with existing models on fuzzy geometries ( see e.g. @xcite ) . it would also be interesting to investigate further the relation between world - volume non - commutativity as introduced in @xcite and non - commuting moduli as discovered by witten @xcite . both phenomena can be traced back to failures in locality properties of boundary fields see @xcite for the case of moduli so that there exists a direct connection between the brane s intrinsic `` fuzziness '' and the way it `` perceives '' its ambient target . + * acknowledgements : * we would like to thank i. brunner , c. chu , r. dijkgraaf , m. douglas , j. frhlich , j. fuchs k. , o. grandjean , p. ho , j. hoppe , c. klimk , n. landsman , g. moore , a. polychronakos , g. reiter , a. schwarz , c. schweigert , s. shatashvili , i.t . todorov and especially j. teschner for useful and stimulating discussions . is grateful to the daad for support and to the aei potsdam for hospitality . * note added : * after this work was completed , another approach to the geometry of branes in wzw models based on exact cft methods was presented in @xcite . here we collect some basic material on clebsch - gordan maps , @xmath106-symbols and the ( quasi-)associativity of various algebras mentioned in the main text . let us denote by @xmath169 the irreducible representation of @xmath136 with spin @xmath170 . by definition , clebsch - gordan maps @xmath171 intertwine between the actions of @xmath136 on the product module @xmath172 and the irreducible module @xmath173 . @xmath106 symbols enter the theory through the basic relation c_q(mk|l ) ( c_q(ij|m ) ^k ) = _ p _ q c_q(ip|l ) ( ^i c_q(jk|p ) ) .[eqa1]they obey a number of fundamental equations . for our purposes , the biedenharn - elliot ( pentagon ) relation is the most important one . with the spin labels set to the values that we need below , it implies _ m _ q y^k_k . [ qfuzzy ] the clebsch - gordan coefficients on the right hand side are obtained from the maps @xmath177 once we have selected a basis in each representation space @xmath178 . associativity of this algebra is rather easy to prove with the help of eqs . ( [ eqa1 ] ) and ( [ eqa2 ] ) : ( y^i_i y^j_j ) y^k_k & = & _ l , l , m , m _ q y^l_l + & = & y^i_i ( y^j_j y^k_k ) for the special case @xmath174 this computation proves the associativity of the world - volume algebra in the limit @xmath116 . when the level @xmath5 is finite and non - rational , however , the defining relation for our algebra @xmath137 from sect . 4 employs the _ undeformed _ clebsch - gordan maps along with the deformed @xmath106 symbols . hence , using relation ( [ eqa1 ] ) for @xmath174 , we generate an undeformed @xmath106 symbol in our computation above . the latter can not be absorbed with the help of the pentagon identity , since we have to deal with a product of one undeformed and two deformed @xmath106 symbols . _ q c(ip|l)(^ic(jk|p ) ) + ( ^-1 ) ^ijk = ( ^i ^j ^k ) & & ( ^-1 ) : v^i v^j v^k v^i v^j v^k . note that this relation involves clebsch - gordan maps of the lie algebra and @xmath147-deformed @xmath106-symbols at the same time . @xmath143 allows to modify the proof we have given for the associativity of the algebra ( [ qfuzzy ] ) such that we obtain the quasi - associativity property ( [ qassoc ] ) . a relation between our quasi - associative algebra @xmath137 and the associative @xmath147-deformation of the fuzzy sphere can be established with the help of drinfeld s twist element @xmath105 . by definition , it maps the deformed and undeformed clebsch gordan maps onto each other , @xmath180 this property becomes crucial in showing that the quasi - associative algebra for non - rational @xmath5 is `` twist - equivalent '' to the associative @xmath147-deformed fuzzy sphere . some details on the notion of twist equivalence can be found e.g. in section 7.3 of @xcite . douglas , c. hull , _ d - branes and the noncommutative torus _ , j. high energy phys . 9802 ( 1998 ) 008 , hep - th/9711165 y .- k.e . cheung , m. krogh , _ noncommutative geometry from d0-branes in a background @xmath2-field _ , nucl . ( 1998 ) 185 , hep - th/9803031 + f. ardalan , h. arfaei , m.m . sheikh - jabbari , _ mixed branes and m(atrix ) theory on noncommutative torus _ , hep - th/9803067 ; , j. high energy phys . 9902 ( 1999 ) 016 , hep - th/9810072 + h. garca - compen , _ on the deformation quantization description of matrix compactifications _ , nucl . * b541 * ( 1999 ) 651 , hep - th/9804188 c. chu and p. ho , _ noncommutative open string and d - brane _ , nucl . * b 550 * ( 1999 ) 151 , hep - th/9812219 v. schomerus , _ d - branes and deformation quantization _ , j. high energy phys . 9906 ( 1999 ) 030 , hep - th/9903205 a.yu . alekseev , v. schomerus , _ d - branes in the wzw model _ , hep - th/9812193 h. garca - compen , j.f . plebaski , _ d - branes on group manifolds and deformation quantization _ , hep - th/9907183 j. frhlich , k. , _ conformal field theory and the geometry of strings _ , crm proceedings and lecture notes vol . * 7 * ( 1994 ) 57 , hep - th/9310187 a.h . chamseddine , j. frhlich , _ some elements of connes non - commutative geometry , and space - time geometry _ , in : chen ning yang , a great physicist of the twentieth century , c.s . liu and s .- t . yau ( eds . ) , international press , 1995 , hep - th/9307012 j. frhlich , o. grandjean , a. recknagel , _ supersymmetric quantum theory , non - commutative geometry , and gravitation _ , les houches lecture notes 1995 , hep - th/9706132 o. grandjean , _ non - commutative differential geometry _ , ph.d . thesis , eth zrich , july 1997 i. brunner , m.r . douglas , a. lawrence , c. rmelsberger , _ d - branes on the quintic _ , hep - th/9906200 c.g . callan , j.a . harvey , a. strominger , _ world sheet approach to heterotic instantons and solitons _ , nucl . * b359 * ( 1991 ) 611 + s .- j . rey , _ the confining phase of superstrings and axionic strings _ , phys . * d43 * ( 1991 ) 526 + s. frste , d. ghoshal , s. panda , _ an orientifold of the solitonic fivebrane _ , phys . * b411 * ( 1997 ) 46 , hep - th/9706057 + m. bianchi , y.s . stanev , _ open strings on the neveu - schwarz pentabrane _ , nucl . phys . * b523 * ( 1998 ) 193 , hep - th/9711069 a. giveon , d. kutasov , n. seiberg , _ comments on string theory on ads@xmath181 _ , adv . theor . * 2 * ( 1998 ) 733 , hep - th/9806194 + j. de boer , h. ooguri , h. robins , j. tannenhauser , _ string theory on ads@xmath181 _ , j. high energy phys . 9812 ( 1998 ) 026 , hep - th/9812046 + d. kutasov , n. seiberg , _ more comments on string theory on ads@xmath181 _ , j. high energy phys . 9904 ( 1999 ) 008 , hep - th/9903219 j. teschner , _ operator product expansion and factorization in the @xmath182-wznw model _ , hep - th/9906215 m. kontsevich , _ deformation quantization of poisson manifolds i _ , q - alg/9709040 a.s . cattaneo , g. felder , _ a path integral approach to the kontsevich quantization formula _ , math/9902090 k. , _ conformal field theory : a case study _ , hep - th/9904145 a.yu . alekseev , a. malkin , e. meinrenken , _ lie group valued moment maps _ , j. diff . geom . * 48 * ( 1998 ) 445 j.l . cardy , _ boundary conditions , fusion rules and the verlinde formula _ , nucl . * b324 * ( 1989 ) 581 g. pradisi , a. sagnotti , y.s . stanev , _ completeness conditions for boundary operators in 2d conformal field theory _ , phys . * b381 * ( 1996 ) 97 , hep - th/9603097 a. recknagel , v. schomerus , _ d - branes in gepner models _ , nucl . * b531 * ( 1998 ) 185 , hep - th/9712186 d.c . lewellen , _ sewing constraints for conformal field theories on surfaces with boundaries _ , nucl . * b372 * ( 1992 ) 654 i. runkel , _ boundary structure constants for the a - series virasoro minimal models _ , nucl . phys . * b549 * ( 1999 ) 563 , hep - th/9811178 l. alvarez - gaum , c. gomez , g. sierra , _ quantum group interpretation of some conformal field theories _ , phys . * b220 * ( 1989 ) 142 j. hoppe , _ diffeomorphism groups , quantization and @xmath183 _ , int . j. mod . phys . * a4 * ( 1989 ) 5235 j. madore , _ the fuzzy sphere _ , class . ( 1992 ) 69 v.g . drinfeld , _ quasi - hopf algebras and knizhnik -zamolodchikov equations _ , in : problems of modern quantum field theory , proceedings alushta 1989 , research reports in physics , springer verlag , 1989 + v.g . drinfeld , _ quasi - hopf algebras _ , leningrad math . j. vol . * 1 * ( 1990 ) no . 6 p. podle , _ quantum spheres _ , lett . * 14 * ( 1987 ) 193 g. mack , v. schomerus , _ action of truncated quantum groups on quasi - quantum planes and a quasi - associative differential geometry and calculus _ , * 149 * ( 1992 ) 513 v. schomerus , _ construction of field algebras with quantum symmetry from local observables _ , * 169 * ( 1995 ) 193 , hep - th/9401042 g. moore , n. reshetikhin , _ a comment on quantum group symmetry in conformal field theory _ , nucl . * b328 * ( 1989 ) 557 a.yu . alekseev , s. shatashvili , _ from geometric quantization to conformal field theory _ , commun . * 128 * ( 1990 ) 197 ; , commun . * 133 * ( 1990 ) 353 j. frhlich , t. kerler , quantum groups , quantum categories , and quantum field theory , lecture notes in mathematics vol . 1542 , springer verlag , 1993 j. fuchs , c. schweigert , _ orbifold analysis of broken bulk symmetries _ , phys . lett . * b447 * ( 1999 ) 266 , hep - th/9811211 ; , hep - th/9902132 ; , hep - th/9908025 + l. birke , j. fuchs , c. schweigert , _ symmetry breaking boundary conditions and wzw orbifolds _ , hep - th/9905038 ; a. connes , noncommutative geometry , academic press , 1994 w. nahm , _ quasi - rational fusion products _ , j. mod . * b8 * ( 1994 ) 3693 , hep - th/9402039 a.yu . alekseev , a. recknagel , v. schomerus , _ generalization of the knizhnik - zamolodchikov equations _ , lett . math . * 41 * ( 1997 ) 169 , hep - th/9610066 n. seiberg , e. witten , _ string theory and noncommutative geometry _ , hep - th/9908142 a. connes , m.r . douglas , a. schwarz , _ noncommutative geometry and matrix theory : compactification on tori _ , j. high energy phys . 9802 ( 1998 ) 003 , hep - th/9711162 m. li , _ comments on supersymmetric yang - mills theory on a noncommutative torus _ , hep - th/9802052 + m. berkooz , _ non - local field theories and the non - commutative torus _ * b430 * ( 1998 ) 237 , hep - th/9802069 i.ya . arefeva , i.v . volovich , _ noncommutative gauge fields on poisson manifolds _ , hep - th/9907114 h. grosse , c. klimk , p. prenajder , _ towards finite quantum field theory in non - commutative geometry _ , int . j. theor . * 35 * ( 1996 ) 231 - 244 , hep - th/9505175 ; , lecture notes clausthal 1995 , hep - th/9510177 + u. carow - watamura , s. watamura , _ noncommutative geometry and gauge theory on fuzzy sphere _ , hep - th/9801195 e. witten , _ bound states of strings and d - branes _ , * b460 * ( 1996 ) 335 , hep - th/9510135 a. recknagel , v. schomerus , _ boundary deformation theory and moduli spaces of d - branes _ , nucl . * b545 * ( 1999 ) 233 , hep - th/9811237 a. recknagel , v. schomerus , _ moduli spaces of d - branes in cft - backgrounds _ , hep - th/9903139 a.yu . alekseev , h. grosse , v. schomerus , _ combinatorial quantization of the hamiltonian chern - simons theory i _ , commun . * 172 * ( 1995 ) 317 , hep - th/9403066 g. felder , j. frhlich , j. fuchs , c. schweigert , _ the geometry of wzw branes _ , hep - th/9909030
|
the geometry of d - branes can be probed by open string scattering .
if the background carries a non - vanishing b - field , the world - volume becomes non - commutative .
here we explore the quantization of world - volume geometries in a curved background with non - zero neveu - schwarz 3-form field strength @xmath0 . using exact and generally applicable methods from boundary conformal field theory
, we study the example of open strings in the su(2 ) wess - zumino - witten model , and establish a relation with fuzzy spheres or certain ( non - associative ) deformations thereof .
these findings could be of direct relevance for d - branes in the presence of neveu - schwarz 5-branes ; more importantly , they provide insight into a completely new class of world - volume geometries . + +
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the large majority of kuiper belt objects ( kbos ) contain no detectable volatile ices on their surfaces , but a small number of the largest objects have been found to have signatures of ch@xmath0 , co , or n@xmath1 , all ices with high vapor pressures at kuiper belt temperatures . after the discovery of volatiles on the surfaces of eris @xcite , makemake @xcite , and sedna , @xcite proposed a simple method for assessing the possibility of volatile retention on kbos . for each relevant ice , they compared the volatile loss due to jean s escape the slowest of many possible escape mechanisms to the total volatile inventory of the object and divided the kuiper belt into objects which could and could not have retained that ice over the age of the solar system . only a handful of objects are massive enough or cold enough to be able to retain volatiles . their model provided a compelling explanation of the low abundance of n@xmath1 on makemake @xcite , which is smaller than pluto and eris , and was also used to successfully predict the presence of methane on quaoar @xcite . to date , the volatile retention model has been completely successful predicting which objects will and which will not have detectable surface volatiles , with the unique exception being the large kbo haumea , which is the parent body of the only collisional family known in the kuiper belt @xcite and clearly had an unusual history . we provide an update to the @xcite calculations in figure 1 . we have used new vapor pressure data from and , where possible , have used measured sizes and masses of the largest kbos . for quaoar , the current measured diameter is 890 @xmath2 70 km implying a density of 4.2 @xmath2 1.3 g @xmath3 @xcite , but we assume the upper limit of size as the smaller sizes lead to physically implausible densities . ( note that even for the smaller size and higher density , however , quaoar is still expected to retain surface methane . ) the size of 2007 or10 is unmeasured , so , as will be justified below , we assume that it has an albedo identical to the 0.18 albedo of quaoar that gives the size that we assume above , though we allow albedo uncertainties of 50% in either direction . as in @xcite , we calculate an `` equivalent temperature '' for each object by integrating the volatile loss through the object s entire orbit and determining the temperature that an object in a circular orbit would have to have to lose the volatile at that rate . for our assumed albedo range , 2007 or10 is somewhere between the fourth and seventh largest object known in the kuiper belt . its potential size spans the range between the small volatile poor objects and the handful of volatile rich objects . 2007 or10 is thus an excellent test object for our understanding of volatile retention in the outer solar system . we explore the surface composition of this object below using a combination of near - ir spectroscopy and multi - wavelength photometry . the low - resolution , near - infrared spectrum of 2007 or10 was obtained on 2010 september 20 ( ut ) using the folded - port infrared echellette ( fire ) spectrograph on the 6.5 m magellan baade telescope @xcite . fire s prism - dispersed mode provides continuous coverage of the 0.852.45 @xmath4 band with a variable resolution of @xmath5 = 250350 . 2007 or10 was acquired and its motion confirmed using fire s @xmath6-band imaging channel . the source was maintained on the 0@xmath76 slit by manual corrections to sidereal tracking . two series of abba dither exposure sequences were obtained with integrations of 120 s at an average airmass of 1.04 . these were followed by a single abba sequence of the g2 v star hd 211544 ( @xmath8=10.9 ) at a similar airmass . exposures of a quartz flat field lamp ( set at 1.2 v and 2.2 v ) and arc lamps ( near ) were obtained for pixel response and wavelength calibration . data were reduced using the methods described in @xcite . the spectrum was converted into relative reflectance as well as corrected for telluric absorption and instrument response by dividing the raw spectrum of 2007 or10 by the spectrum of the solar type star hd 211544 . photometry were obtained with the wide - field camera 3 on the hubble space telescope during cycles 17 ( go program 11644 ) and 18 ( go program 12234 ) . in cycle 17 , two 130 s exposures were taken in the f606w and f814w filters , and two 453 s exposures were taken in the f139 m , and f153 m filters . during the cycle 18 observations , two exposures were acquired in each of the f606w , f775w , f098 m , f110w with exposure times of 128s , 114s , 115s , and 207 s respectively . as well , four exposures of 275 s were acquired in the f127 m filter . for both the cycle 17 and 18 observations , 3 " dithers were applied between image pairs to reduce the effects of cosmic rays and pixel defects , with the exception of the f127 m observations , in which 2 images were taken at each dither position . all observations in a cycle were acquired within a single orbit , minimizing the effect of any light curve or10 may have . all data were processed through calwfc3 version 2.3 , the standard wfc3 image processing pipeline @xcite . circular apertures were used to measure the photometry . _ tiny tim _ version 7.1 psfs @xcitewere used to generate infinite aperture corrections as well as interpolate over any bad pixels flagged during the image reductions . fluxes were converted to relative reflectance by comparing to fluxes computed using the the _ calcphot _ routine for a model solar spectrum @xcite provided as part of the _ iraf _ package _ stsdas.synphot_. approximate absolute reflectances were then obtained by scaling the f606w relative reflectance to a value of 0.18 , our assumed albedo of 2007 or10 . all are shown in table 1 . before calculating relative reflectances , the cycle 18 magnitudes were adjusted upward by 0.03 to account for the difference in the f606w magnitudes between the two epochs . the small magnitude difference is an expected consequence of object rotation . cccc cycle 17 + f606w & 21.68@xmath2 0.02 & -27.00 & 0.18 + f814w & 21.39@xmath2 0.01 & -26.54 & 0.37 + f139 m & 22.06@xmath2 0.01 & -25.34 & 0.59 + f153 m & 22.47@xmath2 0.02 & -25.09 & 0.51 + cycle 18 + f606w & [email protected] & -27.00 & 0.18 + f775w & [email protected] & -26.64 & 0.34 + f098 m & [email protected] & -26.13 & 0.52 + f110w & [email protected] & -25.78 & 0.58 + f127 m & [email protected] & -25.54 & 0.58 + figure 2 shows the fire reflectance spectrum with the wfc3 photometric points overlaid . we scale all point to an albedo of 0.18 in the f606w filter , though we note that the true value of the albedo has not been measured . to increase the signal - to - noise in the fire spectrum , we also plot the median of every 32 spectral channels , oversampled by a factor of two , to simulate how the spectrum would appear to a lower resolution spectrograph . uncertainties on these data points are obtained by calculating the median absolute deviation of eache 32 channel sample , which we multiply by 1.48 to obtain what would be the standard deviation in a normally distributed sample , and then divide by the square root of the number of spectral channels , approximating the standard deviation . the fire spectrum and the wfc3 photometry are in broad agreement in the area of overlap , though the match is imperfect . we suspect that the differences are due to differential refraction in the fire data , which , for these early attempts at tracking a moving object , were not obtained with the slit aligned along the parallactic angle . both data sets show , in particular , a very red optical slope and a distinct absorption around 1.5 @xmath9 m . the fire spectrum shows an additional broad absorption feature near 2.0 @xmath9 m and , potentially , additional features redward . absorptions at 1.5 and 2.0 @xmath9 m are the characteristics features of water ice , which is frequently found on the largest kbos @xcite . figure 3 compares the spectrum of 2007 or10 to a modeled spectrum of a surface consisting of a mixture of water ice and a neutral material . we place no special significance on the precise water ice surface model , as many different types of specific parameters yield similar modeled spectra , but , for concreteness we use the water ice absorption coefficients of @xcite and construct a simple hapke model @xcite with 50@xmath9 m grains at 50 k spatially mixed with equal amounts of a neutral material with an albedo of 80% . the model spectrum is sampled at the same resolution as the smoothed spectrum of the object . even at the low signal - to - noise ratio of the data , the water ice model provides an excellent match to the spectrum redward of 1.4 @xmath9 m . at shorter wavelengths , however , 2007 or10 is significantly redder than water ice . most of the kbos with significant water ice absorption , such as orcus , haumea , and the haumea family members , have nearly neutral optical colors @xcite . the most notable exception to this trend is quaoar , which has water ice absorptions nearly identical in depth to those of 2007 or10 and is almost as red . a spectrum of quaoar @xcite over the same wavelength range is shown , for comparison , in figure 3 . the coloring of quaoar was suggested by @xcite to be due to the effects of methane , which turns red with irradiation @xcite . quaoar , as seen in figure 1 , is barely large enough or cold enough to retain methane on its surface and the amount left is small and detectable only at high signal - to - noise . 2007 or10 , depending on its precise size , could be in a similar regime of volatile retention as quaoar . if true , the quaoar - like optical color of 2007 or10 could be a signature of the retention and irradiation of methane on the kbo . indeed , if 2007 or10 is assumed to have the same albedo as quaoar as the close match in visible and near - infrared spectra might suggest 2007 or10 sits in almost precisely the same volatile loss regime as quaoar . while the perihelion distance of 2007 or10 of only 33.6 au ( compared to quaoar s of 41.6 au ) makes it significantly hotter than quaoar , the significantly larger size of 2007 or10 allows its larger gravitational pull to nonetheless potentially retain methane . if the hypothesis that the extreme red coloration on 2007 or10 is like that of quaoar caused by the irradiation of a small amount of remaining surface methane is correct , we predict that methane absorption should be visible on 2007 or10 as it is on quaoar . figure 3 also shows a simple model of a surface including solid methane and a neutral component . the model is simply the water model from above but with the absorption coefficients of methane @xcite replacing those of water . the spectrum is shifted upward by 0.3 units for clarity . the strongest absorption feature of methane , at 2.3 @xmath9 m does indeed correspond with a large variation from the water ice model . we conclude that while these data do not have sufficient signal - to - noise , particularly in the k - band , to positively detect methane on 2007 or10 , the existence of this volatile is plausible , and would provide a pleasing explanation for the extreme red coloration of the kbo . the large kuiper belt object 2007 or10 provides an excellent test of our understanding of volatile loss and retention on the surfaces of objects in the outer solar system . while the size of 2007 or10 has yet to be measured , the simple assumption that it has an identical albedo to quaoar the object whose spectrum its spectrum most resembles places 2007 or10 into a regime where it would be expected to retain trace amounts of methane on its surface . such an object would be expected to have red optical coloration from methane irradiation , which both quaoar and 2007 or10 do have . in addition , such an object should have detectable signatures of methane if observed at sufficient signal - to - noise . such methane signatures have been detected on quaoar , but require higher signal - to - noise to positively identify on 2007 or10 . while additional measurements of the size and spectrum of 2007 or10 are clearly required , we conclude that volatile retention models @xcite appear to continue to flawlessly predict both the presence and absence of volatiles on all objects in the kuiper belt which have been observed to date . , r. a. , burgasser , a. j. , bernstein , r. a. , bigelow , b. c. , fishner , j. , forrest , w. j. , mcmurtry , c. , pipher , j. l. , schechter , p. l. , & smith , m. 2008 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 7014 , society of photo - optical instrumentation engineers ( spie ) conference series , r. a. , burgasser , a. j. , bochanski , j. j. , schechter , p. l. , bernstein , r. a. , bigelow , b. c. , pipher , j. l. , forrest , w. , mcmurtry , c. , smith , m. j. , & fishner , j. 2010 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 7735 , society of photo - optical instrumentation engineers ( spie ) conference series
|
we present photometry and spectra of the large kuiper belt object 2007 or10 .
the data show significant near - infrared absorption features due to water ice .
while most objects in the kuiper belt with water ice absorption this prominent have the optically neutral colors of water ice , 2007 or10 is among the reddest kuiper belt objects known .
one other large kuiper belt object
quaoar has similar red coloring and water ice absorption , and it is hypothesized that the red coloration of this object is due to irradiation of the small amounts of methane able to be retained on quaoar .
2007 or10 , though warmer than quaoar , is in a similar volatile retention because it is sufficiently larger that its stronger gravity can still retain methane .
we propose , therefore , that the red coloration on 2007 or10 is also caused by the retention of small amounts of methane .
positive detection will require spectra of methane on 2007 or10 will require spectra with higher signal - to - noise .
models for volatile retention on kuiper belt objects appear to continue to do an excellent job reproducing all of the available observations .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the study of both atomic and optical physics problems , often analyzed in the realm of nonlinear schrdinger ( nls ) type equations @xcite , the study of double well potentials has a prominent position . such potentials can be straightforwardly realized in atomic bose - einstein condensates ( becs ) through the combination of a parabolic ( harmonic ) trap with a periodic potential . their experimental realization and subsequent study in becs with self - repulsive nonlinearity has led to numerous interesting observations including tunneling and josephson oscillations for small numbers of atoms in the condensate , and macroscopic quantum self - trapped states for large atom number @xcite and symmetry - breaking dynamical instabilities @xcite . these experimental developments have been accompanied by a larger array of theoretical studies on issues such as finite - mode reductions and symmetry - breaking bifurcations @xcite , quantum effects @xcite , and nonlinear variants of the potentials @xcite . similar features have also emerged in nonlinear optical settings including the formation of asymmetric states in dual - core fibers @xcite , self - guided laser beams in kerr media @xcite , and optically - induced dual - core waveguiding structures in photorefractive crystals @xcite . on the other hand , a theme that has also been progressively becoming of increasing importance within both of these areas of physics is that of long range interactions . in the atomic context , the experimental realization of becs of magnetically polarized @xmath0cr atoms @xcite ( see recent review @xcite and for a study of double well effects @xcite ) , as well as the study of dipolar molecules hetmol , and atoms in which electric moments are induced by a strong external field @xcite have been at the center of the effort to appreciate the role of long range effects . on the other hand , in nonlinear optics , where nonlocal effects have been argued to be relevant for some time now @xcite , numerous striking predictions and observations have arisen in the setting of thermal nonlocal media @xcite . among them , we single out the existence of stable vortex rings @xcite the experimental realization of elliptically shaped spatial solitons @xcite and the observation of potentially pairwise attracting ( instead of repelling as in the standard local cubic media ) dark solitons @xcite . another very important large class of systems displaying a nonlocal nonlinearity consists of materials with a quadratic nonlinearity . in @xcite , it has been shown that , in fact , the quadratic nonlinearity is inherently nonlocal . this implies that nonlocality can be used explain the beautiful x - wave @xcite observations and even the different regimes of soliton pulse compression in quadratic materials @xcite . it is interesting to note that in these quadratic media , not only does the prototypical ingredient of ( effective ) nonlocality arise , but it is also possible for a competition of this feature with the cubic nonlinearity to emerge as is discussed in @xcite . our aim in the present work is to expand on the framework of studies of double well potentials in the presence of nonlocal nonlinear interactions by considering cubic - quintic models . part of the motivation for doing so consists of the fundamental relevance of the cubic - quintic nls . the latter is a model that has been used in a variety of physical settings . these include the light propagation in optical media such as non - kerr crystals @xcite , chalcogenide glasses @xcite , organic materials @xcite , colloids @xcite , dye solutions @xcite , and ferroelectrics @xcite . it has also been predicted that this type of nonlinearity may be synthesized by means of a cascading mechanism @xcite . an additional part of the motivation stems from an interesting set of observations that were made in an earlier work featuring competing _ nonlinearities , one of which was a cubic local and another was a cubic nonlocal one ; see @xcite and the discussion therein . in that work , it was found that for repulsive nonlocal cubic interactions and attractive local ones , it was possible to tune the prefactors determining the competition so as to produce not only a symmetry breaking , but also a symmetry - restoring bifurcation . more recently , a similar conclusion in a local cubic - quintic double well potential was reached in @xcite . here , we present a framework where the competition of cubic and quintic terms can be systematically quantified . in addition , to address the problem from a broader perspective , we consider fully nonlocal interactions both for the cubic and the quintic terms , rendering the local case a straightforward special - case scenario of our study . the specific setup we consider here is partially of interest to the field of cold gases e.g. in the case of repulsive quintic ( but local ) interactions and attractive cubic nonlocal ones . this exactly corresponds to the model of the dipolar tonks - girardeau gas with the dipole moments polarized along the axis , considered earlier in @xcite . the difference here is that in this setting the quintic interaction can not be made nonlocal ( although the relevant mathematical norm form description and physical phenomenology will be essentially the same as presented herein ) . a setup more precisely in tune with the considerations given below arises in the field of nonlinear optics and , more particularly , in the case of thermal optical nonlinearity @xcite but when the heating is provided by the resonant absorption by dopants , in which case the absorption may be saturable . in the appendix , we justify more precisely this connection to the specific model analyzed in what follows . we start our presentation of the theoretical analysis of section ii by developing a two - mode reduction of the system with both the cubic and the quintic terms . we systematically examine all the relevant terms and offer a prescription for assessing the dominant contributions to the resulting dynamics of the left and the right well . following an amplitude - phase decomposition and examining the variables associated with the population imbalance of the two wells , and their relative phase , we construct the hamiltonian normal form of the two - mode reduction of the cubic - quintic double well system . we then _ explicitly illustrate _ how the bifurcation analysis of this normal form encapsulates not only the symmetry breaking but _ also _ the symmetry restoring . we argue that this cubic - quintic realization is the prototypical one where both of these effects can be observed and analytically quantified . subsequently , in section iii , we proceed to test the relevant predictions by means of a computational bifurcation analysis , as well as through direct numerical simulations ( in order to monitor the predicted dynamical instabilities ) . we find very good agreement with the symmetry breaking predictions of the model and even a quite fair agreement with the symmetry restoring ones ( which arise in a highly nonlinear regime and are hence less amenable to a two - mode analysis ) . we also quantify the disparity of the analytical predictions and numerical results for large values of the nonlocality range parameter . finally , section iv contains our conclusions and some directions for future study . as indicated above , our fundamental model will be the 1d nls equation in the presence of two nonlocal terms , namely the cubic and quintic ones : @xmath1 with @xmath2 and the linear operator will be of the standard schrdinger type @xmath3 this encompasses the double - well potential of the form : @xmath4 with @xmath5 being the normalized strength of the parabolic trap and it is @xmath6 in a quasi-1d situation in becs ( here the effective trap frequency is the ratio of the longitudinal trap strength along the condensate over the one of the tightly confined transverse directions ) . in our study we consider a typical experimentally relevant value of @xmath7 , while the generally tunable ( see e.g. @xcite ) parameters of the laser beam forming the light defect are chosen to be @xmath8 and @xmath9 ( which we have found to be fairly typical values representative of the phenomenology to be analyzed below ) . for the kernels @xmath10 , @xmath11 we will focus our considerations on either the gaussian @xmath12 or the exponential @xmath13 while the latter is more specifically relevant to the thermal nonlocal ( optical ) media and to quadratic nonlinear materials @xcite , we also use the former due to the mathematical simplicity of its kernel . in any event , our results will not be significantly different qualitatively between the two cases , although obviously the quantitative details will not be the same . the key parameter here is the range of the nonlocal interaction parametrized by @xmath14 . notice that both kernels in the limit of @xmath15 tend to a genuinely local interaction ( i.e. , @xmath16 ) . we now develop the two - mode approximation in order to obtain a decomposition ( or more accurately a galerkin truncation ) of the solution @xmath17 over the minimal basis of fundamental states . more specifically , we use an orthonormal basis composed by the wave functions @xmath18 , where @xmath19 and @xmath20 ( fig . 1 ) are the ground state and the first excited state , respectively , corresponding to the first two eigenvalues of @xmath21 that are @xmath22 and @xmath23 for our choice of potential parameters above . notice that these two eigenfunctions represent modes with support predominantly on the left and right well , respectively . the eigenfunctions @xmath24 and the rotated basis employed herein of @xmath25 are both shown in fig . [ basis ] . the two - mode approximation is then defined as @xmath26 where @xmath27 and @xmath28 are complex time - dependent amplitudes and the approximation consists of the truncation of the higher modes within the expansion . before substituting into the initial gross - pitaevskii ( gp ) equation , we notice that the action of the linear operator @xmath21 on our basis elements is as follows : @xmath29 where @xmath30 and @xmath31 are linear combinations of the two eigevalues of @xmath21 respectively to the solutions @xmath19 , @xmath20 . subsequently , substitution of our ansatz of eq . ( [ twomode ] ) in the full nonlinear problem of eq . ( [ eq1 ] ) yields : @xmath32\int r(x - x')\phi_l(x')\phi_r(x')dx'\\ + \delta|c_l|^4(c_l\phi_l + c_r\phi_r)\int r_2(x - x')\phi_l^4(x')dx ' + \delta|c_r|^4(c_l\phi_l + c_r\phi_r)\int r_2(x - x')\phi_r^4(x')dx'\\ + \delta\left[(4|c_l|^4|c_r|^4c_l + c_l^3{c_r^*}^2 + c_l^*|c_l|^2c_r^2)\phi_l + ( 4|c_l|^4|c_r|^4c_r + c_r^3{c_l^*}^2 + c_r^*|c_r|^2c_l^2)\phi_r \right ] \cdot\\ \cdot \int r_2(x - x')\phi_l^2(x')\phi_r^2(x')dx'\\ + 2\delta\left[(|c_l|^2c_l^2c_r^ * + |c_l|^4c_r)\phi_l + ( |c_l|^2|c_r|^2c_l + |c_l|^2c_r^2c_l^*)\phi_r\right]\int r_2(x - x')\phi_l^3(x')\phi_r(x')dx'\\ + 2\delta\left[(c_l^2|c_r|^2c_r^ * + |c_l|^2|c_r|^2c_r)\phi_l + ( |c_r|^4c_l + c_r^2|c_r|^2c_l^*)\phi_r \right]\int r_2(x - x')\phi_r^3(x')\phi_l(x')dx'.\end{aligned}\ ] ] in order to project the above equation onto the states @xmath25 we multiply with the respective function ( notice that the eigenfunctions are real due to the hermitian nature of the operator @xmath21 ) and integrate . this involves the following integrals which will play a fundamental role in our considerations below : @xmath33 from the first nonlocal term , as well as @xmath34 from the second nonlocal term . some alternatives that are derived if we interchange the variables @xmath35 and @xmath36 or swap @xmath37 and @xmath38 can also be equivalently considered . a numerical study of the first four intergrals was already conducted in @xcite , where it was found that typically the integrals @xmath39 can be considered as negligible in comparison to @xmath40 which is the dominant term . on the other hand , @xmath41 is close to @xmath39 for near - local interactions ( i.e. , for small values of @xmath14 ) , but becomes comparable to @xmath40 as the latter descreases for wide nonlocal interaction ranges ( i.e. , for large @xmath14 ) . the criterion that we use to determine whether @xmath41 is negligible or not was @xmath42 where @xmath43 . this yields that @xmath41 remains significant until ( i.e. , down to ) a critical value @xmath44 and @xmath45 for the gaussian and exponential kernels , respectively . the dependence of the relevant overlap integrals on the range of the interaction @xmath14 is shown in fig . [ overlap_int ] . taking into regard the second nonlocal term ( which for simplicity we have assumed to share the same range parameter as the first ) , we can see from fig . [ overlap_int2 ] that the integrals @xmath46 are always negligible but @xmath47 appears to be a nontrivial competing term . this is to a certain degree intuitively anticipated , as this represents the dominant term associated with the quintic interaction . adapting the same criterion as in @xcite ( namely @xmath48 ) , we incorporate the relevant @xmath49 for @xmath50 , @xmath51 for the gaussian and exponential kernel , respectively . according to this we may distinguish three cases : * the terms @xmath40 and @xmath49 are considered for @xmath52 ( for the gaussian kernel ) ; * the term with prefactor @xmath41 is added when @xmath53 . * for @xmath54 , @xmath49 is omitted and only @xmath40 , @xmath41 are taken into account . for the first case , the projection of the equation onto the states @xmath25 yields @xmath55 and by introducing madelung representation of action - angle or amplitude - phase decomposition ( @xmath56 ) , we obtain @xmath57 where we have defined the relative phase @xmath58 and the respective equations for @xmath59 and @xmath60 can be obtained by exchanging @xmath37 and @xmath38 and using @xmath61 instead of @xmath62 . focusing now on the steady solutions ( satisfying @xmath63 ) , we need to enforce @xmath64 or @xmath65 for non - zero amplitudes . this leads us to symmetric and antisymmetric ( equal or opposite amplitudes ) _ pairs _ of solutions , namely for @xmath66 we have the symmetric ( only positive ones among the ) solutions @xmath67 with @xmath68 , for @xmath69 ( @xmath70 for @xmath71 ) . also , for @xmath72 , we have ( only the positive amplitude ones among ) the antisymmetric solutions @xmath73 with @xmath74 for @xmath69 ( resp . @xmath75 for @xmath76 ) . for the asymmetric solutions one has to solve the polynomial @xmath77 which can more conveniently be written as a function of the norm of the solutions ( representing the atom number in becs and the optical intensity in optics ) . thus , introducing @xmath78 ( @xmath79 ) yields the quartic polynomial @xmath80 for the second case ( @xmath81 ) the integrals @xmath40 , @xmath41 , @xmath49 are taken into account and the projection equations onto the states @xmath25 , read : @xmath82 here , the amplitude - phase decomposition yields @xmath83 we can , once again , obtain the set of stationary solutions as follows . when @xmath66 ( symmetric case ) the solutions will be ( the positive amplitude ones among ) @xmath84 for @xmath85 for @xmath69 ( @xmath86 for @xmath87 ) and when @xmath88 ( antisymmetric case ) the solutions are ( the positive amplitude ones among ) @xmath89 and exist for @xmath90 for @xmath69 ( @xmath91 for @xmath87 ) . the asymmetric solutions now , directly in norm expression , will be given by the polynomial @xmath92 with @xmath93 here standing for @xmath94 . in the third case , when @xmath54 , the effect of the quintic terms is deemed to be negligible and the situation reverts to the analysis of @xcite and is hence omitted here . in order to derive a more convenient form of the system so that we can proceed to the analysis of the spontaneous symmetry breaking ( ssb ) bifurcation , we introduce the population imbalance between the two wells , @xmath95 where @xmath96 and @xmath97 . together with the relative phase between the two wells @xmath58 , this forms a set of conjugate variables , in which we obtain the dynamical system : @xmath98 this can be written in the hamiltonian form @xmath99 with the hamiltonian function @xmath100 note that @xmath93 stands either for @xmath40 ( @xmath101 ) or for @xmath94 ( @xmath102 ) . the system possesses the stationary solutions ( critical points ) @xmath103 and @xmath104 with @xmath105 , @xmath106 , @xmath107 that correspond to the symmetric and antisymmetric solutions , identified above . furthermore , the stationary solutions representing the asymmetric branches are given by : @xmath108 these branches emerge and merge as bifurcations from and to the symmetric or antisymmetric solutions and they exist for those values of @xmath78 for which @xmath109 . taking @xmath110 , we get that @xmath111 by substituting @xmath112 or @xmath113 we get the same four possible expressions for @xmath78 as a function of @xmath14 that are displayed in fig.@xmath114 and we denote them with @xmath115 , @xmath116 , @xmath117 and @xmath118 ( the subscripts @xmath119 and @xmath120 correspond to the ( - ) signs in the left and right expressions of eq . ( [ ns ] ) , respectively , while the subscripts @xmath121 and @xmath122 to the ( + ) signs ) . one can then see that when @xmath112 and demanding that @xmath123 , one gets that @xmath78 should either lie in the area outside the curves @xmath115 and @xmath116 or in the area inside the curves @xmath117 and @xmath118 . in the case of @xmath124 and @xmath71 , the role of the symmetric and anti - symmetric branches gets exchanged in as far as the bifurcation of the asymmetric branch is concerned ( see also below ) . importantly , it can be observed in fig . [ crit_points ] that @xmath115 is always negative , hence it is omitted for the principal case considered herein , namely @xmath125 and @xmath126 . on the one hand , the _ critical _ conclusion of our analysis is that for @xmath127 , the system is predicted to have for the anti - symmetric branch both a symmetry breaking bifurcation ( at @xmath128 ) and a symmetry restoring one that eliminates the asymmetric branch ( at @xmath129 ) . on the other hand , the right panel suggests that @xmath117 , @xmath118 coincide @xmath130 , beyond which there is only a single ( symmetry breaking ) bifurcation . however , as will be discussed below , for large interaction range @xmath14 this prediction seems to have some discrepancy from what actually happens as we will see that in fact , we observe a symmetry restoring bifurcation while we do not observe a bifurcation at all in the symmetric branch . to the best of our knowledge , this is the first example of an analytical prediction of the existence of a symmetry restoring bifurcation , a feature that is unique to the analysis of the normal form of the bifurcation for the cubic - quintic case ( and can not be predicted e.g. in the purely cubic case two - mode analysis of @xcite ) . the new critical points appear or disappear as a pitchfork bifurcation that emerges from the antisymmetric solutions for @xmath88 respectively . from the symmetric solution , in this case of @xmath125 and @xmath126 , only a single bifurcation arises at @xmath131 . for the opposite case ( to the one principally considered herein ) of @xmath124 and @xmath71 , i.e. , for a focusing cubic nonlinearity , the bifurcations emerge from the symmetric branch , while for @xmath125 , i.e. , for a defocusing cubic term , then the relevant symmetry breakings arose from the anti - symmetric branch . thus , in this case , we expect an asymmetric branch to bifurcate and break the symmetry at @xmath128 , while it returns to the parent symmetric branch restoring the symmetry at @xmath132 . on the other hand , for the anti - symmetric waveform with a focusing cubic nonlinearity , only a single bifurcation arises at @xmath133 . we provide further details of each of these bifurcations and their comparison with the full numerics of the underlying nls model in the next section . from the system of eqs . ( [ system ] ) , one can reduce the dynamical evolution to a single second - order ode : @xmath134 which can also be rewritten in the `` position - momentum '' variables as : @xmath135 this renders the system amenable to the phase plane representation of the form shown in fig . [ phase_plane ] . here we observe that there is a stationary solution @xmath136 which is a fixed point of the center type . however , for the cases when @xmath112 , for @xmath78 crossing the critical point @xmath116 in the case of the symmetric branch and for @xmath137 $ ] in the case of the anti - symmetric branch , there appear two more critical points at @xmath138 and @xmath139 , representing the asymmetric solutions . the point @xmath140 is a fixed point of center type before the bifurcation occurs , but past the relevant critical number of atoms ( or optical intensity ) , it becomes a saddle as the two new ( asymmetric ) fixed points that appear are of center type . [ phase_plane ] shows the phase space of the full system , as well as the vicinity of the critical points for the gaussian kernel with @xmath141 , @xmath142 and @xmath143 . it is worth mentioning at this point that there are no further changes in the stability of the critical points ( and thus in the corresponding stationary solutions ) for other values of @xmath78 except for those reported above . for the sake of simplicity we illustrate this below for the antisymmetric solution bifurcation as a preamble towards the corresponding numerical results of the next section . the antisymmetric solution corresponds the critical point @xmath144 where the jacobian of the linearized version of ( [ system ] ) is @xmath145 and its eigenvalues satisfy @xmath146 for the case where @xmath147 the graph of @xmath148 versus @xmath78 ( illustrated in the left panel of fig . [ eigen ] ) shows clearly that the two initially ( i.e. , close to the linear limit ) purely imaginary eigenvalues become real at @xmath149 , so that the center type equilibrium becomes a saddle until @xmath150 where it turns back to its initial state , restoring the stability on the antisymmetric branch ( symmetry - restoring bifurcation ) with no other changes in between ( or after that ) . for the asymmetric solution that corresponds to the point @xmath151 , where @xmath152 and @xmath88 , the jacobian becomes @xmath153 and for its eigenvalues we obtain @xmath154 again for @xmath147 the graph of @xmath148 versus @xmath78 ( illustrated in the right panel of fig . [ eigen ] ) shows that the eigenvalues are always purely imaginary which corresponds to an equilibrium of the center type . one can observe here that the values of @xmath78 where the eigenvalues of the asymmetric branch touch " the @xmath35-axis ( @xmath155 and @xmath156 ) coincide with the values where the bifurcation occurs ( fig . [ crit_points]-right panel ) therefore the critical points @xmath157 cease to exist before @xmath158 and after @xmath159 . for the values of @xmath78 within this interval , no further change of stability is observed . as it is made clear in the next section , these stability results are in excellent agreement with the corresponding numerical ones . additionally , it will be come transparent therein that additional turning points in the @xmath78 vs. @xmath160 bifurcation diagram do _ not _ correspond to any instabilities in complete agreement with the recent analysis of @xcite . we now turn to the examination of our analysis against the results of numerical bifurcation analysis ( and in the next subsection also compare them to direct numerical simulations ) . we focus here on the case where @xmath125 , @xmath69 , as we are especially interested in the case of _ competing _ interactions ; we will briefly also touch upon the case of @xmath124 and @xmath71 . in our numerical computations , the stationary solutions are obtained by using a fixed - point newton - raphson iteration for a finite difference decomposition of the relevant boundary value problem , with a choice of the grid spacing of @xmath161 and employing a parametric ( and wherever needed a pseudo - arclength ) continuation of the solutions with respect to the chemical potential parameter @xmath160 ( in optics this is the so - called propagation constant ) . the linear stability is analyzed by considering the standard linearization around the stationary solutions @xmath162 in the form @xmath163 this yields the eigenvalue problem @xmath164 where the operators are defined as @xmath165\phi + \\ + s\int_{-\infty}^{+\infty}k(x - x')\psi_0(x)\psi_0^*(x')\phi(x')dx ' + 2\delta\int_{-\infty}^{+\infty}k(x - x')\psi_0(x)\psi_0(x'){\psi_0^*}^2(x')\phi(x')dx'\end{aligned}\ ] ] and @xmath166 for any real function @xmath167 . instability is guaranteed by the existence of any eigenvalues @xmath168 of the linearized operator with @xmath169 in the sense that perturbations along the corresponding eigendirections will deviate exponentially from the corresponding fixed point . recall that this is also the case for all eigenvalues of our hamiltonian system , since when @xmath168 is an eigenvalue , so are @xmath170 , @xmath171 and @xmath172 . in the case where all eigenvalues are found to be purely imaginary , then the solution is found to be marginally stable . in our specific case of competing interactions , we comment on the following . the positive value ( @xmath173 ) denotes the repulsive behavior of the cubic nonlocal term while the negative one @xmath69 leads to attractive behavior of the quintic nonlocal nonlinearity . as we examine the bifurcation problem of nonlinear states from the corresponding linear eigenstates , we expect that for lower values of @xmath78 ( i.e. , weaker nonlinearities ) , the former repulsive term should be dominant , while for larger values of @xmath78 ( i.e. , stronger nonlinearities ) , it is anticipated that the latter attractive term will take over . this is accurately reflected in the numerical bifurcation diagrams that we now show in figs . [ stat1]-[stat3 ] , for three ( distinct by roughly an order of magnitude in each case ) values of the range @xmath14 . the first value of @xmath174 in fig . [ stat1 ] is supposed to reflect the local case ( since the range of interaction is much smaller than any other intrinsic length scale in the system ) . here the agreement with the two - mode approximation is very good quantitatively for low @xmath78 and very good qualitatively ( and even good quantitatively for some features such as chemical potentials of critical points ) for large @xmath78 . the quality of these types of agreements is found to be preserved for an intermediate interaction range of @xmath141 in fig . [ stat2 ] . however , when the interaction range becomes sufficiently large that it competes ( or overcomes ) the length scale of the potential wells , then fundamental disparities are expected to be found and that is the very conclusion of fig . [ stat3 ] for @xmath175 . [ stat ] + in the first case where @xmath147 , the symmetric and antisymmetric branches of nonlinear states emanate from @xmath176 and @xmath177 , as expected , respectively ( @xmath178 and @xmath179 ) , both of them being dynamically stable , for sufficiently small values of @xmath78 . the rightward bending of the branches for small @xmath78 confirms the dominance of the self - repulsive part of the ( cubic ) interactions for small @xmath78 , as indicated above . the antisymmetric branch ( top right panel of fig . [ stat1 ] and see also the zoom of the bottom panel of the figure ) is destabilized and the theoretically predicted asymmetric branch emerges . the numerical value of the chemical potential for the bifurcation point is found to be @xmath180 , whereas the corresponding analytical one is @xmath181 , confirming the quantitative nature of the agreement with the two - mode approximation . for larger @xmath78 , we observe that the asymmetric solution has two apparent turning points ( where the sign of @xmath182 changes , but in fact its stability does not change - which agrees with the theoretical result presented in the previous section ) , before it reaches the anti - symmetric branch at the numerically computed value @xmath183 where we observe the symmetry restoring effect , which , in fact , re - stabilizes the anti - symmetric branch . in our theoretical analysis , we observe the same qualitative behavior and the symmetry restoring occurs at @xmath184 , in reasonable agreement with the full numerical results . two additional observations should be made here . on the one hand , since the symmetry restoring occurs at much larger values of @xmath78 , the relevant agreement is expected to be less adequate quantitatively than for the symmetry breaking occurring at lower @xmath78 . this is because a two - mode expansion is less appropriate of a reduction at such higher nonlinearities . on the other hand , it can indeed be observed that while the overall trend of the two curves is the same ( and even critical / turning points in terms of their chemical potential are rather accurately captured ) , this agreement is not adequate quantitatively e.g. for critical values of @xmath78 ( or for detailed quantitative matching of the curves for large @xmath78 ) . for the symmetric solution of the top left panel of fig . [ stat1 ] , we can observe that it is increasing monotonically until @xmath185 where it sustains a pitchfork bifurcation leading to the emergence of an asymmetric branch and also a subsequent turning point . the symmetric branch becomes unstable thereafter and the asymmetric emerging state is the stable daughter branch . notice that the theoretical analysis is once again quantitatively accurate for small @xmath78 and the agreement becomes more qualitative for higher @xmath78 s . the critical point for the emergence of the asymmetric branch is predicted for @xmath186 in reasonable agreement with the full numerical result . for the case of @xmath187 the effects are similar to those in the previous case . the symmetry breaking of the antisymmetric branch ( top right , as well as zoom in of the bottom panel of fig . [ stat2 ] ) occurs now at @xmath188 according to the numerical results and at @xmath189 in the two - mode approximation , again attesting to its validity for small @xmath78 . after following a similar trajectory with the case @xmath147 , the asymmetric solution merges back to the antisymmetric one at @xmath190 ( numerical value ) or at @xmath191 ( analytical value ) with the antisymmetric branch again regaining its stability past the symmetry restoring bifurcation . the symmetric solution ( top left panel of fig . [ stat2 ] ) again increases monotonically until it sustains a symmetry breaking bifurcation of its own at @xmath192 . the two - mode approximation predicts this bifurcation to arise at @xmath193 . + next , in fig . [ stat3 ] , we increase the interaction range , roughly , another order of magnitude by setting @xmath194 . here , as may be intuitively expected given that the interaction range is wider than the wells of the potential , the results are quite different . for small values of @xmath160 ( and thus atom number @xmath78 or optical power ) we have a quite satisfactory agreement ( even quantititative ) with the two mode approximation , as may be expected . as a demonstration of that , we note that the symmetry breaking of the antisymmetric branch occurs in our analysis at @xmath195 , while numerically it is found to take place at @xmath196 . on the other hand , due to the predicted earlier collision of the critical points @xmath117 and @xmath118 , there is no symmetry restoring taking place in our normal form reduction . nevertheless , we observe that such a restoring , in fact , still takes place in the full numerical bifurcation diagram . furthermore , in this case , we have not been able to detect a symmetry - breaking bifurcation in the case of the symmetric branch , even though such a bifurcation is predicted within the reduction . this illustrates that for such large values of @xmath14 , even the qualitative agreement previously associated with the large @xmath78 case dynamics should not be expected to be present . finally , we examine also one case where we switch the signs of the nonlocal terms to @xmath197 , so now the cubic term is the one that behaves attractively while the quintic one behaves repulsively . this is illustrated in fig . [ stat3a ] . the interaction range @xmath14 is selected here to be @xmath121 and here we see that the same phenomenology appears in a region where the cheminal potential varies from @xmath198 to @xmath199 , thus attaining negative values . as earlier , both states emanate for the same values of @xmath160 and as we decrease its value we observe the symmetry breaking at @xmath200 ( both for numerical and analytical ) this time on the symmetric state which becomes unstable . as we further decrease the chemical potential to negative values of @xmath160 , the symmetry restoring of the asymmetric state towards its parent symmetric branch occurs at @xmath201 ( numerical value ) . the analytical prediction for this critical point is @xmath202 . hence , once again we observe a good qualitative agreement for larger @xmath78 ( although once again slight quantitative disparities exist between the overall curves and the critical points in terms of @xmath78 ) . a look at the antisymmetric branch now shows us that a bifurcation occurs at the point where the solution changes slope ( @xmath182 ) , precisely at @xmath203 ( numerical ) and is theoretically predicted to arise at @xmath204 ( analytical ) with the antisymmetric branch becoming unstable past this critical point . once again the zoom of the bottom panel confirms the quantitative nature of the analytical - numerical agreement for small values of @xmath78 , which retains its qualitative value even for larger @xmath78 . + + finally , we briefly turn to the dynamics of the system , in order to observe the implications of the dynamical instability due to the symmetry breaking . the relevant evolution of the unstable solutions for @xmath205 and @xmath206 , in the case of @xmath141 ( recall that @xmath125 and @xmath126 ) are shown in fig . [ evolution ] . in both cases , it can be seen that the weak perturbation added on top of the exact numerical solution in the initial conditions has a projection along the unstable eigenmode . this projection , for sufficiently long times ( about @xmath207 in the left panel and about @xmath208 in the right panel ) , gets amplified and eventually leads to a visible ( i.e. , of order unity ) symmetry breaking in the profile of the state . while the space - time evolution of the density ( in the atomic case ; optical intensity in the optical case ) is shown in fig . [ evolution ] , an interesting alternative way to visualize the instability was proposed recently by @xcite . in the latter work , the pde dynamics was , in fact , projected to the phase plane of the two - mode approximation and visualized therein . an example of such a visualization for the case of @xmath205 can be seen in fig . [ pde_phase_plane ] . from both the phase plane curves and the profiles illustrated underneath of the solution at different times , we can extract some interesting conclusions . in particular , in the one degree of freedom reduction of our theoretical analysis , the trajectory occurs over iso - contours of the energy . hence , the kind of phase plane picture shown in fig . [ pde_phase_plane ] would only be possible by `` conglomerating '' many distinct orbits . however , it is important to appreciate that the pde has infinitely many degrees of freedom . in that capacity , it is possible for the `` subspace '' of our two - mode approximation to _ dissipate _ energy towards ( or possibly regain energy from ) higher energy states ( of the point spectrum of the system ) . in so doing , it appears as if the system visits further and further inward trajectories of lower energy , because indeed the excess energy has been imparted to other degrees of freedom . this yields a clear illustration of how the subspace of our two - modes is a _ closed system _ for the ode reduction , but instead is an _ open system _ for the full pde evolutionary dynamics . in the present work , we examined double well potentials in the presence of nonlocal interactions both in the cubic and in the quintic part of the nonlinearity . we attempted to address such settings by means of a two - mode decomposition that has the notable advantage that nonlocality is not substantially different to handle therein , as the nonlocal kernels merely contribute to relevant overlap integrals that need some systematic book - keeping , but are otherwise not considerably harder than is the locally nonlinear case . there are some particularly important attributes of the quintic case that we were able to extract via a normal form reduction and phase plane visualization ( under suitable circumstances of `` competition '' e.g. for a defocusing cubic but focusing quintic nonlinearity ) . one such is that contrary to the purely cubic case , the reduction is able to predict not only a symmetry breaking bifurcation , but _ also _ a symmetry restoring one ( at least for a suitable interval of range parameters for the interaction kernel ) . another unusual characteristic is that symmetry breaking bifurcations are encountered _ both _ for the symmetric and the antisymmetric branch , again differently than is the case for the cubic nonlinearity in the double well setting . these features were tested against numerical bifurcation results and good agreement was found where appropriate ( e.g. low atom numbers and a suitable range of the interaction range ) . disparities arising for high @xmath78 and large @xmath14 were systematically explained . finally , the instability dynamics was visualized not only by space - time density evolution plots but also by offering its projection to the phase plane of the double well theoretical reduction and assessing the similarities and differences therein of the ode approximation and full pde result . there are numerous possibilities for the extension of the present results to more elaborate contexts . on the one hand , even in the one - dimensional setting , one could envision a study of different interaction ranges between the cubic and quintic terms ( or , for that matter , combinations of local and nonlocal nonlinearities within the cubic and/or quintic terms ) . on the other hand , extensions to one dimensional settings with more wells would bring along a richer phenomenology ( in that setting the three - well local case has been studied @xcite and was recently revisited in @xcite ) , while in higher dimensional settings such as 2d , four well settings in a square configuration @xcite or other configurations exploiting the geometry of the system would be interesting to study . pgk gratefully acknowledges support from the national science foundation under grants dms-0806762 and cmmi-1000337 , as well as by the alexander von humboldt foundation through a research fellowship , the alexander s. onassis public benefit foundation ( grant rzg 003/2010 - 2011 ) and the binational science foundation ( grant 2010239 ) . the work of pat was partially supported by the state scholarships foundation in greece . vmr gratefully acknowledges support from research council of auth ( grant 87872 ) . this research has been co - financed by the european union ( european social fund - esf ) and greek national funds through the operational program `` education and lifelong learning '' of the national strategic reference framework ( nsrf ) - research funding program : thales . investing in knowledge society through the european social fund . the standard 1d model of the thermal optical nonlinearity is based on the following system ( see , e.g. , refs . @xcite):@xmath209where @xmath210 is the squared correlation length of the nonlocal nonlinearity , the real field @xmath211 is a local perturbation of the refraction index , and @xmath212 is the coefficient of the optical absorption which leads to heating of the medium , so that @xmath213 is the local source in the effective heat - conductivity equation ( [ m ] ) . if the heating is provided by the resonant absorption by dopants , the absorption may be saturable . the saturation may be described , in the simplest approximation , by the following modification of eq . ( [ m]):@xmath214 s. raghavan , a. smerzi , s. fantoni , and s. r. shenoy , phys . a * 59 * , 620 ( 1999 ) ; s. raghavan , a. smerzi , and v. m. kenkre , phys . rev . a * 60 * , r1787 ( 1999 ) ; a. smerzi and s. raghavan , phys . rev . a * 61 * , 063601 ( 2000 ) . c. par and m. florjaczyk , phys . a * 41 * , 6287 ( 1990 ) ; a. i. maimistov , kvant . elektron . * 18 * , 758 ( 1991 ) [ in russian ; english translation : sov . j. quantum electron . * 21 * , 687 ; w. snyder , d. j. mitchell , l. poladian , d. r. rowland , and y. chen , j. opt . b * 8 * , 2102 ( 1991 ) ; p. l. chu , b. a. malomed , and g. d. peng , j. opt . b * 10 * , 1379 ( 1993 ) ; n. akhmediev , and a. ankiewicz , phys . lett . * 70 * , 2395 ( 1993 ) ; b. a. malomed , i. skinner , p. l. chu , and g. d. peng , phys . e * 53 * , 4084 ( 1996 ) . a. griesmaier , j. werner , s. hensler , j. stuhler , and t. pfau , phys . rev . lett . * 94 * , 160401 ( 2005 ) ; j. stuhler , a. griesmaier , t. koch , m. fattori , t. pfau , s. giovanazzi , p. pedri , and l. santos , _ ibid_. * 95 * , 150406 ( 2005 ) ; j. werner , a. griesmaier , s. hensler , j. stuhler , and t. pfau , _ ibid_. * 94 * , 183201 ( 2005 ) ; a. griesmaier , j. stuhler , t. koch , m. fattori , t. pfau , and s. giovanazzi , _ ibid_. * 97 * , 250402 ( 2006 ) ; a. griesmaier , j. phys . . phys . * 40 * , r91 ( 2007 ) ; t. lahaye , t. koch , b. frhlich , m. fattori , j. metz , a. griesmaier , s. giovanazzi , and t. pfau , nature ( london ) * 448 * , 672 ( 2007 ) . t. khler , k. gral , and p. s. julienne , rev . phys . * 78 * , 1311 ( 2006 ) ; j. sage , s. sainis , t. bergeman , and d. demille , phys . lett . * 94 * , 203001 ( 2005 ) ; c. ospelkaus , l. humbert , p. ernst , k. sengstock , and k. bongs , _ ibid_. * 97 * , 120402 ( 2006 ) ; j. deiglmayr , a. grochola , m. repp , k. mrtlbauer , c. glck , j. lange , o. dulieu , r. wester , and m. weidemller , _ ibid_. * 101 * , 133004 ( 2008 ) ; f. lang , k. winkler , c. strauss , r. grimm , and j. h. denschlag , _ ibid_. * 101 * , 133005 ( 2008 ) . m. marinescu and l. you , phys . rev . lett . * 81 * , 4596 ( 1998 ) ; s. giovanazzi , d. odell , and g. kurizki , phys . rev . lett . * 88 * , 130402 ( 2002 ) ; i. e. mazets , d. h. j. odell , g. kurizki , n. davidson , and w. p. schleich , j. phys . b 37 , s155 ( 2004 ) ; r. lw , r. gati , j. stuhler and t. pfau , europhys . lett . * 71 * , 214 ( 2005 ) . w. krlikowski , o. bang , j. j. rasmussen , and j. wyller , phys . e * 64 * , 016612 ( 2001 ) ; o. bang , w. krlikowski , j. wyller and j. j. rasmussen , phys . e * 66 * , 046619 ( 2002 ) ; j. wyller , w. krlikowski , o. bang and j. j. rasmussen , phys . e * 66 * , 066615 ( 2002 ) . b. l. lawrence and g. i. stegeman . two - dimensional bright spatial solitons stable over limited intensities and ring formation in polydiacetylene para - toluene sulfonate . _ optics letters _ , * 23 * , 8 ( 1998 ) 591593 . f. smektala , c. quemard , v. couderc , and a. barthlmy , j. non - cryst . solids 274 , 232 ( 2000 ) ; g. boudebs , s. cherukulappurath , h. leblond , j. troles , f. smektala , and f. sanchez , opt . commun . 219 , 427 ( 2003 ) .
|
in the present work , we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential . employing a two - mode approximation ,
we systematically develop two cubic - quintic ordinary differential equations and assess the contributions of the long - range interactions in each of the relevant prefactors , gauging how to simplify the ensuing dynamical system .
finally , we obtain a reduced canonical description for the conjugate variables of relative population imbalance and relative phase between the two wells and proceed to a dynamical systems analysis of the resulting pair of ordinary differential equations . while in the case of cubic and quintic interactions of the same kind ( e.g. both attractive or both repulsive ) , only a symmetry breaking bifurcation can be identified , a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions
is a `` symmetry restoring '' bifurcation .
namely , in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the anti - symmetric state , there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the anti - symmetric parent one .
the relevant bifurcations , the stability of the branches and their dynamical implications are examined both in the reduced ( ode ) and in the full ( pde ) setting .
the model is argued to be of physical relevance , especially so in the context of optical thermal media .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
two outstanding mysteries in particle theory are the cause of electroweak symmetry breaking and the origin of flavor symmetry breaking by which the quarks and leptons obtain their diverse masses . the standard model of particle physics , based on the gauge group @xmath0 accommodates both symmetry breakings by including a fundamental weak doublet of scalar ( `` higgs '' ) bosons @xmath1 with potential function @xmath2 . however the standard model provides no explanation of the dynamics responsible for the generation of mass . furthermore , the scalar sector suffers from two serious problems . the scalar mass is unnaturally sensitive to the presence of physics at any higher scale @xmath3 ( e.g. the planck scale or a grand - unification scale ) : this is known as the gauge hierarchy problem . in addition , if the scalar must provide a good description of physics up to arbitrarily high scale ( i.e. , be fundamental , not composite ) , the scalar s self - coupling ( @xmath4 ) is driven to zero at finite energy scales . that is , the scalar field theory is free ( or `` trivial '' ) . then the scalar can not fill its intended role : if @xmath5 , the electroweak symmetry is not spontaneously broken . it is , thus , necessary to seek the origin of mass in physics that lies beyond the standard model . one intriguing possibility is to introduce supersymmetry @xcite . the gauge structure of the minimal supersymmetric version of the standard model ( mssm ) is identical to that of the standard model , but each ordinary fermion ( boson ) is paired with a new boson ( fermion ) called its `` superpartner '' and two higgs doublets are needed to provide mass to all the ordinary fermions . each loop of ordinary particles contributing to the higgs boson s mass is now countered by a loop of superpartners . if the masses of the ordinary particles and superpartners are sufficiently close , the gauge hierarchy can be stabilized @xcite . in addition , supersymmetry relates the scalar self - coupling to gauge couplings , so that triviality is not a concern . another interesting class of models involve dynamical electroweak symmetry breaking@xcite . in these theories , a new strong gauge interaction with @xmath6 ( e.g technicolor ) breaks the chiral symmetries of a set of massless fermions @xmath7 at a scale @xmath8tev . if the fermions carry appropriate electroweak quantum numbers , the resulting condensate @xmath9 breaks the electroweak symmetry as desired . the logarithmic running of the strong gauge coupling renders the low value of the electroweak scale ( i.e. the gauge hierarchy ) natural . the absence of fundamental scalar bosons obviates concerns about triviality . how is one to choose among the various models ? consider a rough graph of the masses of the known fermions and gauge bosons : = = = = = = .5cme 1.27 cm w z .67 cm t + + + the top quark is singled out@xcite : it is the heaviest known elementary particle , with a mass of order the electroweak scale , @xmath10gev , and is far heavier than its weak partner ( @xmath11 ) . this suggests that the top quark may afford us insight about existing models of electroweak physics and may even play a special role in electroweak and flavor symmetry breaking . the large mass of the top quark has illuminated aspects of existing theories of electroweak and flavor physics . we review some opportunities and constraints the top quark has provided for supersymmetric models and theories of dynamical electroweak symmetry breaking . a challenge for supersymmetric models is to explain why the higgs scalar develops a negative mass - squared ( so that the electroweak symmetry breaks ) while the scalar partners of the ordinary fermions do not ( so that color and electromagnetism are preserved ) . the heavy top quark provides a solution . in many types of models ( e.g. the constrained mssm ( cmssm ) , models of dynamical supersymmetry breaking ) the mass - squared of the higgs is related to that of the sfermions , and is therefore positive at scales ( @xmath12 ) well above the weak scale@xcite @xmath13 moving towards lower scales , the masses of the higgs ( @xmath14 ) and of the top squarks ( @xmath15 ) evolve under the renormalization group@xcite : @xmath16 clearly , the top quark s large yukawa coupling @xmath17 is important and the mass - squared of the higgs @xmath18 is affected more than that of the squarks . the approximate solution for the light higgs mass - squared at scale @xmath19 is @xmath20 for a top quark mass @xmath21 gev , the higgs mass - squared is driven negative near the electroweak scale @xcite , while those of the squarks are not . as desired , the electroweak symmetry breaks while color and electromagnetism survive . after electroweak symmetry breaking , the higgs spectrum of the mssm includes two neutral scalar bosons . the tree - level upper bound on the mass of the lighter one ( @xmath18 ) is @xmath22 . this would appear to forbid @xmath23 and lies quite close to the experimental lower bound on @xmath24 . radiative corrections involving the heavy top quark and its superpartners provide a resolution . for @xmath25 , the bound on @xmath14 becomes @xcite @xmath26 so that @xmath27 gev and @xmath23 ( the @xmath28 symmetric limit @xcite or `` light gaugino - higgsino window '' @xcite ) is still viable . extended technicolor ( etc ) is an explicit realization of dynamical electroweak symmetry breaking and fermion mass generation . one starts with a strong gauge group ( technicolor ) felt only by a set of new massless fermions ( technifermions ) and extends the technicolor gauge group to a larger ( etc ) group under which ordinary fermions are also charged . at a scale @xmath29 , etc breaks to its technicolor subgroup and the gauge bosons coupling ordinary fermions to technifermions acquire a mass of order @xmath29 . at a scale @xmath30 the technicolor coupling becomes strong enough to form a technifermion condensate and break the electroweak symmetry . because the massive etc gauge bosons couple the ordinary fermions to the condensate , the ordinary fermions acquire mass too . the top quark s mass , e.g. , comes from and its size is @xmath31 @xmath32 . thus the scale @xmath29 must satisfy @xmath33 tev in order to produce @xmath34 gev . several difficulties arise when one tries to balance the need to create a wide range of ordinary fermion masses against the requirement of keeping the oblique correction @xmath35 small . first are the so - called `` direct '' contributions@xcite to @xmath35 . the etc sector must violate weak isospin in order to make @xmath36 . this can induce dangerous technifermion ( @xmath37 ) contributions to @xmath35 : .5 cm @xmath38 to satisfy the experimental constraint @xmath39 , one might consider making the etc boson heavy . this requires @xmath40 , which is too heavy to produce @xmath21gev . a better alternative@xcite is to arrange for separate dynamical sectors to break the electroweak symmetry and produce the bulk of @xmath41 . then the technifermions contributing to @xmath35 can satisfy @xmath42gev . there are also `` indirect '' contributions@xcite to @xmath35 from isospin- violating .1 cm @xmath43 at low energies @xmath44 . however the @xmath45 and @xmath11 quarks must feel some additional strong interaction not shared by the light fermions or technifermions , which can generate @xmath46 as sketched in figure [ figyy ] . we will revisit these ideas later . while the contributions to @xmath35 just discussed are physically distinct from generation of @xmath41 , the very etc boson responsible for @xmath41 makes potentially large contributions to @xmath48 . consider the simplest etc models , those in which the etc and weak gauge groups commute and the etc bosons carry no weak charge . the etc boson responsible for generating the top quark mass couples to the current @xmath49 where @xmath50 is an etc clebsch , @xmath51 , and @xmath52 is a technifermion doublet . the top quark mass comes from and is of order @xmath53 . this implies @xcite that the size of the typical vertex correction arising from exchange of this etc boson is proportional to the top mass : @xmath54 . in particular , this etc boson causes a radiative correction to the @xmath47 vertex .1 cm @xmath55 we next consider whether the top quark s large mass implies that the @xmath45 quark has unique new interactions . such scenarios provide alternative mechanisms of electroweak and flavor symmetry breaking and are experimentally testable . to start , we return to extended technicolor . our discussion of @xmath48 did not rule out the possibility of `` non - commuting '' models in which @xmath56 is embedded in @xmath57 so that the etc bosons carry weak charge . to build such a model , one must balance the requirements of providing a range of quark masses against the need to ensure that the weak interactions are universal at low energies . as discussed in @xcite , this leads to separate weak interactions for the 3rd generation fermions ( @xmath58 ) and the light fermions ( @xmath59 ) and the symmetry - breaking pattern @xmath60 the result is a model where the top quark has non - standard weak interactions . our first concern is the value of @xmath48 this model predicts . the etc boson .1 cm @xmath61 not make @xmath48 compatible with experiment . but in addition , mixing of the @xmath62 .1 cm @xmath63 non - standard top quark weak interactions may be detectable in single top - quark production at tev33 @xcite . the ratio of cross - sections @xmath64 can be measured ( and calculated ) to an accuracy @xcite of at least @xmath65 . a non - commuting etc model might alter @xmath66 in several ways . mixing of the two @xmath67 bosons alters the light @xmath67 s couplings to the final state fermions . exchange of both heavy and light @xmath67 bosons contributes to the cross - sections . but exchange of the etc boson that generates @xmath41 does * not * modify the @xmath68 vertex , because the boson does not couple to all of the required fermions : ( @xmath69 ) . in the case of @xmath48 , the vertex and boson mixing effects canceled , leaving @xmath48 at the standard model value ; here the boson mixing effects are not canceled and can yield a visible * increase * in @xmath66 ( see figure [ figxx ] ) . to build a dynamical symmetry breaking model that provides both electroweak symmetry breaking and a large @xmath41 , it has been suggested @xcite that all or some of electroweak symmetry breaking could be caused by a top quark condensate ( @xmath70 ) . one way to implement this is to start with a spontaneously broken strong gauge interaction that distinguishes top from the other quarks . suppose the model includes an @xmath71 for the @xmath45 ( and @xmath11 ) and an @xmath72 for the other quarks which break to their diagonal subgroup ( identified with @xmath73 at a scale @xmath29 . at energies below @xmath29 , exchange of the heavy gauge bosons yields a new four - fermion interaction can cause top quark condensation . @xmath74 the simplest `` topcolor - assisted technicolor '' model@xcite incorporating top condensates has the following gauge group and symmetry - breaking pattern . @xmath75 the groups @xmath76 and @xmath56 are ordinary technicolor and weak interactions ; the strong and hypercharge groups labeled `` h '' couple to 3rd - generation fermions and have stronger couplings than the `` l '' groups coupling to light fermions the separate @xmath77 groups ensure that the bottom quark will not condense when the top quark does . below the scale @xmath29 , the lagrangian includes effective interactions for @xmath45 and @xmath11 : @xmath78 ^ 2 -{{4\pi \kappa_1}\over{m^2 } } \left[{1\over3}\overline{\psi_l}\gamma_\mu \psi_l + { 4\over3}\overline{t_r}\gamma_\mu t_r -{2\over3}\overline{b_r}\gamma_\mu b_r \right]^2\ ] ] so long as the following relationship is satisfied ( where the critical value is @xmath79 in the njl approximation @xcite ) @xmath80 only the top quark will condense and become very massive@xcite . topcolor - assisted technicolor models have several appealing features@xcite@xcite . technicolor causes most of the electroweak symmetry breaking , with the top condensate contributing a decay constant @xmath81 gev ; this prevents @xmath35 from being too large , as mentioned earlier . so long as the @xmath82 charges of the technifermions are isospin - symmetric , they cause no additional large contributions to @xmath35 . etc dynamics at a scale @xmath83tev generates the light fermion masses and contributes about a gev to the heavy fermions masses ; this does not generate large corrections to @xmath48 . the top condensate can , then , provide the bulk of the top quark mass . precision electroweak data constrain the mass of the extra @xmath62 boson in these models to weigh at least 1 - 2 tev @xcite . finally , we return to supersymmetric models . consider the mass matrix for the supersymmetric partners of the top quark : _ t ^2 = the presence of @xmath41 in the off - diagonal entries shows that a large top quark mass can drive one of the top squarks to be quite light . if the top squark is light enough ( which experiment allows @xcite ) the decay @xmath84 becomes possible ; that is , the top quark may be the only quark able to decay to its own superpartner . this idea can be tested in top quark production experiments . the simplest test is to see whether the measured top mass and cross - section match , since the latter depends on how the top is assumed to decay . cdf and do data indicate that assuming the top decays only through the standard channel @xmath85 gives a good fit to the data @xcite . if additional decay channels for the top existed , the production cross - section measured in the @xmath86 final states would be lower than the standard model prediction . however , this effect could be balanced @xcite in supersymmetric models either by production of states that decay to top quarks @xmath87 or by production of states whose decays mimic those of top quarks @xmath88 . checking these possibilities would require seeking , specific signatures of the presence of supersymmetric particles ; for instance , while qcd produces mainly top anti - top pairs , gluino pair production with subsequent decay to top quarks and squarks also produces top / top and anti - top / anti - top pairs in the ratio [ @xmath89 . the top quark s large mass singles it out in several ways . it may play a special role in electroweak symmetry breaking ( through its effects on rg running in supersymmetric models or through formation of top quark condensates ) . it has potentially large effects on precision electroweak observables like @xmath35 or @xmath48 . in some cases it has a strong influence on the masses of other particles such as higgs bosons or superpartners . finally , the top quark may be subject to non - standard interactions that distinguish it from the up and charm quarks . as a result , the top quark has already made a difference in our attempts to understand electroweak and flavor physics . is the top quark actually different in any of the ways outlined here ? time and experiment will tell ! a recent review is s. dawson , hep - ph/9602229 . g. anderson , d. castano , and a. riotto , _ phys.rev . _ * d55 * , 2950 ( 1997 ) ; h. murayama and m. peskin , _ ann nucl . part . a recent review is r.s . chivukula , hep - ph/9701322 . recent reviews of top quark properties and prospects include r. frey et al . , hep - ph/9704243 ; s. frixione et . al , hep - ph/9702287 . h. baer et al . _ * d54 * , 5866 ( 1996 ) and * d53 * , 6241 ( 1996 ) and * d52 * , 2746 ( 1995 ) ; m. machacek and m. vaughn , _ nucl . phys . _ * b222 * , 83 ( 1983 ) ; c. ford et al . , _ nucl . phys . _ * b395 * , 17 ( 1995 ) ; m. dine , a. nelson , and y. shirman , _ phys . rev . _ * d51 * , 1362 ( 1995 ) ; m. dine . et al . , _ phys . rev . _ * d53 * , 2658 ( 1996 ) ; j. amundson et al . , hep - ph/9609374 . l. ibanez , _ nucl . phys . _ * 218*,514 ( 1983 ) ; _ phys . lett . _ * b110 * , 215 ( 1982 ) ; j. ellis , d. nanopoulos , and k. tamvakis , _ phys . lett _ * b121 * , 123 ( 1983 ) ; l. alvarez - gaume , j. polchinski , and m. wise , _ nucl . b221 _ , 495 ( 1983 ) ; b. ananthanarayan , g. lazarides , and q. shafi , _ nucl . phys . _ * d44 * , 1613 ( 1991 ) . j. ellis , g. ridolfi and f. zwirner , _ phys . lett . _ * b257 * , 83 ( 1991 ) ; h.e . haber and r. hempfling , _ phys . rev . lett . _ * 66 * , 1815 ( 1991 ) . l.j . hall and l. randall , _ nucl . phys . _ * b352 * , 289 ( 1991 ) ; l. randall and n. rius , _ phys . lett . _ * b286 * , 299 ( 1992 ) ; n. rius and e.h . simmons , _ nucl . * b416 * , 722 ( 1994 ) ; e.h . simmons and y. su , _ phys . rev . _ * d54 * , 3580 ( 1996 ) . j.l . feng , n. polonsky , and s. thomas , _ phys.lett . _ * b370 * , 95 ( 1996 ) . t. appelquist et al . , _ phys . lett _ * 53 * , 1523 ( 1984 ) and _ phys . rev . _ * d31 * , 1676 ( 1985 ) . chivukula , s.b . selipsky and e.h . simmons , _ phys * 69 * , 575 ( 1992 ) . lep electroweak working group and sld heavy flavor group ( d. abbaneo et al . ) , `` a combination of preliminary electroweak constraints on the standard model , '' cern - ppe-96 - 183 , dec 1996 . r.s . chivukula , e.h . simmons , and j. terning , _ phys . * b331 * , 383 ( 1994 ) ; ibid . _ phys . rev . _ * d53 * , 5258 ( 1996 ) . t. stelzer and s. willenbrock , _ phys . * b357 * , 125 ( 1995 ) ; e.h . simmons , _ phys . * d55 * , 5494 ( 1997 ) . heinson , hep - ex/9605010 ; a.p . heinson , a.s . belyaev and e.e . boos , hep - ph/9612424 ; m.c . smith and s. willenbrock , _ phys . rev . _ * d54 * , 6696 ( 1996 ) . miransky , m. tanabashi and k. yamawaki , _ phys . lett . _ * b221 * , 177 ( 1989 ) and _ mod . lett . _ * a4 * , 1043 ( 1989 ) ; y. nambu , efi-89 - 08 ( 1989 ) unpublished ; w.j . marciano , _ phys . lett . _ * 62 * , 2793 ( 1989 ) ; w.a . bardeen , c.t . hill and m. lindner , _ phys . _ * d41 * , 1647 ( 1990 ) ; c.t . hill , _ phys . lett . _ * b266 * , 419 ( 1991 ) . hill , _ phys . _ * b345 * , 483 ( 1995 ) . y. nambu and g. jona - lasinio , _ phys . rev . _ * 122 * , 345 ( 1961 ) . k. lane and e. eichten , _ phys . lett . _ * b352 * , 382 ( 1995 ) ; r.s . chivukula , b.a . dobrescu , and j. terning , _ phys.lett . _ * b353 * , 289 ( 1995 ) ; g. buchalla et al . , _ phys.rev . * d53 * , 5185 ( 1996 ) . chivukula and j. terning , _ phys.lett . _ * b385 * , 209 ( 1996 ) . d0 collaboration ( s. abachi et al . ) . _ phys.rev.lett . _ * 76 * , 2222 ( 1996 ) . m. paulini ( for the cdf and d0 collaborations ) , hep - ex/9701019 . g.l . kane and s. mrenna , _ phys.rev.lett . _ * 77 * , 3502 ( 1996 ) .
|
experiment shows that the top quark is far heavier than the other elementary fermions .
this finding has stimulated research on theories of electroweak and flavor symmetry breaking that include physics beyond the standard model .
efforts to accommodate a heavy top quark within existing frameworks have revealed constraints on model - building .
other investigations have started from the premise that a large top quark mass could signal a qualitative difference between the top quark and other fermions , perhaps in the form of new interactions peculiar to the top quark .
such new dynamics may also help answer existing questions about electroweak and flavor physics .
this talk explores the implications of the heavy top quark in the context of weakly - coupled ( e.g. susy ) and strongly - coupled ( e.g. technicolor ) theories of electroweak symmetry breaking .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
classical novae are rather frequently observed in our galaxy ( liller & mayer 1987 , shafter 1997 ) , and have also been studied in external galaxies ; typically @xmath2 34 per year are detected in our galaxy ( duerbeck 1995 , warner 1995 ) . most of the discoveries and observations of galactic novae have been made by amateur astronomers with little access to spectroscopic and photometric equipment . sky coverage has been episodic and extremely hard to calculate . classification attempts have also been hindered . as a result , many of the most basic properties involving their global rate and distribution are surprisingly uncertain . for example , a number of arguments suggest that the galactic rate of novae must be much higher than @xmath3 : \(a ) the typical limiting apparent magnitude obtainable with amateur apparatus and methods has been increasing steadily in recent years , but for the period covered by this paper may be taken to be @xmath4 , within a very wide range , and with extremely uneven coverage . application of the expanding - photosphere method to a subset of relatively nearby and bright novae has yielded the empirical relation @xmath5 ( warner 1995 ) for the absolute magnitude , where @xmath6 ( the _ speed class _ ) is the time taken for @xmath7 to increase by 2 from discovery . it follows that the distance out to which amateur astronomers are detecting typical novae is @xmath8 kpc , or only about one - half the volume of the galaxy . furthermore , the rate of discoveries at the faintest magnitudes ( @xmath9 ) is greater than what would be extrapolated from brighter novae . this indicates that a new population presumably associated with the galactic bulge rather than the disk is present and poorly sampled ( duerbeck 1990 ; see below ) . \(b ) even within that part of the galaxy which is effectively searched for novae , the discovery rate is blatantly incomplete . not only does the discovery rate for novae with @xmath10 fall below the extrapolated rate for brighter events ( thus , in contrast to the preceding argument , suggesting that many events in this range are missed : duerbeck 1990 ) , but there is a marked deficiency of discoveries in the southern celestial hemisphere ( warner 1995 ) . this is relevant to our work , since the tgrs detector is permanently pointed at the southern sky ( 2.1 ) . during its period of operation ( 19951997 ) five novae were discovered in the southern hemisphere ( harris et al . 1999 , hereafter paper i ) , but there is no way of knowing how many were missed . a few days ) which rise and fall in between successive visits to a given location . ] the possibility of detecting undiscovered novae as bright as @xmath11 ( marginally within tgrs s capabilities ) is one of the justifications for the present work . \(c ) in galactic latitude , the distribution of classical novae is somewhat concentrated toward the equatorial plane ( scale heights for disk and bulge populations 125 and 500 pc respectively : duerbeck 1984 , 1990 ) . they must therefore be affected to some degree by interstellar extinction , and a deficiency of discoveries close to the plane is indeed observed ( warner 1995 ) . in terms of the composition of their ejecta , novae are classified into co - rich and one - rich ; it is thought that the distinction reflects the composition of the underlying white dwarf material , with the one class coming from more massive progenitors whose cores burned beyond the early he - burning stage which yields c and o. different levels of positron annihilation line flux are expected from each class ( 4 ) . if the progenitors of the one subclass are really more massive , they will tend to lie closer to the galactic plane , and the resulting novae will be more strongly affected by extinction and relatively under - represented in the discovered sample ( of which they compose @xmath12 : gehrz et al . evidence of this has been detected by della valle et al . ( 1992 ) . \(d ) the three preceding factors would all tend to enhance the true galactic nova rate above that observed . however , a second , quite distinct approach to the problem tends to produce systematically lower rates . in this approach , several external galaxies ( particularly the magellanic clouds , m31 and m33 ) have been monitored for novae , and their observed rates extrapolated in some fashion to the milky way ( ciardullo et al . 1987 , della valle & livio 1994 ) . the usual basis for extrapolation is absolute blue luminosity ( della valle & claudi 1990 ) . as can be seen in table 1 , the results from this approach are systematically smaller than attempts to correct for the missing galactic novae directly . the original explanation for this effect was provided by duerbeck ( 1990 ) , who postulated two different classes of event by spatial distribution disk and bulge novae . it was claimed that the bulge population has a systematically slower speed class , and is therefore generally less luminous by equations ( 13 ) , which might account for the discrepancy , given a larger bulge in the main external source of novae , m31 . as will be seen ( 4.1 ) , our search method is probably relevant only to a disk population . a third approach to the problem is theoretically possible , by which classical nova outbursts are assumed to be part of a life - cycle of which other cataclysmic variables are manifestations . the galactic nova rate is then derived from the assumed space densities of these related objects , together with some model for the outburst recurrence time ( warner 1995 ) . this approach is more reliable at predicting the galactic space density rather than the global rate , which is more directly related to the measurements we shall present . it is important to correct for and combine these various factors into an overall global galactic nova rate , which would govern the input of novae into galactic chemical evolution , dust grains and interstellar radioactivity ( gehrz et al . 1998 ) . however attempts to do so have yielded wildly discordant results , ranging from 11260 novae yr@xmath13 ( see table 1 ) . we have therefore adopted in this work yet a fourth ( and simplest ) approach which is to make an _ unbiased _ search for novae in our galaxy . the detection of @xmath14-ray lines from radioactive decays of the nucleosynthesis products produced in novae is such an approach ; these decays in general emit positrons , whose annihilation with electrons produces a line at 511 kev . an obvious advantage of this approach is the very small absorption of @xmath14-rays in the galaxy . we will also see that problems of uneven coverage and sensitivity are minimal . these advantages are realised when the @xmath14-ray detector tgrs , on board the _ wind _ mission , is used ( 2.1 ) . in paper i we determined that tgrs does indeed have the capability to perform a sky survey for classical novae . the target of paper i was to detect the positron annihilation line in five known novae ; although none was detected , the viability of such a method was established . the key to the method ( see 2 below ) is that the line arises in nova material expanding towards the observer , and is therefore broadened and blueshifted ( leising & clayton 1987 ) . its peak is therefore shifted away from a strong background line at exactly 511 kev , which arises in the instrument itself from decays of unstable nuclei produced by cosmic ray spallation . in the next section we give a brief description of the detector and data , and of our analysis . none of these is substantially different from that of paper i , where the reader may find a more detailed description . the tgrs experiment is very well suited to a search for the 511 kev line , for several reasons . first , it is located on board the _ wind _ spacecraft whose orbit is so elliptical that it has spent virtually all of its mission since 1994 november in interplanetary space , where the @xmath14-ray background level is relatively low . second , these backgrounds are not only low but very stable over time . third , tgrs is attached to the south - facing surface of the rotating cylindrical _ wind _ body , which points permanently toward the south ecliptic pole . the detector is unshielded , and tgrs therefore has an unobstructed view of the entire southern ecliptic hemisphere . taken together , these three facts make possible a continuous and complete survey of the southern sky . fourth , and most importantly , the tgrs ge detector has sufficient spectral resolution to detect a 511 kev line which is slightly doppler - shifted away from the background 511 kev line mentioned in 1 . the doppler blueshift in the nova line , for the epochs @xmath1512 hr which we consider , is predicted to be 25 kev ( hernanz 1999 , kudryashov 2000 ) , which compares with the tgrs energy resolution at 511 kev of 34 kev fwhm ( harris et al . 1998 and paper i ) . the tgrs detector is a radiatively cooled 35 @xmath16 n - type ge crystal sensitive to energies between 20 kev8 mev . since the launch of _ wind _ in 1994 november , tgrs has accumulated count rates continuously in this energy range . the few gaps in the data stream are due either to perigee passes , which are rare ( lasting @xmath17 d at several month intervals ) thanks to _ wind _ s very eccentric orbit , or to memory readouts following solar flare or @xmath14-ray burst triggers , which may last for @xmath18 hr . the data were binned in 1 kev energy bins during 24 min intervals . we searched in data covering a period of nearly three years , from 1995 january to 1997 october . in the fall of 1997 the performance of the detector began to degrade seriously , and the energy resolution became too coarse to resolve the 511 kev background and nova lines . this degradation is believed to result from crystal defects induced by accumulated cosmic ray impacts , which trap semiconductor holes and reverse the impurity charge status . a region of the crystal thus becomes undepleted and the effective area is reduced ( kurczynski et al . we terminated our search of the data when the photopeak effective area at 511 kev fell below an estimated 80% of its original value . the total live time accumulated was about @xmath19 s , which was nominally 88% of the entire interval . in fact , the distribution of live times among the 6 hr intervals was such that 41% of all intervals had the full 6 hr of live time , and almost 99% of intervals contained some live time . our analysis procedure relies heavily on the most recently theoretically - predicted properties of the 511 kev line ( hernanz et al . 1999 , kudryashov 2000 ) , mainly its light - curve , energy and shape . the timescale over which the background spectra described above are summed is set by the predicted @xmath14-ray light - curve from the `` thermonuclear flash '' which powers a nova . in this process a degenerate accreted h layer on the surface of a co or one white dwarf ignites proton capture reactions involving both accreted material and some material dredged up from the interior of the white dwarf . the timescale for this process is set by the @xmath0-decay timescales of the unstable nucleosynthesis products of rapid proton capture on c , o and ne . these unstable species fall into two groups , one having very rapid decays ( @xmath20 minutes : e.g. @xmath21n , @xmath22o , @xmath23f ) and the more slow - decaying @xmath24f ( @xmath25 min ) . the light - curve results from the convolution of these decays with the reduction of opacity to 511 kev @xmath14-rays due to envelope expansion ; it thus tends to be double - peaked at values @xmath2010100 s and @xmath2036 hr ( gmez - gomar et al . 1998 ) , with significant emission lasting for @xmath26 hr ( hernanz et al . 1999 ) . the 10100 s peak is ultimately due to the decay of the very short - lived group of isotopes , and is thus especially prominent in the co nova light - curve ( though these isotopes are essential to the energetics of both classes ) . the 36 hr peak reflects the survival of slower - decaying @xmath24f in both classes ( gmez - gomar et al . 1998 ) . with these timescales in mind , we summed the 24-min background spectra into 6 hr intervals the 4005 resulting 6-hr spectra were fit by a model ( described in paper i ) containing the strong background 511 kev line at rest , and a broadened blueshifted nova line . the energies of the nova line were fixed at the predicted values ( 516 kev after 6 hr , dropping to 513 kev after 12 hr : gmez - gomar et al . 1998 , hernanz et al . 1999 , kudryashov 2000 ) . the widths were taken to be 8 kev fwhm and the shapes to be gaussian , as in paper i ; the shapes are poorly documented in published models , but the approximation is probably reasonable at an epoch of a few hours ( leising & clayton 1987 ) . instrumental broadenings of these lines and of the background 511 kev line were very small during 19951997 ( harris et al . although our analysis is somewhat sensitive to the departure of the actual line parameters from these predictions , we believe that it should be adequate to detect lines in the parameter range appropriate for fast novae . for example , we estimate that lines with energies in the range 513522 kev are detected with @xmath27% of true amplitude , corresponding to expansion velocities 12006500 km s@xmath13 which bracket the range observed in fast novae ( warner 1995 ) . the 4005 count spectra were fit to the above model ( plus an underlying constant term ) and the line amplitudes were divided by the photopeak effective area at 511 kev . this photopeak efficiency was determined from monte carlo simulations as a function of energy and zenith angle ( seifert et al . 1997 ) , taking into account the effects of hole - trapping in the detector ( 2.1 ) ; we found that the efficiency remained extremely stable until the fall of 1997 , whereupon it rapidly fell to 80% of its value at launch . the effective area is a slowly varying function of the zenith angle of the source . to calculate the average effective area , we assumed the galactic distribution of the synthetic population of several thousand novae computed by hatano et al . ( 1997 ) , for the southern part of which the mean tgrs zenith angle is @xmath28 , corresponding to an effective area 13.6 @xmath16 . the fits were performed by the standard method of varying the model parameters to minimize the quantity @xmath29 , with errors on the parameters computed from the parameter range where @xmath29 exceeded the minimum by + 1 ( paper i ) . with a sufficiently large sample of spectra , there is a probability that a fitted line of any given amplitude may be produced by chance . we therefore imposed a rather high value of significance as the threshold above which a detection would be established . if the significances are normally distributed ( see 3 below ) then our sample size of 4005 spectra implies that a threshold level of @xmath30 yields a probability @xmath31% of a single false detection by chance ( abramowitz & stegun 1964 ) . a typical fit to a 6 hr spectrum is shown in figure 1 ( there is a more detailed discussion in paper i ) . the fits are generally acceptable , with values of @xmath29/d.o.f . close to 1 . the amplitudes of the nova lines are significantly positive in all cases ( see below ) . this arises from a significant departure of the blue wing of the 511 kev background line from the gaussian shape assumed in the fits , whose origin is unclear ( paper i ) . the full series of measurements for a nova line of fwhm 8 kev and blueshift 5 kev , ( parameters corresponding to typical predicted values after 6 hr : hernanz et al . 1999 and paper i ) is shown in figure 2 . it can be seen that the systematic positive offset mentioned above was extremely stable throughout the mission ; there are very weak linear trends on @xmath20year timescales which are almost invisible in fig . we subtracted this quasi - constant systematic value from all nova line measurements . a very similar time series was obtained for a nova line at position predicted for 12 hr after the explosion ( 513 kev ) , except that the error bars were very much larger ( see paper i , 3.4 ) . each fitted 6 hr line was combined with the following 12 hr fit in the proportions suggested by the light curve of hernanz et al . the results closely resembled those of fig . 2 after subtraction of the quasi - constant systematic , since the 12 hr lines contributed little on account of their large error bars . it is also clear from inspection of fig . 2 that there are no highly significant line amplitudes lying above the mean . we further show in figure 3 that the distribution of significant deviations from the mean is very close to normal . the variability in the error bars comes almost entirely from the variability in live times , which is small ( 2.1 ) . there is therefore a well - defined mean @xmath32 error of @xmath33 photon @xmath34 s@xmath13 ( compare results of paper i for zenith angle @xmath28 ) . the @xmath30 threshold based on this average error is shown by a dotted line in fig . the only points lying above this line are a few 6 hr periods with low live time and large errors . we therefore conclude that no previously - undetected novae were discovered by tgrs during 19951997 , in an almost unbiased search covering a live time of @xmath19 s. recent developments in the theory of nucleosynthesis in classical novae ( hernanz et al . 1999 ) have been discouraging for our purpose of a positron annihilation @xmath14-ray search , since new measurements of nuclear reaction rates have led to much lower predictions of the flux in this line after 6 and 12 hr . the discussion in paper i of the capability of constraining the global galactic nova rate using our present results was therefore over - optimistic . nevertheless , we will discuss the application of our method in general terms , so that even though important constraints can not now be derived , it may be useful for more sensitive future experiments ( e.g. _ integral _ ) or for more optimistic theoretical predictions . a formal expression for the number of novae detectable by tgrs is @xmath35 where @xmath36 is the galactic nova rate ; @xmath37 is a given ( time varying ) threshold flux for detection by tgrs ; @xmath38 is the fraction of the mass of the galaxy within tgrs s detection radius @xmath39 and @xmath40 ; @xmath41 is the total tgrs live time ; @xmath42 is the fraction of tgrs live time for which @xmath43 ; @xmath44 is the distribution of white dwarf masses in classical novae ; and @xmath45 is the predicted 511 kev line flux at 1 kpc for mass @xmath46 . the white dwarf mass distribution in novae , @xmath47 , is very poorly known . whereas field white dwarf masses appear to peak at @xmath48 and to decline in number for higher masses up to the chandrasekhar limit @xmath49 ( warner 1990 ) , the mass distribution in nova systems must be weighted towards higher masses . this is because the thermonuclear runaway occurs when the basal pressure of the material accreted onto the white dwarf exceeds some critical value . the critical pressure is proportional to the -4 power of the white dwarf radius , to the white dwarf mass , and to the accreted mass . since white dwarf radii decrease with increasing white dwarf mass , the accreted mass necessary to reach critical pressure is a strongly decreasing function of white dwarf mass . if the accretion rate from the secondary star is roughly independent of white dwarf mass , it follows that explosions on more massive white dwarfs will recur after much shorter intervals ( gehrz et al . there have not been reliable measurements of this effect , although theory indicates that the ratio of one : co novae of 1:2 is compatible with a distribution peaking at about @xmath50 ( truran & livio 1986 ) . further , the mass ranges corresponding to the co and one compositions are poorly known and may well overlap ( livio & truran 1994 ) . theoretical predictions of 511 kev line emission are only available for a few values of @xmath46 . in table 2 we show the parameters of the most recent models suitable for use in eq . ( 4 ) ( hernanz et al . 1999 ) . earlier models suggest that emission from lower - mass co white dwarf events is considerably less ( gmez - gomar et al . 1998 ) . in view of the remarks above about the one : co ratio , we will make the crude assumption that the ratio of `` low mass '' co objects to `` high mass '' co objects to one objects is 1:1:1 , where `` high mass '' co objects have the properties given in table 2 and `` low mass '' co objects are assumed to produce no 511 kev line emission at all . this eliminates the integral over _ m _ in eq . the remaining integral in eq . ( 4 ) can be approximated by the value of the integrand when @xmath37 has its mean value this follows from our result in 2.1 , 3 that the variation of live times in our sample ( and therefore of the errors in fig . 2 ) is very small . for a given model in table 2 , therefore , taking @xmath51 photon @xmath34 s@xmath13 , the problem is reduced to the computation of the fraction @xmath52 of galactic mass which lies within the radius @xmath53 kpc . as an example , let us consider the hernanz et al . ( 1999 ) model of a @xmath54 co nova from table 2 . here @xmath55 photon @xmath34 s@xmath13 , so that @xmath56 kpc . within this value of @xmath39 we determined that 0.61% of the galaxy s mass resides , according to the widely used bahcall - soneira model of the galaxy ( bahcall & soneira 1984 ) : one - half of this value ( i.e. the southern hemisphere ) gives @xmath57 . for the @xmath54 co model , eq . ( 1 ) then reduces to @xmath58 . our upper limit for @xmath59 is @xmath60 , with 63% probability ( for poisson - distributed events : gehrels 1986 ) , and the live time is @xmath19 s ( 2.1 ) , giving us a 63% upper limit on the rate of `` high - mass '' co novae of @xmath61 yr@xmath13 . this is quite close to the exact value 123 yr@xmath13 obtained by explicitly integrating eq . ( 4 ) over @xmath37 ( table 2 ) . in the same way we obtain an upper limit of 238 yr@xmath13 on the rate of novae occurring on one white dwarfs from hernanz et al.s ( 1999 ) prediction . our best result comes from the co model in table 2 , which we have assumed to be one - third of the total , from which we derive a global galactic nova rate of @xmath62 yr@xmath13 . uncertainties in this value clearly arise from uncertainties in the nova models , in the fraction of white dwarfs in nova systems of each type , in the bahcall - soneira model , and in the possibility of distinct spheroid and disk nova populations having differing rates , since our typical detection radius @xmath31 kpc includes almost none of the bahcall - soneira spheroid . since our results do not significantly constrain previous measurements of the nova rate ( table 1 ) , we do not make estimates of these errors , which will require attention from other , more sensitive experiments ( see next section ) . an attempt has been made , using the batse instrument on the _ compton _ observatory , to detect 511 kev line emission from a recent nearby nova ( v382 vel ) by a similar method to that used here and in paper i ( hernanz et al . the advantage of observing with batse over tgrs is its much larger effective area . its disadvantages are much poorer energy resolution with a nai spectrometer , and a background varying on very short ( @xmath63 min ) timescales . the sensitivities achieved are comparable to those obtained here . degradation of the ge detector ( 2.1 ) prevented tgrs from achieving comparable sensitivity on v382 vel ( harris et al . 2000 ) , so future efforts in this field will rely on batse and on the _ integral _ mission , which is scheduled for launch in 2001 september carrying a ge spectrometer ( spi ) with resolution comparable to tgrs but a much larger effective area . hernanz et al . ( 1999 ) estimated that spi could detect the model novae of table 2 out to @xmath64 kpc . however they also pointed out that the short duration of the 511 kev line emission would make it difficult for _ integral _ to slew to a candidate event . thus the detection rate would be limited to novae within the spi field of view , which is @xmath65 fwhm . the search method which we have used , i.e. an _ ex post facto _ search in background spectra , ought to be perfectly feasible with _ integral_. the chief requirements for this method are very high energy resolution and a sufficiently low and stable background . while the spi detector has excellent resolution , the background level in it has not yet been rigorously computed . nevertheless , qualitative arguments suggest that the background will be no worse than that in tgrs . like _ wind _ , _ integral _ will be in a high - altitude elliptical orbit which avoids extensive exposure to earth s trapped radiation belts and to albedo @xmath14-rays from earth s atmosphere . the main disadvantages of _ integral _ for nova detection are the small spi field of view and the planned observing strategy which cuts down the amount of time spent pointing towards the main concentration of novae near the galactic center . we can make use of the planned program of integral observations of the central galactic radian in the first year of operation ( winkler et al . 1999 ) to estimate the rate at which novae might be detected in the spi data . as previously , we assume that novae follow the bahcall & soneira ( 1984 ) galactic distribution . the planned first - year _ integral _ observations may be approximated by a @xmath66 grid with @xmath67 spacing between @xmath68 and @xmath69 , the exposure to each point being 1180 s per pass , with 12 passes per year covering the whole grid . thus the live time for the whole grid is 0.153 yr . from the bahcall - soneira model , the pointing geometry , and the spi aperture @xmath65 we calculated that the _ integral _ detection radius @xmath64 kpc intercepts @xmath200.75% of the galactic nova distribution . the live time 0.153 yr is then multiplied by a typical galactic nova rate @xmath70 yr@xmath13 , ( of which 2/3 are practically detectable , as assumed in 4.1 ) , and by the intercepted fraction , to imply that _ integral _ ought to detect 0.04 novae yr@xmath13 . unless theoretical estimates of the 511 kev line flux turn out to be considerably larger , the prospects for such a detection appear to be small . the same conclusion probably applies to a different method of detecting 511 kev line emission indirectly , by observing the 170470 kev continuum produced by compton scattering in the nova envelope using spi s large - area csi shield ( jean et al . 1999 ) . we are grateful to m. hernanz and a. kudryahov for helpful discussions and for providing pre - publication results , and to j. jordi ( the referee ) for constructive comments . peter kurczynski ( university of maryland ) helped in assessing the instrument performance . theresa sheets ( lhea ) and sandhia bansal ( hstx ) assisted with the analysis software . abramowitz , m. , & stegun , i. a. 1964 , handbook of mathematical functions ( washington , dc : nbs ) allen , c. w. 1954 , mnras , 114 , 387 bahcall , j. n. & soneira , r. m. 1984 , apjs , 55 , 67 ciardullo , r. , ford , h. , neill , j. d. , jacoby , g. h. , & shafter , a. w. 1987 , apj , 318 , 520 della valle , m. , & claudi , r. 1990 , in physics of classical novae , ed . a. cassatelli & r. viotti , springer lecture notes in physics 369 ( springer - verlag , berlin ) , 53 della valle , m. , & livio , m. 1994 , a&a , 286 , 786 della valle , m. , bianchini , a. , livio , m. , & orio , m. 1992 , a&a , 266 , 232 duerbeck , h. w. 1984 , ap&ss , 99 , 363 duerbeck , h. w. 1990 , in physics of classical novae , ed . a. cassatelli & r. viotti , springer lecture notes in physics 369 ( springer - verlag , berlin ) , 34 duerbeck , h. w. 1995 , in cataclysmic variables , ed . a. bianchini , m. della valle , & m. orio , ( kluwer , dordrecht ) , 39 gehrels , n. 1986 , apj , 303 , 336 gehrz , r. d. , truran , j. w. , williams , r. e. , & starrfield , s. 1998 , pasp , 110 , 3 gmez - gomar , j. , hernanz , m. , jos , j. & isern , j. 1998 , mnras , 296 , 913 harris , m. j. , teegarden , b. j. , cline , t. l. , gehrels , n. , palmer , d. m. , ramaty , r. , & seifert , h. 1998 , apj , 501 , l55 harris , m. j. , naya , j. e. , teegarden , b. j. , cline , t. l. , gehrels , n. , palmer , d. m. , ramaty , r. , & seifert , h. 1999 , apj , 522 , 424 ( paper i ) harris , m. j. , palmer , d. m. , naya , j. e. , teegarden , b. j. , cline , t. l. , gehrels , n. , ramaty , r. , & seifert , h. 2000 , in proc . fifth compton symposium , ed . m. mcconnell , in press hatano , k. , branch , d. , fisher , a. , & starrfield , s. 1997 , mnras , 290 , 113 hernanz , m. , jos , j. , coc , a. , gmez - gomar , j. , & isern , j. 1999 , apj , 526 , l97 hernanz , m. , smith , d. m. , fishman , g. j. , harmon , b. a. , gmez - gomar , j. , jos , j. , isern , j. , & jean , p. 2000 , in proc . fifth compton symposium , ed . m. mcconnell , in press higdon , j. c. , & fowler , w. a. 1987 , apj , 317 , 710 jean , p. , et al . 1999 , lett . & comm . , 38 , 421 kudryashov , a. d. 2000 , in proc . fifth compton symposium , ed . m. mcconnell , in press kurczynski , p. , et al . 1999 , methods phys . res . a , 431 , 141 leising , m. d. , & clayton , d. d. 1987 , apj , 323 , 159 liller , w. , & mayer , b. 1987 , pasp , 99 , 606 livio , m. , & truran , j. w. 1994 , apj , 425 , 797 seifert , h. , naya , j. e. , sturner , s. j. , & teegarden , b. j. 1997 in aip conf . 410 , proc . fourth compton symposium , ed . c. d. dermer , m. s. strickland , & j. d. kurfess ( new york : aip ) , 1567 shafter , a. w. 1997 , 487 , 226 sharov , a. s. 1972 , sov . astr . , 16 , 41 truran , j. w. , & livio , m. 1986 , apj , 308 , 721 van den bergh , s. 1988 , comments astrophys . , 12 , 131 warner , b. 1989 , in classical novae , ed . m. f. bode & a. evans ( wiley : new york ) , 1 warner , b. 1990 , in physics of classical novae , ed . a. cassatelli & r. viotti , springer lecture notes in physics 369 ( springer - verlag , berlin ) , 24 warner , b. 1995 , cataclysmic variable stars ( cup : cambridge ) winkler , c. , gehrels , n. , lund , n. , schnfelder , v. , & ubertini , p. 1999 , astrophys . lett . & comm . , 39 , 361 lcc method & rate yr@xmath13 & reference + m31 , m33 , lmc comparison & @xmath71 & ( 1 ) + extrapolate from known nearby novae & @xmath72 & ( 2 ) + correct incompleteness & extinction & @xmath73 & ( 3 ) + correct incompleteness & extinction & @xmath74 & ( 4 ) + correct incompleteness & extinction & 260 & ( 5 ) + monte carlo simulation & @xmath75 & ( 6 ) + m31 comparison & 46 & ( 7 ) + extrapolate luminosity function & @xmath76 & ( 8) + external galaxies comparison & 1146 & ( 9 ) + m31 comparison & @xmath77 & ( 10 ) + _ references _ + 1 . dellavalle & livio 1994 . warner 1989 . liller & mayer 1987 . shafter 1997 . sharov 1972 . hatano et al . higdon & fowler 1987 . allen 1954 . ciardullo et al . van den bergh 1988 . present work . + lcc model & co & one + white dwarf mass , @xmath78 & 1.25 & 1.15 + line flux at 1 kpc , & @xmath79 & @xmath80 + photon @xmath34 s@xmath13 & & + detection radius @xmath39 , kpc & 0.9 & 0.7 + galactic rate , yr@xmath13 & @xmath81 & @xmath82 +
|
the good energy resolution ( 34 kev fwhm ) of the transient gamma ray spectrometer ( tgrs ) on board the _ wind _ spacecraft makes it sensitive to doppler - shifted outbursts of 511 kev electron - positron annihilation radiation , the reason being that the doppler shift causes the cosmic line to be slightly offset from a strong instrumental background 511 kev line at rest , which is ubiquitous in space environments .
such a cosmic line ( blueshifted ) is predicted to arise in classical novae due to the annihilation of positrons from @xmath0-decay on a timescale of a few hours in an expanding envelope . a further advantage of tgrs its broad field of view , containing the entire southern ecliptic hemisphere has enabled us to make a virtually complete and unbiased 3year search for classical novae at distances up to @xmath1 kpc .
we present negative results of this search , and estimate its implications for the highly - uncertain galactic classical nova rate and for future space missions .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
nearly 30 years ago , toomre & toomre ( 1972 ) elegantly demonstrated that the tails and bridges emanating from many peculiar galaxies may arise kinematically from dynamically cold disk material torn off of the outer regions of galaxies experiencing strong gravitational interactions . early spectroscopic studies of gas within the tidal tails of merging galaxies provided observational support for this hypothesis by showing the tails to have the kinematics expected for a gravitational origin ( e.g. stockton 1974a , b ) . h@xmath0i mapping studies are particularly well suited to such studies , as the tidally ejected disk material is usually rich in neutral hydrogen and can be traced to very large distances from the merging systems ( e.g. van der hulst 1979 ; simkin _ et al . _ 1986 ; appleton _ et al . _ 1981 , 1987 ; yun _ et al . _ 1994 ) . once mapped , the tidal kinematics can be used either alone , to disentangle the approximate spin geometry of the encounter ( stockton 1974a , b ; mihos _ et al . _ 1993 ; hibbard & van gorkom 1996 , hereafter hvg96 ; mihos & bothun 1998 ) , or in concert with detailed numerical models , to constrain the full encounter geometry ( e.g. combes 1978 ; combes _ et al . _ 1988 ; yun 1992 , 1997 ; hibbard & mihos 1995 ; gardiner & noguchi 1996 ) . however , not all systems can be easily explained by purely gravitational models such as those used by toomre & toomre . for example , gravitational forces by themselves should not lead to differences between stellar and gaseous tidal components . numerical models which include hydrodynamical effects do predict a decoupling of the dissipative gaseous and non - dissipative stellar components ( e.g. noguchi 1988 ; barnes & hernquist 1991 , 1996 ; weil & hernquist 1993 ; mihos & hernquist 1996 ; appleton , charmandaris & struck 1996 ; struck 1997 ) , but only in the inner regions or along bridges where gas orbits may physically intersect ( see e.g. fig . 4 of mihos & hernquist 1996 ) . decoupling of the gaseous and stellar components within the tidal tails is not expected . nonetheless , differences between the optical and gaseous tidal morphologies have been observed . these differences can be subtle , with the peak optical and h@xmath0i surface brightnesses simply displaced by a few kpc within the tails ( e.g. ngc 4747 , wevers _ et al . _ 1984 ; ngc 2782 smith 1994 ; ngc 7714/4 smith _ et al . _ 1997 ; arp 295a , ngc 4676b , and ngc 520 southern tail , hibbard 1995 , hvg96 ) , or they can be extreme , with extensive h@xmath0i tidal features apparently decoupled from , or even anti - correlated with , the optical tidal features . it is this latter category of objects that we wish to address in this paper . in particular , we address the morphology of the tidal gas and starlight in the merging systems ngc 520 ( arp 157 ) , arp 220 , and arp 299 ( ngc 3690 ) . the three systems were observed as part of our on - going studies on the tidal morphologies of optically and ir selected mergers ( hibbard 1995 , hvg96 , hibbard & yun 1996 and in prep . ) . these studies involve moderate resolution ( @xmath3 ) vla h@xmath0ispectral - line mapping observations and deep optical @xmath4 and @xmath5broad - band imaging with large format ccds using the kpno 0.9 m ( ngc 520 ) and the university of hawaii 88 telescopes . the h@xmath0i and optical observations , reduction , and data products have been presented in hibbard ( 1995 ) and hvg96 for ngc 520 , in hibbard & yun ( 1999 , hereafter hy99 ) for arp 299 , and in yun & hibbard ( 2000 ; see also hibbard & yun 1996 ) for arp 220 . we refer the reader to these papers for details of the observations and data reduction . these systems are extremely disturbed , and we can not hope to offer a full description of their peculiarities here . for more information we refer the reader to the above references . figures [ fig : n520mos][fig : a220mos ] show the optical and atomic gas morphologies of each of the three systems discussed here . for ngc 520 and arp 220 only the inner regions are shown in order to highlight the differences we wish to address . panel * ( a ) * presents a greyscale representation of the optical morphology of each system with features of interest labeled . panel * ( b ) * shows the h@xmath0i distribution . contours indicate the distribution of h@xmath0imapped at low - resolution ( @xmath6 ) , whereas the greyscales show the h@xmath0i mapped at higher resolution ( @xmath3 ) . the former is sensitive to diffuse low column density ( @xmath7 ) neutral hydrogen , while the latter delineates the distribution of the higher column density h@xmath0i . the central region of each h@xmath0i map appears to have a hole ( indicated by the dotted contours ) , which is due to h@xmath0i absorption against the radio continuum associated with the disk - wide starbursts taking place in each galaxy ( see condon _ et al . _ 1990 ) . in panel * ( c ) * , we again present the optical morphology in greyscales , and the higher resolution h@xmath0i distribution as contours . finally , panel * ( d ) * presents a smoothed , star - subtracted @xmath5-band image contoured upon a greyscale representation of the high - resolution h@xmath0i map . in the final panels of figs . [ fig : n520mos][fig : a220mos ] dashed lines labeled `` slice '' indicate the locations from which h@xmath0iand optical intensity profiles have been extracted ; these profiles are plotted in figure [ fig : slices ] . arrows labeled superwind " indicate the position angle ( p.a . ) of h@xmath8 or soft x - ray plumes , believed to arise from a starburst - driven outflow or galactic superwind in each system . such outflows are common in other ir bright starbursts ( e.g. heckman , armus & miley 1987 , 1990 hereafter ham90 ; armus , heckman , & miley 1990 ; lehnert & heckman 1996 ) , and are thought to arise when the mechanical energy from massive stars and supernovae in the central starburst is sufficient to drive the dense interstellar medium outward along the minor axis ( e.g. chevalier & clegg 1985 ; joseph & wright 1985 ; suchkov _ et al . _ 1994 ) . often , such starbursts are powerful enough to drive a freely expanding wind of hot plasma completely out of the galaxy ( blowout " ; ham90 ) . in the following subsections we briefly discuss what is known about the dynamical state of each system , and describe the differences between the stellar and gaseous tidal morphologies . throughout this paper distances and other physical properties are calculated assuming @xmath9 . ngc 520 ( arp 157 , ugc 966 ) is an intermediate - stage merger , with the two progenitor nuclei separated by 40 ( 5.8 kpc for @xmath10= 30 mpc , 1 = 145 pc ) and embedded within a common luminous envelope ( hvg96 and references therein ; see fig . [ fig : n520mos]a ) . there is a bright optical tidal tail stretching 24 kpc to the southeast ( henceforth referred to as the s tail ) which bends sharply eastward and connects onto a broad optical plume . this plume continues to the north and west for 60 kpc before it appears to connect onto extended light surrounding the dwarf galaxy ugc 957 ( outside the region plotted in fig . [ fig : n520mos]a ; see stockton & bertola 1980 ) . the primary nucleus ( the easternmost nucleus in fig . [ fig : n520mos]a ) possesses a massive ( @xmath11 ) 1 kpc - scale rotating molecular gas disk ( sanders _ et al . _ 1988 ; yun & hibbard 1999 ) . an h@xmath0idisk is kinematically centered on this molecular disk , and extends to a radius of @xmath12 20 kpc ( labeled `` inner disk '' in fig . [ fig : n520mos]b ) . beyond this there is an intermediate ring of h@xmath0i with a mean radius of @xmath12 30 kpc ( i.e. , the material which contains the feature labeled `` n clump '' in fig . [ fig : n520mos]b ) , and a nearly complete outer ring of h@xmath0i with a mean radius of 60 kpc , which extends smoothly through the dwarf galaxy ucg 957 ( only partially seen in fig . [ fig : n520mos]b ) . there is a kinematic and morphological continuity between the molecular gas disk , the inner h@xmath0i disk , and outer h@xmath0i ring ( yun & hibbard 1999 ) , which suggests that all of this material is associated with the primary nucleus . the observations suggests that the ngc 520 interaction involved a prograde - retrograde or prograde - polar spin geometry : the linear morphology of the optical tail - to - plume system is typical of features produced by a disk experiencing a prograde encounter ( i.e. , the disk rotates in the same direction as the merging systems orbit each other ) . the disk - like morphology and rotational kinematics of the large - scale hi and the lack of any aligned linear tidal features , on the other hand , are more typical of polar or retrograde encounter geometries ( i.e. , disk rotation either perpendicular to or opposite the direction of orbital motion ) . such encounters fail to raise significant tails ( toomre & toomre 1972 ; barnes 1988 ) , and much of the disk material remains close to its original rotational plane . neither the intermediate nor the outer h@xmath0i ring has an optical counterpart ( @xmath13@xmath14 27 mag arcsec@xmath15 ) . despite smooth rotational kinematics , this outer h@xmath0i has a very clumpy and irregular morphology , with notable gaps near the optical minor axis ( labeled `` ne gap '' and `` sw gap '' in fig . [ fig : n520mos]c ) . this figure shows that the outer h@xmath0i and optical structures are anti - correlated , with the peak h@xmath0i column densities ( associated with the n clump ) located to one side of the optical plume . in fig . [ fig : n520mos]d the h@xmath0i clump appears to be bounded on three sides by the optical contours . in fig . [ fig : slices]a we present an intensity profile at the location indicated by the dotted line in fig . [ fig : n520mos]d , showing that the gas column density increases precisely where the optical light decreases . while the h@xmath0i features exhibit a clear rotational kinematic signature , the well - defined edges and nearly - linear structure of the optical plume suggests that its constituent stars are moving predominantly along the plume , rather than in the plane of the sky : any substantial differential rotation would increase the width of the plume and result in a more disk - like morphology . we therefore conclude that the gas rings and optical plume are both morphologically and kinematically distinct entities . this suggests that the observed gas / star anti - correlation is either transient ( and fortuitous ) or actively maintained by some process . a deep h@xmath8 image of ngc 520 shows plumes of ionized gas emerging both north and south along the minor axis and reaching a projected height of 3 kpc from the nucleus ( hvg96 ) . it has been suggested that this plume represents a starburst - driven outflow of ionized gas ( hvg96 , norman _ et al . _ the position angle of this plume is indicated by an arrow in fig . [ fig : slices]d ( p.a . this direction corresponds to the most dramatic h@xmath0i / optical anti - correlations mentioned above , and in the following we suggest that this region of the optical tail actually lies directly in the path of the out - flowing wind . arp 299 ( ngc 3690/ic 694 , ugc 6471/2 , mrk 171 , vv 118 ) is also an intermediate stage merger , with two disk systems ( ic 694 to the east , ngc 3690 to the west see figure [ fig : a299mos]a ) in close contact but with their respective nuclei separated by 20 ( 4.7 kpc for @xmath10= 48 mpc , 1 = 233 pc ) . a long , narrow , faint ( @xmath13@xmath16 26mag arcsec@xmath15 ) tidal tail stretches to the north to a radius of @xmath12 125 kpc . h@xmath0i imaging of this system by hy99 ( see also nordgren et al . 1997 ) shows a rotating gas - rich disk within the inner regions , and a pair of parallel h@xmath0i filaments extending to the north . from the h@xmath0i morphology and kinematics , hy99 deduce that arp 299 is the result of a prograde - retrograde or prograde - polar encounter between two late - type spirals , with the inner h@xmath0i disk associated with the retrograde disk of ic 694 , and the northern optical tail and tidal h@xmath0ifilaments ejected by the prograde disk of ngc 3690 . the parallel - filament or bifurcated morphology of the tidal h@xmath0i is quite unlike that of the optical tail . the inner h@xmath0i filament ( so labeled in fig . [ fig : a299mos]b ) is of lower characteristic column density ( @xmath7@xmath17 8) and is associated with the low surface brightness stellar tail ( fig . [ fig : a299mos]c ) . the gas in this filament has a more irregular morphology than that in the outer filament ( e.g. , the `` gap '' and `` knot '' in fig . [ fig : a299mos]b ) , and much of this material is detectable only after a substantial smoothing of the data ( fig . [ fig : a299mos]b ) . the outer filament is characterized by a higher h@xmath0i column density ( @[email protected] ) but has no optical counterpart ( @xmath13@xmath14 27.5 mag arcsec@xmath15 ) . this filament is displaced by approximately 20 kpc ( in projection ) to the west of the inner filament for much its length , after which the filaments merge together in a feature labeled the `` n clump '' in fig . [ fig : a299mos]b . the parallel filaments have nearly identical kinematics along their entire lengths , and join smoothly at the n clump . this implies that these features form a single physical structure . based on preliminary numerical simulations , hy99 suggest that a bifurcated morphology can arise quite naturally during tail formation . this occurs when the optically faint , gas - rich outer regions of the progenitor disk are projected adjacent to optically brighter regions coming from smaller initial radii ( see also mihos , 2000 ) . however , this scenario does not explain why the inner filament , presumably drawn from optically bright but still gas - rich material within the optical disk of the progenitor , should lack accompanying h@xmath0i . as in ngc 520 , there is an anti - correlation between the optical and gaseous column densities across the n clump , with the highest gas column densities ( 23 ) located on either side of the optical tail . this is illustrated in fig . [ fig : slices]b , where we plot a profile along the position indicated by the dotted line in fig . [ fig : a299mos]d . the optical tail emerges above the n clump , and appears to curve exactly around the northern edge of the n clump ( labeled `` hook '' in fig . [ fig : a299mos]c ) . also labeled in fig . [ fig : a299mos]c are the three regions with anomalously high h@xmath0ivelocity dispersions noted by hy99 ( @xmath18 1320 km s@xmath19 compared with @xmath20 710 km s@xmath19 for the remainder of the tail ; see fig . 7d of hy99 ) ; we will will refer to these regions in the discussion ( [ sec : rpswind ] ) . within the main body of arp 299 , vigorous star formation is taking place , with an inferred star formation rate ( sfr ) of 50 @xmath21 ( ham90 ) . recent x - ray observations reported by heckman _ _ ( 1999 ) show evidence for hot gas emerging from the inner regions and reaching 25 kpc to the north , which the authors interpret as evidence for a hot , expanding superwind . the position angle of this feature ( p.a.=25 ) is indicated by the arrow in fig . [ fig : a299mos]d , and points towards the inner tidal filament and n clump . arp 220 ( ugc 9913 , ic 4453/4 ) is the prototypical ultraluminous infrared galaxy with @xmath22 ( soifer et al . 1984 ) . it is an advanced merger system with two radio and infrared nuclei separated by 0.9 ( 345 pc for @xmath10= 79 mpc ) , and a bright optical plume extending 35 kpc to the nw ( fig . [ fig : a220mos]a ) . each of the two nuclei has its own compact molecular disk . the two nuclear disks are in turn embedded in one larger 1 kpc scale molecular gas disk ( see scoville , yun , & bryant 1997 , downes & solomon 1998 , sakamoto _ et al . _ 1999 and references therein ) . the spin axis of the eastern nucleus is aligned with that of the kpc - scale disk while the western nucleus rotates in the opposite direction . these observations suggest that arp 220 is the product of a prograde - retrograde merger of two gas rich spiral galaxies ( scoville , yun , & bryant 1997 ) . an irregular disk - like distribution of neutral hydrogen extends over a 100 kpc diameter region surrounding the optical galaxy ( yun & hibbard 1999a ) . the overall h@xmath0i kinematics indicates that this material has a component of rotation in the same sense as that for the eastern nucleus and the molecular gas disk , and opposite the rotation of the western nucleus . this suggests that the h@xmath0i disk and eastern nucleus originated from the retrograde progenitor , while the the western nucleus and nw optical plume ( fig . [ fig : a220mos]a ) arose from the prograde progenitor . because of the vigorous star formation occurring within arp 220 ( sfr=340 @xmath21 , ham90 ) , much of the h@xmath0i within the optical body of the system is seen only in absorption against the bright radio continuum emission from the central starburst . beyond this , the h@xmath0i has high column densities ( @[email protected] ) , but only to the ne and sw . most notably , there are local h@xmath0i minima to the nw and se ( see gaps in fig . [ fig : a220mos]b ) . comparison of the h@xmath0i map with the optical image ( fig . [ fig : a220mos]c ) shows that the nw gap occurs exactly at the location of the optical tail . the relationship between the optical and h@xmath0i surface brightness levels across this feature are illustrated by an intensity profile measured along the dotted line shown in fig . [ fig : a220mos]d , and plotted in fig . [ fig : slices]c . as in ngc 520 and arp 299 , the gas column density increases precisely where the optical light from the tail begins to fall off . there is a similar h@xmath0i gap to the se , but in this case there is no corresponding optical feature associated with it . at even larger radii , the h@xmath0i is more diffuse ( @xmath7@xmath123 ) and has no optical counterpart down to @xmath13=27 mag arcsec@xmath15 . an x - ray image obtained with the rosat hri camera ( heckman _ et al . _ 1996 ) reveals an extended central source that is elongated along p.a.=135(indicated by arrows in fig . [ fig : a220mos ] ) . a deep h@xmath8 + [ n@xmath0ii ] image of arp 220 reveals ionized gas with a bright linear morphology at this same position angle ( heckman , armus & miley 1987 ) . the optical emission line kinematics are suggestive of a bipolar outflow ( ham90 ) , and the physical properties of the warm and hot gas strongly support the superwind scenario for this emission ( ham90 , heckman _ et al . _ 1996 ) . as in ngc 520 and arp 299 , the position angle of the putative expanding superwind is in the same direction as the h@xmath0i minima , i.e. nw and se . figures [ fig : n520mos][fig : slices ] provide evidence for both small- and large - scale differences in the distributions of the tidal gas and stars in these three systems . the small - scale differences are of the type illustrated in fig . [ fig : slices ] , whereby the gas column density falls off just as the optical surface brightness increases at various edges of the tidal features . in ngc 520 and arp 220 , the large - scale differences are between the outer h@xmath0i rings and disks ( which have no associated starlight ) and the optical tails and plumes ( which have no associated h@xmath0i ) . although these features are kinematically decoupled at present ( with the gas rings and disks predominantly in rotation and the optical tails and plumes predominantly in expansion ) , it is possible that they had a common origin and have subsequently decoupled and evolved separately . in arp 299 , on the other hand , the h@xmath0i filaments and optical tail have similar morphologies and continuous kinematics and are therefore part of the same kinematic structure . in this system we believe the bifurcated tidal morphology results from a progenitor with a warped gaseous disk ( [ sec : a299 ] & hy99 ) , and we seek to understand why the inner filament is gas - poor , given that its progenitor was obviously gas - rich . in this section we investigate a number of possible explanations for these observations . in particular , we discuss the possible role played by : differences in the initial radial distribution of the gas and stars ( [ sec : radial ] ) , dust obscuration ( [ sec : dust ] ) , kinematic decoupling of the gas due to collisions within the developing tidal tail ( [ sec : coll ] ) , ram pressure stripping of the gas , either by a halo or by a galactic scale wind ( [ sec : rps ] ) , and photoionization of the gas , either by the starburst or by local sources ( [ sec : ioniz ] ) . in interacting systems the h@xmath0i is often more widely distributed than the optical light ( see , e.g. the h@xmath0i map of the m81 system by yun _ et al . _ 1994 ; see also van der hulst 1979 ; appleton , davies & stephenson 1981 ) . these gas - rich extensions frequently have no associated starlight down to very faint limits ( e.g. simkin _ et al . _ 1986 ; hvg96 ) . a natural explanation is that such features arise from the h@xmath0i - rich but optically faint outer radii of the progenitor disks . the relatively short lifetimes of luminous stars and the larger velocity dispersions of less luminous stars , especially with respect to the gas , will further dilute the luminous content of this material , and the h@xmath0i - to - light ratio of the resulting tidal features will increase with time ( hibbard _ et al . _ gaseous tidal extensions with very little detectable starlight would seem to be the natural consequence . the outer h@xmath0i rings in ngc 520 and arp 220 and the gas - rich outer filament in arp 299 are all likely to have arisen in this manner . however , gas - rich outer disks can not give rise to gas - poor optical structures , such as the optical plume in ngc 520 , the optical tail in arp 220 , or the inner filament in arp 299 . since these features presumably arise from optically brighter regions of the progenitor disks ( regions which are characterized by h@xmath0i column densities higher than that of the outer disks ) one would have expected a priori that these features should also be gas - rich . it is possible that the disks which gave rise to the plumes in ngc 520 and arp 220 were gas - poor at all radii . however , this would not account for the discontinuities in the outer gaseous features that project near these optical features ( i.e. , ne gap in ngc 520 and nw gap in arp 220 ) . we therefore seek other explanations for these structures . the correspondence between rising gas column density and falling optical surface brightness ( fig . [ fig : slices ] ) suggests that dust associated with the cold gas may attenuate the optical light . to address this possibility , we calculate the expected extinction in the @xmath23band for a given column density of h@xmath0i . we adopt the milky way dust - to - gas ratio determined by bohlin , savage , & drake ( 1978 ; @xmath24= 4.8 mag@xmath19 ) , which is supported by direct imaging of the cold dust in the outer regions of eight disk galaxies ( alton _ et al . _ this is combined with the galactic extinction law of odonnell ( 1994 ; @xmath25 , from table 6 of schlegel _ et al._1998 ) to yield an expected extinction in the @xmath5-band of @xmath26 mag . from fig . [ fig : slices ] , the peak h@xmath0i column densities on either side of the optical features are @xmath123 . the predicted extinction is therefore of order 0.2 mag in the @xmath5-band . from fig . [ fig : slices ] we see that the mean light level drops by about 1.0 mag arcsec@xmath15 for arp 299 ( from 26.5 mag arcsec@xmath15 to below 27.5 mag arcsec@xmath15 ) , about 1.5 mag arcsec@xmath15 for ngc 520 ( from 25 mag arcsec@xmath15 to below 26.5 mag arcsec@xmath15 ) , and by about 2.5 mag arcsec@xmath15 for arp 220 ( from 23.5 mag arcsec@xmath15 to below 26 mag arcsec@xmath15 ) along the extracted slices . to produce this amount of extinction , the tidal gas would have to have a dust - to - gas ratio that is ten times that in the milky way . the above analysis assumes that the measured neutral gas column density represents the total gas column density . however , the sharp drop in h@xmath0i column density observed in many tidal features ( hvg96 , hibbard & yun in preparation ) suggests that the tidal gas may be highly ionized by the intergalactic uv field ( see also references in [ sec : ioniz ] ) . since large dust grains should survive in the presence of this ionizing radiation , the opacity per atom of neutral hydrogen ( @xmath27 ) should increase in regions of increasing ionization fraction . observations of ngc 5018 ( hilker & kissler - patig 1996 ) , in which blue globular clusters are absent in a region underlying an associated h@xmath0i tidal stream , may support a high @xmath27 ratio for tidal gas . nevertheless , the lack of obvious reddening of the @xmath28 colors along the slices in arp 299 and ngc 520 ( hibbard 1995 ; hy99 ) argues against a much higher extinction in these regions . we conclude that extinction might be important for shaping the morphology of the faintest optical features ( e.g. , the `` hook '' and the end of the optical tail of arp 299 , fig . [ fig : a299mos ] , which has @xmath13 near the detection limit of 28 mag arcsec@xmath15 ) , but is insufficient to greatly affect the overall tidal morphology . however , an anomalously high tidal dust - to - gas ratio remains a possibility . this question could be resolved by the direct detection of cold dust in tidal tails with sub - millimeter imaging . during the tail formation process , the leading - edge of the tail is decelerated with respect to the center of mass of the progenitors , while the trailing - edge is accelerated , and the two edges move towards each other ( see toomre & toomre 1972 , fig . eventually , the two edges appear to cross , forming a caustic ( wallin 1990 ; struck - marcell 1990 ) . in most cases , the caustics are simply due to projection effects . only for low - inclination encounters will these crossings correspond to physical density enhancements , and numerical experiments suggest that in these cases the density will increase by factors of a few ( wallin 1990 ) . it has been suggested that collisions experienced by the crossing tidal streams in such low - inclination encounters may lead to a separation between the dissipational ( gas ) and non - dissipational ( stellar ) tidal components ( wevers 1984 ; smith _ et al . _ 1997 ) . the present data do not allow us to directly address this question , since the kinematic decoupling presumably took place long ago . however , several arguments lead us to suspect that this collisional process is not important in tidal tails : ( 1 ) large scale decoupling between the stellar and gaseous tidal morphologies is not seen in many systems known to have experienced low - inclination encounters ( e.g. ngc 4038/9 , `` the antennae '' , hibbard _ et al . _ in preparation ; ngc 7252 `` atoms for peace '' hibbard _ et al . _ 1994 ; ngc 4676 `` the mice '' hvg96 ) ; ( 2 ) the broad plume - like morphologies of the optical features in arp 220 and ngc 520 suggest rather inclined encounters ( [ sec : n520 ] , [ sec : a220 ] ) , in which case wide - spread collisions are not expected ; and ( 3 ) the parallel filaments in the arp 299 tail have identical kinematics , whereas one would expect kinematic differences between the stripped and unstripped material . therefore while gaseous collisions and dissipation might result in differences between gas and stars during tidal development ( particularly along tidal bridges , where the gas streamlines are converging ; e.g. struck 1997 ; ngc 7714/5 smith _ et al . _ 1997 ; arp 295 hvg96 ) , we believe that they are not likely to lead to a wide - spread decoupling in the outer regions . if the tidal features pass through a diffuse warm or hot medium , or if such a medium passes through the tidal features , it is possible that the tidal gas exchanges energy and momentum with this medium due to collisions . such effects have been proposed to explain the stripping of the cool interstellar medium from spiral galaxies as they move through the hot igm in clusters ( gunn & gott 1972 ) , and is referred to as ram pressure stripping ( rps ) . tidal features should be relatively easily stripped , as they lack the natural restoring forces present in disk galaxies , except possibly at a small number of self - gravitating regions . in this case , the momentum imparted due to ram pressure is simply added to or subtracted from the momentum of the gaseous tidal features , and a separation of stellar and gaseous components might be expected . in the next two subsections , we investigate two possible sources for ram pressure : an extended halo associated with the progenitors ( [ sec : rpshalo ] ) ; and an expanding starburst driven superwind ( [ sec : rpswind ] ) . our own galaxy is known to have an extended halo of hot gas ( pietz _ et al._1998 ) . the existence of similar halos around external galaxies has been inferred from observations of absorption line systems around bright galaxies ( e.g. lanzetta _ et al . _ these halos may have sufficient density to strip any low column density gas moving through them . several investigators have suggested that such stripping is responsible for removing gas from the magellanic clouds as they orbit through the galaxy s halo , producing the purely gaseous magellanic stream ( e.g. meurer , bicknell & gingold 1985 ; sofue 1994 ; moore & davis 1994 ) . sofue & wakamatsu ( 1993 ) and sofue ( 1994 ) specifically stress that stripping by galaxy halos should also play an important role in the evolution of h@xmath0i tidal tails . the tidal features in each of our systems have h@xmath0i column densities and velocities similar to those assumed in the numerical models of sofue ( 1994 ) and moore & davis ( 1994 ) , which resulted in rather extreme stripping of the h@xmath0i clouds . although this may seem to provide an immediate explanation for our observations of gas / star displacements , we point out that these column densities and velocities are typical of all of the tails thus far imaged in h@xmath0i , the great majority of which do not show the extreme displacements we describe here . there is no reason to believe that the halo properties of ngc 520 , arp 299 and arp 220 are any different from , or that the encounters were any more violent than similar mergers which do not exhibit such dramatic displacements ( e.g. , ngc 4038/9 , ngc 7252 , ngc 4676 hvg96 ; ngc 3628 dahlem _ et al . _ 1996 ; ngc 2623 , ngc 1614 , mrk 273 hibbard & yun in preparation ; ngc 3256 english _ et al . _ in fact , in light of the stripping simulations mentioned above , one wonders why such displacements are not more common . a possible solution to this puzzle is suggested by the results of numerical simulations of major mergers . in these simulations , the material distributed throughout the halos of the progenitors is tidally distended along with the tails , forming a broad sheath around them ( see e.g. the video accompanying barnes 1992 ) . this sheath has similar kinematics as the colder tail material , resulting in much lower relative velocities than if the tail was moving through a static halo , thereby greatly reducing any relative ram pressure force . in summary , while halo stripping might be effective for discrete systems moving through a static halo ( such as the lmc / smc through the halo of the galaxy , or disks through a hot cluster igm ) , the lack of widespread h@xmath0i / optical decoupling in mergers suggests that it is not very effective for removing gas from tidal tails , and it does not appear to be a suitable explanation of the present observations . the three systems under discussion host massive nuclear starbursts with associated powerful outflows or superwinds " . optical emission lines and/or x - ray emission reveal that the observed outflows extend for tens of kpc from the nuclear regions . theoretical calculations suggest that the observed gas plumes represent just the hottest , densest regions of a much more extensive , lower density medium ( wang 1995 ) . in each of these three systems , the most extreme gaps in the h@xmath0i distribution appear along the inferred direction of the expanding hot superwind ( figs . [ fig : n520mos]d , [ fig : a299mos]d , [ fig : a220mos]d ) . a very similar anti - correlation between an out - flowing wind and tidally disrupted h@xmath0i has been observed in the m82 system ( yun _ et al . _ 1993 ; strickland _ et al . _ 1997 ; shopbell & bland - hawthorn 1998 ) , the ngc 4631 system ( weliachew _ et al . _ 1978 ; donahue _ et al . _ 1995 ; wang _ et al . _ 1995 ; vogler & pietsch 1996 ) , and possibly the ngc 3073/9 system ( irwin _ et al . _ 1987 ; filippenko & sargent 1992 ) . it has been suggested that this anti - correlation is due to an interaction between the blown - out gas of the superwind and the cold gaseous tidal debris , either as the wind expands outward into the debris , or as the tidal debris passes through the wind ( chevalier & clegg 1985 ; heckman , armus & miley 1987 , and references above ) . figure [ fig : cartoon ] presents the suggested geometry for the case of arp 299 . this figure is constructed from our preliminary efforts to model the northern tail of arp 299 using n - body simulations similar to those presented in hibbard & mihos ( 1995 ; i.e. no hydrodynamical effects are included ) . we found that we could not match the morphology and kinematics of both filaments simultaneously , but could match either one separately . [ fig : cartoon ] presents the results of combining these two solutions . in this sense this figure is _ not a self - consistent fit to the data _ , but simply a cartoon which illustrates the proposed relative placement of the tidal tail and the expanding wind . in this figure , the wind opening angle is illustrated by the cone , the gas - rich regions of the tails are represented by dark and light grey circles , and the gas - poor regions of the tails are represented by white circles . the figure illustrates how the restricted opening angle of such a wind ( or of an ionization cone , cf . [ sec : ionizsb ] ) may intersect only a portion of the ribbon - like tail . here we estimate whether ram pressure stripping by the nuclear superwind can exert sufficient pressure on gas at the large distances typical of tidal tails . we use equation ( 5 ) from heckman , lehnert & armus ( 1993 ; see also chevalier & clegg 1985 ) to calculate the expected ram pressure ( @xmath29 ) of the superwind far from the starburst as a function of its bolometric luminosity ( @xmath30 ) : @xmath31 this equation has been shown to fit the pressure profile derived from x - ray and optical emission line data of arp 299 ( heckman _ et al . _ 1999 ) . it should provide a lower limit to the ram pressure , since it assumes that the wind expands spherically , while observations suggest that the winds are limited in solid angle . this pressure can be compared with the pressure of the ambient medium in the tidal tail ( @xmath32 ) , given by the energy per unit volume : @xmath33 where @xmath34 is a constant , @xmath35 is the mass density of the gas , and @xmath36 is the velocity dispersion of the gas . for an equation of state of the form @xmath37 , the constant @xmath34 is equal to @xmath38 , and @xmath36 corresponds to gas sound speed , while for a self - gravitating cloud , @xmath39 and @xmath36 is the one - dimensional velocity dispersion of the cloud . in both cases , we assume that the observed line - of - sight velocity dispersion of the h@xmath0i is a suitable measure of @xmath36 . the mass density of the tidal gas is given by @xmath40 , where the numerical constant accounts for the presence of he , @xmath41 is the mass of a hydrogen atom , and @xmath42 is the number density of atomic hydrogen . if we assume the gas is uniformly distributed with a column density @xmath7 along a length @xmath43 , we have @xmath44 , where the fiducial values are typical of tidal features ( hvg96 , hibbard & yun in preparation ) . we rewrite eqn . ( [ eq : perfgas ] ) in terms of the observables : @xmath45 the maximum radius out to which we expect material to be stripped ( @xmath46 ) is then given by the requirement that @xmath29(r ) @xmath47(r ) at r@xmath48 . we replace @xmath30 in eqn . ( [ eq : prps ] ) by the ir luminosity ( @xmath49 ) , under the assumption that the ir luminosity arises from reprocessed uv photons from the starburst ( lonsdale , persson & matthews 1984 , joseph & wright 1985 ) . the very high ir luminosities of these systems ( @xmath50 ) make it likely that this is indeed the case , and we are probably making an error of @xmath17 10% ( e.g. heckman , lehnert & armus 1993 ) . equating eqns . ( [ eq : prps ] ) and ( [ eq : ptidal ] ) we find : @xmath51 in table [ tab : results ] we provide estimates of @xmath46 for the systems considered here . in calculating @xmath46 we made the very conservative assumption that the values of @xmath7 and @xmath52 for the stripped gas are equal to the maximum values found within the tidal tails ( see table [ tab : results ] ) . the results of these calculations indicate that , in all cases , @xmath46 is larger than the radii of the observed gaps in the tidal h@xmath0i distributions . therefore , in principle , the wind should be able to strip the gas from any tidal material in its path . the above derivation assumes that the tidal gas is at rest with respect to the wind . it can be easily generalized to the case of a wind impacting an expanding tidal feature by reducing the wind ram pressure by a factor @xmath53 . for ngc 520 and arp 220 , the tidal gas is primarily in rotation ( _ i.e. _ , moves perpendicular to superwind ) , so we expect the gas to feel the full ram pressure given above . for arp 299 , heckman _ et al . _ ( 1999 ) find @xmath54 = 800 km s@xmath19 , and we estimate a maximum @xmath55 = 240 km s@xmath19 ( hy99 ) . therefore , the ram pressure could be reduced by 50% , reducing @xmath46 by 70% from that listed in table [ tab : results ] , _ i.e. _ @xmath56 kpc for arp 299 . this is still large enough to reach to the region of the n clump in fig . [ fig : a299mos ] . the regions of high h@xmath0ivelocity dispersion indicated in fig . [ fig : a299mos]c may be due to the influence of such a wind . we note that these regions occur on the side of the h@xmath0i features that face the starburst region . however , no such kinematic signatures are visible in the gas near the wind axis in ngc 520 or arp 220 . the lack of gaseous / stellar displacements in the tidal tails of many superwind systems might seem to provide a strong argument against the scenario outlined above . however , there are two conditions needed to produce wind - displaced tidal features : the starburst must be of sufficient energy and duration to achieve blowout " , and the tidal h@xmath0i must intersect the path of the expanding wind material . the second condition is not met for the blowout systems ngc 4676 , ngc 3628 , ngc 2623 , ngc 1614 , and ngc 3256 ( references given in [ sec : rpshalo ] ) . in these systems the tidal tails appear to lie at large angles with respect to the blowout axis , and their tidal tails should not intersect the wind . both conditions are met for m82 and ngc 4631 , which both show extreme h@xmath0i / optical displacements . for these two systems , the high - latitude h@xmath0i appears to be accreted from nearby disturbed companions ( m81 and ngc 4656 , respectively ; yun _ et al._1993 , weliachew _ et al . _ 1978 ) , while for the three major mergers under study here the h@xmath0i appears to intercept the path of the wind as a result of a highly inclined encounter geometry . a high inclination encounter geometry is therefore a prerequisite for such displaced morphologies , and the host of the superwind should be the disk with a retrograde or polar spin geometry . nevertheless , ram pressure stripping can not provide a complete explanation of the observations . since the stars are unaffected by the wind , it would be an unusual coincidence for the edges of the optical plumes to correspond with the edge of the cold gas which is presently being ablated . nor does it seem likely that the wind could be sufficiently collimated to bore " into the northern h@xmath0i clump in arp 299 just where the optical tail appears projected upon it . therefore a second process is still needed to explain the small - scale anti - correlations . in conclusion , the expanding wind _ should _ affect any tidal h@xmath0i in its path ; however this effect alone can not explain all the details of the observations . disk galaxies are known to exhibit a precipitous drop in neutral hydrogen column density beyond column densities of a few times @xmath57 cm@xmath15 ( corbelli , schneider & salpeter 1989 ; van gorkom 1993 ; hoffman _ et al . _ this drop has been attributed to a rapid change in the ionization fraction of the gas due to influence of the intergalactic uv field ( maloney 1993 , corbelli & salpeter 1993 , dove & shull 1994 ; see also felten & bergeron 1969 , hill 1974 ) , rather than a change in the total column density of h. tidal tails are assembled from the outermost regions of disk galaxies . since this material is redistributed over a much larger area than it formerly occupied , its surface brightness must decrease accordingly . therefore , if the progenitors were typical spirals , with h@xmath0i disks extending to column densities of a few times @xmath57 cm@xmath15 , then the resulting tidal tail _ must _ have gas at much lower column densities . however , tidal tails exhibit a similar edge in their column density distribution _ at a similar column density _ ( hvg96 ; hibbard & yun in preparation ) . this is one of the most compelling pieces of evidence for an abrupt change in the phase of the gas at low h@xmath0i column density . the outer tails mapped in h@xmath0i should therefore be the proverbial `` tip of the iceberg '' of a lower column density , mostly ionized medium . with the tidal gas in this very diffuse state , fluctuations in the incident ionizing flux might be expected to produce accompanying fluctuations in the neutral gas fraction . given these considerations , we examine the possibility that the total hydrogen column density does not change at the regions illustrated in figs . [ fig : n520mos][fig : slices ] , but that the neutral fraction does , _ i.e. _ that the gas in the regions under study has a higher ionization fraction than adjacent regions . the intergalactic uv field should be isotropic , and would not selectively ionize certain regions of the tails . here we examine the possibility of two non - isotropic sources of ionizing flux : ( 1 ) leakage of uv flux from the circumnuclear starburst ; ( 2 ) ionization by late b stars and white dwarfs associated with the evolved stellar tidal population . our procedure is to compare the expected ionizing flux density shortward of 912 to the expected surface recombination rate of the gas . we assume that in the area of interest the gas is at a temperature of @xmath59 k , for which a case - b recombination coefficient of @xmath60 is appropriate ( spitzer 1956 ) . we assume that the hydrogen is almost completely ionized , so @xmath61 . we further assume that the density of ionized gas is the same as the density of the neutral gas in the adjacent regions , @xmath62 , where @xmath42 is calculated as above ( @xmath63 , [ sec : rpswind ] ) . the detailed ionization state will depend sensitively on the clumpiness of the gas , but a full treatment of this problem is beyond the scope of this paper . here we wish to investigate if these processes are in principle able to create effects similar to those observed . here we consider the case that the superwind does not affect the tidal gas by a direct interaction , but influences it by providing a direct path from the tidal regions to the starburst , free of dust and dense gas ( see also fig . [ fig : cartoon ] ) . through these holes , ionizing photons from the young hot stars stream out of the nuclear regions and are quickly absorbed by the first neutral atoms they encounter . following felton & bergeron ( 1969 ; see also mahoney 1993 ) , we solve the equation : @xmath64 the right hand side of this equation represents the total ionizing radiation escaping the starburst region along a direction that has been cleared of obscuring material by the superwind . we express this in terms of the total ionizing flux of a completely unobscured starburst of a given bolometric luminosity @xmath30 by introducing the factor @xmath65 to account for the fact that only a fraction of the photons emitted into a solid angle @xmath66 find their way out of the starburst region , the total fraction of ionized photons emerging from a starburst , and @xmath65 , the fraction emerging along a particular sightline . @xmath67 is the total angle averaged fraction , i.e. @xmath67 is the integral of @xmath68 over all solid angles , while @xmath65 is the integral over a solid angle cleared by the wind . most studies in the literature quote values for @xmath67 . ] . the expected ionizing flux for a starburst of a given bolometric luminosity @xmath30 is calculated from the population synthesis models of bruzual & charlot ( 1993 ; 1995 ) , assuming continuous star formation with a duration longer than 10 myr ( long enough for the burst to achieve blowout ) , a salpeter imf with @xmath69 and @xmath70 , and solar metallicity . this yields @xmath71 . again making the standard assumption that most of the starburst luminosity is emitted in the far infrared ( _ i.e._@xmath72 , cf . [ sec : rpswind ] ) , we rearrange eqn . ( [ eq : photosb ] ) to solve for the radius , @xmath73 , out to which the starburst is expected to ionize a given column density of h@xmath0iof thickness @xmath43 : @xmath74 resulting values for @xmath73 are listed in table [ tab : results ] . for this computation , we have adopted a value of 10% for @xmath65 . this is equal to the total fraction of ionizing photons , @xmath67 , escaping from a normal disk galaxy as calculated by dove , shull & ferrara ( 1999 ) . even higher values of @xmath67 are expected in starburst systems ( dove _ et al . _ since we stipulate that a higher fraction of ionizing photons escape along sightlines above the blowout regions than are emitted along other directions , it follows that @xmath75 , and as a result the values of @xmath73 calculated in table [ tab : results ] should be conservative estimates . table [ tab : results ] shows that under these simplified conditions , @xmath73 is of the order of , or larger than , the tidal radii of interest . we therefore conclude that the starburst seems quite capable of ionizing tidal h@xmath0i , if indeed there is an unobstructed path from the starburst to the tidal regions . this might explain the lack of h@xmath0ialong the wind axis in ngc 520 and arp 220 , and the absence of h@xmath0ialong the optical tidal tail in arp 299 . this process is especially attractive since it can potentially explain the lack of h@xmath0i at the bases of otherwise gas - rich tidal tails in ngc 7252 ( hibbard _ et al . _ 1994 ) , arp 105 ( duc _ et al . _ 1997 ) , and ngc 4039 ( hibbard , van der hulst & barnes in preparation ) . these systems do _ not _ show evidence for expanding superwinds , which rules out the possibility that rps is playing a role . and each of these systems possesses a level of star formation that , according to eqn . [ eq : rion ] , is capable of ionizing gas out to the necessary radii . however , photoionization by the central starburst does not seem capable of explaining all of the observations . as with the wind hypothesis above , it would be an unusual coincidence for the edges of the optical plumes to correspond with the edge of ionization cone . therefore a second process is still needed to explain the small - scale anti - correlations . the fact that the h@xmath0i column density falls off just as the optical surface brightness increases at the edges of various tidal features ( fig . [ fig : slices ] ) leads us to suspect that there may be local sources of ionization within the stellar features themselves . for ngc 520 and arp 220 , we believe the outer h@xmath0i is in a disk structure which is intersected by the tidal stellar plumes and we wish to investigate whether ionization by evolved sources within the stellar plumes , such as late b stars and white dwarfs , could be responsible for decreasing the neutral fraction of the diffuse outer h@xmath0i . for arp 299 the geometry is more complicated , and we refer the reader to fig . [ fig : cartoon ] . here we suggest that part of the purely gaseous tidal filament ( the light grey filament in fig . [ fig : cartoon ] ) is ionized by evolved sources near the end of the stellar tail ( the dark grey filament in fig . [ fig : cartoon ] , especially those regions nearest the gas - rich filament in the right hand panel of this figure ) . as in the previous section , we balance the surface recombination rate with the expected ionizing flux density ( eqn . [ eq : photosb ] ) . in this case , we calculate the ionizing flux density for an evolved population of stars of a given @xmath5-band luminosity density ( @xmath76 , in @xmath77 ) . in order to approximate the stellar populations in the tidal tails , we assume that the tails arise from the outer edges of an sbc progenitor , and that star formation ceased shortly after the tails were launched . we again use the models of bruzual & charlot ( 1993 , 1995 ) for a salpeter imf over the mass range 0.1125 @xmath78 , and adopt an exponentially decreasing sfr with a time constant of 4 gyr ( typical of an sbc galaxy , bruzual & charlot 1993 ) , which is truncated after 10 gyr and allowed to age another 500 myr . this simulates the situation in which star formation within the disk is extinguished as the tail forms , and the ejected stellar population passively fades thereafter . while tidal tails frequently exhibit _ in - situ _ star formation ( e.g. schweizer 1978 ; mirabel , lutz & maza 1991 ) , it is usually not widespread . under these assumptions , a population with a projected @xmath5-band surface brightness of 1 @xmath79 pc@xmath15 should produce an ionizing flux of @xmath80 . therefore eqn . ( [ eq : photosb ] ) becomes @xmath81 , which can be rewritten as : @xmath82 noting that 1 @xmath77 corresponds to @xmath13=25.9 mag arcsec@xmath15 , we rewrite this as a condition on the surface brightness of the tidal features : @xmath83 \label{eq : mur}\end{aligned}\ ] ] referring to fig . [ fig : slices ] , we see that only the northern plume of arp 220 is bright enough to ionize nearby tidal h@xmath0iat the appropriate column densities . neither the optical plume in the ngc 520 system nor the northern tail in the arp 299 system appears bright enough to ionize the necessary columns of hydrogen unless the tidal features are unreasonably thick ( @xmath1260 kpc ) . however , since we have no other explanation for the small scale h@xmath0i / optical differences illustrated in fig . [ fig : slices ] , we are hesitant to abandon this explanation too quickly . a possible solution is to invoke continued star formation even after the tails are ejected . for instance , if we do not truncate the star formation rate after 10 gyr , instead allowing the star formation rate to continue its exponential decline as the tail expands , then the ionizing flux per 1 @xmath79 pc@xmath15 is 70 times higher than the value of @xmath80 used above . this would lower the fiducial surface brightness in eqn . ( [ eq : mur ] ) from 23.0 mag arcsec@xmath15 to 27.5 mag arcsec@xmath15 , in which case the faint tidal features in ngc 520 ( @xmath84 25 mag arcsec@xmath15 ) and arp 299 ( @xmath84 26.5 mag arcsec@xmath15 ) could indeed ionize the necessary column densities of adjoining h@xmath0i . the observed broad - band colors of the tidal tails are not of sufficient quality to discriminate between these two star formation histories , since the expected color differences are only of order @xmath28=0.1 mag . however , whether or not the gas is more highly ionized in the regions of interest can be addressed observationally . the expected emission measure ( @xmath85 ) can be parameterized as : @xmath86 since emission measures of order 0.2 @xmath87 pc have been detected with modern ccd detectors ( e.g. donahue _ et al . _ 1995 ; hoopes , walterbos & rand 1999 ) , there is some hope of being able to observationally determine if regions of the tidal tails are significantly ionized . if the gas has a clumpy distribution , then there should be some high density peaks which might be sufficiently bright to yield reliable emission line ratios . such ratios would allow one to determine the nature of the ionizing source , e.g. photoionization vs. shocks . therefore , while we can not assert unequivocally that photoionization plays a role in shaping the outer tidal morphologies , it is possible to test this hypothesis with future observations . the hypothesis that ionizing flux from a stellar tidal feature may ionize gas in a nearby gaseous tidal feature is not necessarily at odds with the observations that many stellar tails are gas - rich . this is because tails with cospatial gas and stars arise from regions originally located within the stellar disk of the progenitors , while the optical faint gas - rich tidal features likely arise from regions beyond the optical disk ( [ sec : radial ] ) . in normal disk galaxies , the h@xmath0iwithin the optical disk is dominated by a cooler component with a smaller scale height and velocity dispersion , while the h@xmath0i beyond the optical disk is warmer and more diffuse , with a larger scale height and velocity dispersion ( braun 1995 , 1997 ) . as a result , @xmath43 should be considerably larger for purely gaseous tidal features than for optically bright tidal features . in this paper we have described differences between gaseous and stellar tidal features . there are large - scale differences , such as extensive purely gaseous tidal features ( the outer disks in ngc 520 and arp 220 and the outer filament in arp 299 ) and largely gas - poor optical features ( tidal plumes in ngc 520 and arp 220 and the inner filament in arp 299 ) . and there are smaller - scale differences : the anti - correlation between the edges of gaseous and optical features depicted in fig . [ fig : slices ] . a similar anti - correlation is observed between h@xmath0i and optical shells in shell galaxies ( schiminovich _ et al . _ 1994a , b , 1999 ) , many of which are believe to be more evolved merger remnants . we have examined a number of possible explanations for these observations , including dust obscuration , differences in the original distribution of gas and starlight in the progenitor disks , gas cloud collisions within the developing tails , ram pressure stripping due to an extensive hot halo or an expanding superwind , and photoionization by either the central starburst or evolved sources in the tidal tails themselves . however , no one model easily and completely explains the observations , and it is conceivable that all explanations are playing a role at some level . the most likely explanation for the lack of starlight associated with the outer tidal h@xmath0iis that such features arise from the h@xmath0i - rich but optically faint outer radii of the progenitor disks . the relatively short lifetimes of luminous stars and the large velocity dispersions of less luminous stars , especially with respect to the gas , will further dilute the luminous content of this material , and the h@xmath0i - to - light ratio of the resulting tidal features will increase with time ( hibbard _ et al . _ gaseous tidal extensions with very little detectable starlight would seem to be the natural consequence . the outer h@xmath0i rings in ngc 520 and arp 220 and the gas - rich outer filament in arp 299 are all likely to arise from these gas - rich regions of their progenitor disks . for the gas - poor tidal features we suggest that the starburst has played an important role in shaping the gaseous morphology , either by sweeping the features clear of gas via a high - pressure expanding superwind , or by excavating a clear sightline towards the starburst and allowing ionizing photons to penetrate the tidal regions . the primary supporting evidence for this conclusion is rather circumstantial : the five galaxies with the most striking h@xmath0i / optical displacements ( the three systems currently under study here , and the h@xmath0i accreting starburst systems m82 and ngc 4631 ) host massive nuclear starbursts with associated powerful outflows or superwinds aligned with the direction of the most extreme h@xmath0i / optical displacements . ngc 520 , arp 299 , and arp 220 each experienced prograde / polar or prograde / retrograde encounters . this relative geometry may be a pre - requisite for the morphological differences reported here . retrograde and polar encounters do not raise extensive tidal tails ( e.g. barnes 1988 ) , leaving large gaseous disks in the inner regions . these disks should help collimate and `` mass - load '' the superwind ( heckman , lehnert & armus 1993 ; suchkov _ et al . _ 1996 ) , which in turn leads to denser and longer - lived winds . simultaneously , the combination of opposite spin geometries provides the opportunity for the tidal tail from the prograde system to rise above the starburst region in the polar or retrograde system , where it may intersect the escaping superwind or uv radiation . if this suggestion is correct , only systems hosting a galactic superwind and experiencing a high - inclination encounter geometry should exhibit such extreme differences between their h@xmath0i and optical tidal morphologies . the observations do not allow us to discriminate between either the rps or the photoionization models : simple calculations suggest that either is capable of affecting the diffuse outer gas if the geometry is right . there might be some evidence for the effects of an impinging wind on the outer material in arp 299 from the increased velocity dispersion at several points ( hy99 ) ; however ngc 520 and arp 220 show no such signatures . photoionization is an attractive solution , as it offers a means of explaining the lack of tidal h@xmath0i found at the base of otherwise gas - rich tidal tails in mergers which show no evidence of a superwind ( e.g. ngc 7252 , arp 105 , ngc 4039 ; see [ sec : ionizsb ] ) . since any ionized hydrogen will emit recombination lines , both explanations can be checked observationally . the expected emission measure is given by eqn . ( [ eq : em ] ) , which predicts detectable features at the column densities of interest . the morphology of the ionized gas should reveal the nature of the ionizing source : photoionized gas should be smoothly distributed , while gas excited by rps should be concentrated in dense shocked regions on the edges of the h@xmath0i that are being compressed by the superwind , _ i.e. _ on the edges nearest the wind axis in figs . [ fig : n520mos]d , [ fig : a299mos]d & [ fig : a220mos]d . if the gas is clumpy , there may be regions bright enough to allow line ratios to be measured , which should further aid in discriminating between photoionization or shock excitation . only two scenarios are offered to explain the small - scale anti - correlations : dust obscuration and photoionization due to evolved sources in the optical tails . dust obscuration likely affects the apparent tidal morphologies at the lowest light levels , but we suspect that the dust content is too low to significantly obscure the brighter tidal features . however , if the tidal tails are highly ionized , with the neutral gas representing only a small fraction of the total hydrogen column density , it is possible that we are grossly underestimating the expected amount of absorption . this question can be investigated directly with submm imaging of the cold dust in tidal tails . the other possibility is that the uv flux from evolved sources in the optical tails is responsible for ionizing nearby diffuse outer h@xmath0i . a simple calculation suggest that the tidal tail in arp 220 is bright enough to ionize nearby h@xmath0i , but the expected ionization flux from the optical tails in ngc 520 and arp 299 is too low to explain the observed differences , unless significant star formation continued within these features after their tidal ejection . if this is indeed the case , then the regions where the neutral gas column density drops rapidly ( see fig.[fig : slices ] ) should contain ionized gas which would emit recombination radiation . the expected levels of emission should be observable with deep imaging techniques ( see above ) . this situation requires that the gas and stellar features are physically close , and not just close in projection , which can tested with detailed numerical simulations . we would like to thank lee armus tim heckman for sharing of unpublished results , and rhodri evans , jacqueline van gorkom , dave schiminovich , and josh barnes for useful discussions . we thank the referee , chris mihos , for a thorough and useful report . aaronson , m. huchra , j. , mould , j. , schechter , p. l. & tully , r. b. 1982 , apj , 258 , 64 alton , p. b. , trewhella , m. , davies , j. i. , evans , r. , bianchi , s. , gear , w. , thronson , h. , valentijn , e. & witt , a. 1998 , a&a , 335 , 807 appleton , p. n. , charmandaris , v. & struck , c. 1996 , apj , 468 , 532 appleton , p. n. , davies , r. d. & stephenson , r. j. 1981 , mnras , 195 , 327 appleton , p. n. , ghigo , f. d , van gorkom , j. h. , schombert , j. m. , & struck - marcell , c. 1987 , nature , 330 , 140 armus , l. , heckman , t. m , & miley , g. k. 1990 , apj , 364 , 471 barnes , j. e. 1988 , apj , 331 , 699 barnes , j. e. 1992 , apj , 393 , 484 barnes , j. e. & hernquist , l. 1991 , apj , 370 , l65 barnes , j. e. & hernquist , l. 1996 , apj , 471 , 115 bohlin , r. c. , savage , b. d. & drake , j. f. 1978 , apj , 224 , 132 braun , r. 1995 , a&as , 114 , 409 braun , r. 1997 , apj , 484 , 637 bruzual , a. g. & charlot , s. 1993 , apj , 405 , 538 bruzual , a. g. & charlot , s. 1995 , personal communication chevalier , r. a. & clegg a. w. 1985 , nature , 317 , 44 combes , f. 1978 , a&a , 65 , 47 combes , f. , dupraz , c. , casoli , f. , & pagani , l. 1988 , a&a ( lett . ) , 203 , 9 condon , j. j. , helou , g. , sanders , d. b. , & soifer , b. t. 1990 , apjs , 73 , 359 corbelli , e. & salpeter , e. e. 1993 , apj , 419 , 104 corbelli , e. , schneider , s. e. & salpeter , e. e. 1989 , aj , 97 , 390 dahlem , m. , heckman , t. m. , fabbiano , g. , lehnert , m. d. , & gilmore , d. 1996 , apj , 461 , 724 donahue , m. , aldering , g. , & stocke , j. t. 1995 , apj ( lett . ) , 450 , l45 dove , j. b. & shull , j. m. 1994 , apj , 423 , 196 dove , j. b. & shull , j. m. & ferrara 1999 , apj , in press ( astro - ph/9903331 ) downes , d. & solomon , p. m. 1998 , apj , 507 , 615 duc , p. -a . , brinks , e. , wink , j. e. , & mirabel , i. f. 1997 , a&a , 326 , 537 english , j _ et al . _ 1999 , aj , submitted felton , j. e. & bergeron , j. 1969 , apjl , 4 , 155 filippenko , a. v. & sargent , w. l. w. 1992 , aj , 103 , 28 gardiner , l. t. & noguchi , m. 1996 , mnras , 278 , 191 gunn , j. e. & gott , j. r. 1972 , apj , 176 , 1 heckman , t. m. , armus , l. , miley , g. k. 1987 , aj , 93 , 276 heckman , t. m. , armus , l. , miley , g. k. 1990 , apjs , 74 , 833 * ( ham90 ) * heckman , t. m. , armus , l. , weaver , k. a. & wang , j. 1999 , apj , 517 , 130 heckman , t. m. , dahlem , m. , eales , s. a. , fabbiano , g. , & weaver , k. 1996 , apj , 457 , 616 heckman , t. m. , lehnert , m. & armus , l. 1993 , in `` the evolution of galaxies and their environment '' , edited by h. a. thronson and j. m. shull ( kluwer , dordrecht ) , p. 455 hibbard , j. e. 1995 , ph . d. thesis , columbia university hibbard , j. e. & van gorkom , j. h. 1996 , aj , 111 , 655 * ( hvg96 ) * hibbard , j. e. , guhathakurta , p. , van gorkom , j. h. , & schweizer , f. 1994 , aj , 107 , 67 hibbard , j. e. & mihos , j. c. 1995 , aj , 110 , 140 hibbard , j. e. & yun , m. s. 1996 , in `` cold gas at high redshift '' , edited by m. bremer , h. rottgering , p. van der werf , and c. l. carilli ( kluwer , dordrecht ) , p. 47 hibbard , j. e. & yun , m. s. 1999 , aj , 118 , 162 * ( hy99 ) * hilker , m. & kissler - patig , m. 1996 , a&a , 314 , 357 hill , j. k. 1974 , a&a , 34 , 431 hoffman , l. g. , lu , n. y. , salpeter , e. e. , farhat , b. , lamphier , c. & roos , t. 1993 , aj , 106 , 39 hoopes , c. , walterbos , r. & rand , r. 1999 , apj , 522 , 669 van der hulst , j. m. 1979 , a&a , 75 , 97 irwin , j. a. , seaquist , e. r. , taylor , a. r. & duric , n. 1987 , apj ( lett . ) , 313 , l91 joseph , r. d. & wright , g. s. 1985 , mnras , 214 , 87 lanzetta , k. m. , bowen , d. v. , tytler , d. & webb , j. k. 1995 , apj , 442 , 538 lehnert , m. d. & heckman , t. m. 1996 , apj , 462 , 651 lonsdale , c. j. , persson , s. e. , & matthews , k. 1984 , apj , 287 , 95 maloney , p. 1993 , apj , 414 , 41 meurer , g. r. , bicknell , g. v. & gingold , r. a. 1985 , pasa , 6 , 195 mihos , j. c. 2000 , in preparation mihos , j. c. & bothun , g. d. 1998 , apj , 500 , 619 mihos , j. c. , bothun , g. d. , richstone , d. o. 1993 , apj , 418 , 82 mihos , j. c. & hernquist , l. 1996 , apj , 464 , 641 mirabel , i. f. , lutz , d. & maza , j. 1991 , a&a , 243 , 367 moore , b. & davis , m. 1994 , mnras , 270 , 209 noguchi , m. 1988 , a&a , 201 , 37 nordgren , t. e. , chengalur , j. n. , salpeter , e. e. , & terzian , y. 1997 , aj , 114 , 77 norman , c. , bowen , d. v. , heckman , t. , blades , c. , & danly , l. 1996 , apj , 472 , 73 odonnell , j. e. 1994 , apj , 422 , 1580 pietz , j. , kerp , j. , kalberla , p.m.w . , burton , w.b . , hartmann , d. & mebold , u. 1998 , a&a , 332 , 55 sakamoto , k. , scoville , n. z. , yun , m. s. , crosas , m. , genzel , r. , tacconi , l. j. 1999 , apj , 514 , 68 sanders , d. b. , scoville , n. z. , sargent , a. i. & soifer , b. t. 1988 , apj , 324 , l55 sanders , d. b. , scoville , n. z. , & soifer , b. t. 1991 , apj , 370 , 158 schiminovich , d. , van gorkom , j. h. , van der hulst , j. m. , & kasow , s. 1994 , apj , 423 , l101 schiminovich , d. , van gorkom , j. h. , van der hulst , j. m. , & malin , d. f. 1995 , apj , 444 , l77 schiminovich , d. , van gorkom , j. h. , & van der hulst , j. m. 1999 , aj , submitted schlegel , d. j. , finkbeiner , d. p. & davis , m. 1998 , apj , 500 , 525 . schombert , j. m. , wallin , j. f. & struck - marcell , c. 1990 , aj , 99 , 497 schweizer , f. 1978 , the structure and properties of nearby galaxies " , iau symp . 77 , edited by e. m. berkhuijsen and r. wielebinski ( reidel , dordrecht ) , p. 279 scoville , n. z. , sargent , a. i. , sanders , d. b. , & soifer , b. t. 1991 , apj , 366 , l5 scoville , n. z. , yun , m. s. , & bryant , p. m. 1997 , apj , 484 , 702 shopbell , p. l. , & bland - hawthorn , j. 1998 , apj , 493 , 129 . simkin s. m. , van gorkom , j. h. , hibbard , j. e. , hong - jun , s. 1987 , science , 235 , 1367 . smith , b. j. 1994 , aj , 107 , 1695 . smith , b. j. , struck , c. , & pogge , r. w. 1997 , apj , 483 , 754 . sofue , y. 1994 , apj , 423 , 207 . sofue , y. , & wakamatsu , k. 1993 , a&a , 273 , 79 . soifer , b. t. , helou , g. , lonsdale , c. j. , neugebauer , g. , hacking , p. , houck , j. r. , low , f. j. , rice , w. , & rowan - robinson , m. 1984 , apj , 283 , l1 spitzer , l. 1956 , apj , 124 , 20 . stanford , s. a. , & balcells , m. 1991 , apj , 370 , 118 stockton , a. 1974a , apj , 187 , 219 stockton , a. 1974b , apj , 190 , l47 stockton , a. & bertola , f. 1980 , apj , 235 , 37 strickland , d. k. , ponman , t. j. , & stevens , i. r. 1997 , a&a , 320 , 378 struck , c. 1997 , apj ( supp . ) , 113 , 269 struck - marcell , c. 1990 , aj , 99 , 71 suchkov , a. a. , berman , v. g. , heckman , t. m. & balsara , d. s. , 1996 , apj , 463 , 528 suchkov , a. a. , balsara , d. s. , heckman , t. m. , leitherer , c. 1994 , apj , 430 , 511 toomre , a. , & toomre , j. 1972 , apj , 178 , 623 van gorkom , j. h. 1993 , in `` the evolution of galaxies and their environment '' , edited by h. a. thronson and j. m. shull ( kluwer , dordrecht ) , p. 345 vogler , a. , & pietsch , w. 1996 , a&a , 311 , 35 wallin , j. f. 1990 , aj , 100 , 1477 wang , b. 1995 , apj , 444 , 590 wang , q. d. , walterbros , r. a. m. , steakley , m. f. , norman , c. a. & braun , r. 1995 , apj , 439 , 176 weil , m. l. , & hernquist , l. 1993 , apj , 405 , 142 weliachew , l. , sancisi , r. , & guelin , m. 1978 , a&a , 65 , 37 wevers , b. h. m. r. , appleton , p. n. , davies , r. d. , & hart , l. 1984 , a&a , 140 , 125 yun , m. s. 1992 , ph.d . thesis , harvard university yun , m. s. 1997 , in _ `` galaxy interactions at low and high redshfit '' _ , iau symposium no . 186 , eds . d. sanders & j. barnes , in press yun , m. s. , ho , p. t. p. , & lo , k. y. 1993 , apjl , 411 , l17 yun , m. s. , ho , p. t. p. , & lo , k. y. 1994 , nature , 372 , 530 yun , m. s. & hibbard , j. e. 2000 , in preparation yun , m. s. & hibbard , j. e. 1999 , apj , submitted lrlllc @xmath88 & ( km s@xmath19 ) & 2260 & 3080 & 5400 & + @xmath10 & ( mpc ) & 30 & 48 & 79 & @xmath89 + @xmath49 & ( l@xmath90 ) & @xmath91 & @xmath92 & @xmath93 & @xmath94 + @xmath95 & ( cm@xmath15 ) & @xmath96 & @xmath97 & @xmath97 & @xmath98 + @xmath99 & ( km s@xmath19 ) & 16 & 20 & 40 & @xmath98 + @xmath100 & ( kpc ) & 25 & 70 & 30 & @xmath101 + + @xmath46 & ( kpc ) & 50 & 140 & 95 & @xmath102 + @xmath103 & ( kpc ) & 25 & 95 & 130 & @xmath104 + [ tab : results ]
|
as part of several h@xmath0i synthesis mapping studies of merging galaxies , we have mapped the tidal gas in the three disk - disk merger systems arp 157 ( ngc 520 ) , arp 220 , and arp 299 ( ngc 3690 ) .
these systems differ from the majority of the mergers mapped in h@xmath0i , in that their stellar and gaseous tidal features do not coincide .
in particular , they exhibit large stellar tidal features with little if any accompanying neutral gas and large gas - rich tidal features with little if any accompanying starlight
. on smaller scales , there are striking anti - correlations where the gaseous and stellar tidal features appear to cross .
we explore several possible causes for these differences , including dust obscuration , ram pressure stripping , and ionization effects .
no single explanation can account for all of the observed differences .
the fact that each of these systems shows evidence for a starburst driven superwind expanding in the direction of the most striking anti - correlations leads us to suggest that the superwind is primarily responsible for the observed differences , either by sweeping the features clear of gas via ram pressure , or by excavating a clear sightline towards the starburst and allowing uv photons to ionize regions of the tails . if this suggestion is correct , only systems hosting a galactic superwind and experiencing a high - inclination encounter geometry ( such that tidal gas is lifted high above the starburst regions ) should exhibit such extreme differences between their h@xmath0i and optical tidal morphologies .
= cmcsc10 at 12pt = cmcsc10 = cmr12 = cmr10 # 1@xmath1 # 1@xmath2
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
reference frames are one of the most basic notions in physics . almost every time a measurement is performed or a formula is written down , a reference frame has been implicitly used . the choice of reference frame can either be thought of as abstract labelling of space - time , or a description relative to some physical laboratory with a particular set of rulers and clocks . in the latter case , the laboratory is always taken to be much heavier than the system under observation , with a well defined velocity and position . in other words , it is assumed to be fundamentally _ classical_. however , in their seminal papers @xcite , aharonov and susskind showed that the concept of reference frame can be suitably accommodated in quantum theory . although our everyday reference frames are classical for all practical purposes , aharonov and susskind have shown that theoretically nothing prevents the idea of a _ quantum reference frame_. these early works were followed by @xcite and culminated in the work of aharonov and kaufherr @xcite that is of particular relevance to us here . since then much progress has been made by investigating the role of quantum reference frames from both foundational and applied viewpoints . several works have investigated fundamental properties of quantum reference frames @xcite . also , quantum reference frames have been studied in quantum information @xcite , in particular in the context of quantum communication @xcite and entanglement detection @xcite . in the present paper we go back to the original problem discussed in @xcite , namely that of relative positions and momenta , and ask what physics looks like from an isolated quantum reference frame . in particular , we consider states composed of a small number of quantum systems , then promote one of these systems to the role of a _ quantum observer _ and examine the description of the state from its internal perspective . for example , we will consider experiments performed within a freely - floating quantum rocket , from the perspective of the rocket itself , making no mention of any external reference frame whatsoever . we find that we are easily lead into paradoxes when trying to analyse experiments from the perspective of such a quantum observer . by careful reanalysis , we resolve these paradoxes and in doing so learn much about the physics of quantum reference frames . let us consider a simple interference experiment , in which a particle is sent in superposition towards an interferometer . although such experiments usually involve photons , for simplicity here we consider the particle to be non - relativistic . the interferometer consists of two mirrors , a beam splitter , and two detectors , as depicted in fig . [ f : mz ] ( a ) . we take these components to be rigidly fixed together , but free to move relative to the laboratory ( which we treat as a classical external reference frame ) . we initially place the centre of the interferometer at the origin of the laboratory , and denote the position of the left and right path of the particle by @xmath0 and @xmath1 respectively . the initial state of the interferometer and particle relative to the laboratory can then be written @xmath2 where @xmath3 denotes a narrow wavepacket for the interferometer / particle centred at @xmath4 ( in the horizontal coordinate ) . this description of the experiment is from the point of view of the laboratory , but what we are actually interested in is the description of this experiment from the perspective of the interferometer and the particle themselves . since neither the interferometer nor the particle interacts with the laboratory we should not need to use it to describe the experiment . to this end we ask the following questions : a ) what does the interferometer see ? b ) what does the particle see ? let us begin by first clarifying precisely what we mean by our terminology that the the interferometer or particle ` sees ' . we use this rather colloquial terminology to mean the description of the physics relative to the system in question , i.e. the relative positions and momenta of the rest of the system . assuming that the interferometer is much heavier than the particle , the answers are as we might expect . the interferometer sees the particle approaching it in a superposition of positions @xmath0 and @xmath1 , while the particle sees the inverse the interferometer approaching it in a superposition of positions @xmath1 and @xmath0 . now consider the slightly different experimental setup shown in fig . [ f : mz ] ( b ) . in this case , the particle is positioned at the origin of the laboratory , while the centre of the interferometer is placed in a superposition of positions @xmath0 and @xmath1 . the state of the joint system for this second experiment is @xmath5 one might think that as far as the particle and interferometer are concerned this second situation would look the same as the first experiment , with the interferometer seeing the particle approaching it in a superposition of positions @xmath0 and @xmath1 , and the particle seeing the interferometer in a superposition of @xmath1 and @xmath0 as before . since it is only the particle and interferometer that are relevant , and not the external frame , we would expect the two experiments to yield identical results . indeed , if we were to place the detectors instead at the location of the mirrors then both situations would be equivalent ; in both cases both detectors would be equally likely to click . surprisingly , however , if we follow the evolution in both cases we see a dramatic difference . this is illustrated from the perspective of the laboratory in figure 2 . in the first experiment , the two particle paths interfere at the beam splitter and only the left detector clicks , while in the second experiment there is no interference between the photon paths and both detectors are equally likely to click . as all perspectives must agree on the probability of detector clicks , the second experiment must also look different to the first from the perspective of the particle and the interferometer . in other words , experiments ( a ) and ( b ) are not equivalent , contrary to what we expected . but why not ? to find out , we consider in more detail the correct description of the particle relative to the interferometer for both experiments . to do this we change from laboratory co - ordinates to the relative coordinate of the particle as seen by the interferometer , @xmath6 , and the centre of mass , @xmath7 . these are given by @xmath8 where @xmath9 and @xmath10 are the mass and position of the interferometer / particle , and @xmath11 . one should note that when changing co - ordinates subtle effects due to the finite width of the wavepackets emerge . consider a state of the form @xmath12 , by which we actually mean @xmath13 , i.e. two wavepackets , centred around @xmath14 and @xmath15 , hence the centre of mass is located around @xmath16 and the relative co - ordinate is centred around @xmath17 . however , in this state there is an uncertainty in the position of each particle , hence there is an uncertainty in the position of the centre of mass and relative distance ; in fact the centre of mass and relative position will in general be entangled . more precisely , in relative co - ordinates the state @xmath13 becomes @xmath18 which will in general not factorise as there will be entanglement between the _ fluctuations _ of the centre of mass and relative coordinate . entanglement between the centre of mass and relative distance will turn out to be crucial in what follows . however , it is not this entanglement between the fluctuations but the entanglement that appears when we consider superpositions of localised wavepackets , such as @xmath19 that is crucial in our paper . in other words , our concern is the entanglement between the _ mean _ positions of the centre of mass and relative co - ordinates , not the entanglement of the fluctuations . for the sake of clarity , from here on we thus work under the following approximation , @xmath20 , which amounts to ignoring the entanglement between the fluctuations . importantly , this approximation has no consequences on the physics we present below ; for a more detailed analysis , including finite width effects , see the appendix . given the above , in relative co - ordinates the two experimental situations are described by the states @xmath21 the crucial point is to realise that there are degrees of freedom that are accessible to the internal observer and degrees of freedom which are not and to distinguish between them . on the one hand , it is clear that the position of the centre of mass is a degree of freedom that is not accessible inside as it explicitly refers to the external reference frame , i.e. the laboratory . indeed the position of the centre of mass can neither be determined nor altered by any internal observer that has no access to any external frame . on the other hand , the relative position between the particle and interferometer is certainly accessible as it does not refer to the laboratory ( or any other external frame ) . therefore to correctly describe the physics of the internal observer we must trace over the inaccessible degrees of freedom , i.e. the centre of mass . it is the reduced state , @xmath22 which accurately describes the state of the particle relative to the interferometer . similarly by taking instead @xmath23 we find the state of the interferometer relative to the particle . now , if we consider the regime where the mass of the interferometer is much greater than that of the particle , that is @xmath24 , we require @xmath25 then we see that the states ( 5 ) and ( 6 ) become , approximately , @xmath26 thus we see that in experiment ( a ) upon tracing over the centre of mass we are left in the pure state @xmath27 . in this situation the interferometer sees the particle in a coherent superposition , hence we observe interference and only one of the detectors clicks . on the other hand however , in experiment ( b ) upon tracing over the centre of mass we are left in a mixed state @xmath28 representing an equal chance of the particle impinging upon the left and right mirrors . here no interference can take place and hence both detectors are equally likely to click . note that the relative state of the interferometer as seen by the particle is exactly the same , up to the sign of the relative position , and thus the same conclusions also hold from this perspective . hence both internal observers will make the same predictions as the external observer about the experimental outcome . we see that the reason why the internal observers make different predictions in the two experiments is due to the fact that relative states in each experiment are different ; one is a coherent superposition , whilst the other is a mixture . we are thus led to realise that the centre of mass plays a crucial role when considering what the interferometer or particle sees , for it is the entanglement of the relative coordinate with the centre of mass coordinate which makes the two states ( 5 ) and ( 6 ) completely different . below , we provide another thought - experiment which shows explicitly how interference disappears whenever the centre of mass is entangled with the internal degrees of freedom . this is in order to make it clear that the disappearance of interference is due to the entanglement between the centre of mass and relative co - ordinate which was already present in the initial state of the system , and not due to its subsequent interactions , such as the reflection of the particle by the mirrors . consider the situation depicted in fig . [ rocket ] in which a particle of mass @xmath29 moves freely inside a stationary rocket of mass @xmath30 . after the preparation of their initial state , the particle and the rocket no longer interact with the external world . for an external observer this is a _ quantum _ rocket , i.e. the position of the rocket s centre of mass is given by a narrow wavepacket centred at the origin of the external frame , with a quantum uncertainty @xmath31 . suppose the particle has been prepared in a coherent superposition relative to the external observer . the two wave packets composing the superposition move towards each other with constant average momentum @xmath32 . initially , they are symmetrically located at a distance @xmath1 apart from the origin of the external frame . a quantum rocket is prepared in a well localised state , whose centre of mass is here depicted by the narrow red wavepacket . inside the rocket is a particle in superposition , here depicted as the blue wavepackets , initially located a distance @xmath1 from the centre of the rocket , with momemtum @xmath32 towards the centre . at a time @xmath33 when the wavepackets interfere either ( a ) the centre of mass is sufficienty localised to resolve the fringes or ( b ) the centre of mass is too uncertain to resolve them . [ rocket ] ] after a period of time @xmath34 , given by @xmath35 the two wave packets meet at the origin and from the external observer s perspective , they form an interference pattern corresponding to the wavelength @xmath36 , in units such that @xmath37 . then the question arises whether an observer inside the rocket would be able to see the interference pattern . intuitively we expect the uncertainty in the rocket s position at the time the two wavepackets interfere to be the limiting factor on the ability of the internal observer to measure the interference , that is the rocket can only resolve details significantly larger than its uncertainty @xmath38 relative to the external frame . thus , this implies that the internal observer can only see the interference pattern when the uncertainty in the rocket s position is much smaller than the wavelength @xmath39 of the particle , namely @xmath40 where @xmath38 denotes the variance in the position of the rocket at the instant @xmath34 . this is situation ( a ) in fig . [ rocket ] . on the other hand , in situation ( b ) when @xmath41 , although from the external frame of reference there is still interference , from inside the rocket this can no longer be seen . what we shall now see is that whether we end up in situation ( a ) or ( b ) depends only upon whether or not the centre of mass of the rocket is entangled with the relative co - ordinates . let us assume that the rocket is initially in a minimum uncertainty wave packet ( @xmath42 ) . using the approximation @xmath43 we now re - write eq . as @xmath44 note that both terms on the left - hand side are positive , hence each must be much smaller than the right - hand side . using equation this implies that @xmath45 this means that the interference pattern can indeed be resolved by an observer in the rocket provided that the initial uncertainty in the rocket s position obeys these relations . given that we take the mass of the rocket to be much larger than the particle , then @xmath46 and @xmath47 , and eq . implies that @xmath48 here @xmath49 represents by how much the centre of mass is displaced from the origin when the particle is prepared to the left or to the right . therefore the meaning of ( 13 ) is that this displacement is smaller than the original uncertainty in the centre of mass , hence the centre of mass is insensitive to whether the particle is to the left or the right , i.e. it decouples from the relative position . again we see that the entanglement between the centre of mass and the relative co - ordinates determines whether or not there will be interference . however , this is by no means the end of the story , as we shall see in the next section . in the previous section we have seen that the centre of mass , in particular its entanglement to the relative co - ordinates , plays a central role in the physics of quantum reference frames . however , we shall see now that this is not the only issue in understanding quantum reference frames . contrary to the situation considered so far we shall envisage a thought experiment in which the centre of mass is physically irrelevant , yet we run into an even more intriguing paradox . let us consider now two particles , of mass @xmath50 and @xmath51 , prepared in the following entangled state : @xmath52 where the state @xmath53 denotes that particle @xmath54 is sharply localized at position @xmath4 relative to the laboratory , and @xmath55 is an arbitrary relative phase . we choose the positions @xmath14 and @xmath15 such that @xmath56 and hence the centre of mass of the particles is at the origin of our co - ordinate system , as shown in fig . [ f : particles]a . we also denote the distance between the particles by @xmath57 , which allows us to write @xmath58 so far the description of the system has been given relative to an external reference frame . again we would like to promote particle 1 to the role of a _ quantum observer _ and understand how _ it _ would describe the state of particle 2 . ( a ) the two particles are in an entangled state , such that the centre of mass of the system is unentangled from the relative co - ordinates . ( b ) a third , rather innocuous , particle is added in the picture , leaving the centre of mass unentangled with the relative co - ordinates . ] thus we move again from absolute to relative positions . we therefore describe our system using the position of the centre of mass @xmath59 and the relative position @xmath60 , given by @xmath61 we can rewrite the state of the system in these new co - ordinates as @xmath62 now we are in position to describe what particle 1 would observe without access to the external reference frame , i.e. by tracing over the centre of mass . first , from eq . we see that the position of the centre of mass and the relative position actually decouple , i.e. the global state is a product state . therefore we conclude that particle 1 sees particle 2 in a pure state . importantly this implies that particle 1 can get access to the phase @xmath55 by interacting with particle 2 alone , i.e. without access to the external reference frame . now let us bring a third , and rather innocuous , particle into the game . this particle has mass @xmath63 , and is located at position @xmath64 relative to the external frame ( see fig . particles 1 and 2 are exactly as previously . in the external reference frame , the global state of the system is now given by @xmath65 in order to understand what particle 1 would observe , we move again to relative co - ordinates , given by @xmath66 where @xmath67 . note that @xmath60 and @xmath68 are the positions of particle 2 and 3 relative to particle 1 . the state of the system , expressed in these new co - ordinates is @xmath69 again , let us distinguish those degrees of freedom which are accessible to particle 1 from those which are not . first of all , we note that the inaccessible degree of freedom , i.e. the position of the centre of mass , factorizes once again . however , the surprising fact now is that the accessible degrees of freedom the positions of particles 2 and 3 relative to particle 1 are now entangled . consequently it appears that particle 1 must do a joint measurement on particles 2 and 3 to determine the phase @xmath55 . if particle 1 can not access particle 3 , for instance when these two particles are far apart , it seems that we should trace over @xmath59 and @xmath68 , giving the reduced state @xmath70 which contains no information about the phase whatsoever . here the paradox emerges . since the innocuous particle 3 is completely uncorrelated from particles 1 and 2 , adding it into the description of the system should clearly not change the physics . however from the above discussion , it appears that the addition of particle 3 leads to dramatically different situations from the perspective of the internal observer . on the one hand , when considering only particles 1 and 2 , we concluded that particle 1 can gain information about the phase @xmath55 . on the other hand , when particle 3 is included ( but inaccessible to particle 1 ) , it appears that particle 1 can not extract any information about the phase @xmath55 . clearly this is in flagrant conflict with the predictions of the external observer , who would argue that particle 1 should be able to extract exactly the same information about @xmath55 in both cases , as it is a property of particles 1 and 2 alone . in the following subsections we shall identify the optical illusion which is actually present and hence solve the paradox . in doing so we will learn more about the nature of quantum observers . one may suspect that the root of the problem resides in the phase measurement ; indeed above we never explicitly showed how the phase could be measured . in the following we address this aspect in detail . we focus initially on the case of two particles , for which we concluded that particle 1 has access to the phase , and see how particle 1 would proceed . let us consider again the state of the system in relative co - ordinates , i.e. eq . . measuring the relative position of particle 2 provides no information about the phase . instead , the information about the phase is encapsulated in the relative momentum , @xmath71 where @xmath72 is the reduced mass . however , this information is encoded in an intricate way : the expectation values of the moments of momentum , @xmath73 , are all independent of the relative phase . the function of momentum whose expectation is sensitive to the phase is the relative shift operator @xmath74 where @xmath75 @xcite . this operator has the effect of shifting the position of particle 2 versus particle 1 by a distance @xmath76 , causing one wavepacket in the original state to overlap the other in the shifted state , and revealing information about @xmath55 in its expectation value : @xmath77 where in the first line we use the approximation . note that the shift operator is non - hermitian , the corresponding hermitian operators are @xmath78 and @xmath79 . in practice instead of measuring them directly , one would perform an interference experiment . from the perspective of the external observer , we can also understand this result , by noting that @xmath80 corresponding to a shift of particle 1 by @xmath81 , and particle 2 by @xmath82 in the external reference frame . from equation ( [ phi ] ) we see that @xmath83 so far everything appears perfectly consistent . now let us move to the case of three particles . clearly , particle 1 can still measure the relative velocity of particle 2 , even when it has no access to particle 3 . thus particle 1 can still measure the relative shift operator @xmath84 . furthermore , still holds in the three particle case , hence using the state as seen by the external observer , we have @xmath85 this fits with the view that particle 1 can measure the phase without interacting with particle 3 . the paradox arises when we look at the same measurement in terms of the relative state . if @xmath86 only acts on the relative coordinate of particle 2 , it will see only the reduced state given by , which has no dependence on the phase . we now present the solution to the paradox . the crucial ( and surprising ) observation is that @xmath86 actually shifts the relative coordinate of particle 3 as well as that of particle 2 . the root of the problem is that we have been considering relative position and momentum operators which are not canonically conjugate to each other . when we considered the case of only two particles @xmath87 was the momentum canonically conjugate to @xmath88 . however , when we move to the situation with three particles , this is no longer the case , as @xmath87 does not commute with @xmath89 . indeed it can be immediately checked that @xmath90 = i\frac{\mu_{12}}{m_1 } \neq 0 . \end{aligned}\ ] ] at this point it is important to recall that canonical observables play a fundamental role , because they are the building blocks of the hilbert space and underlie its structure . specifically , the tensor product structure of the hilbert space is defined through a set of canonical observables ; the eigenstates of these observables define the natural basis . from this point of view , it now appears judicious to identify the relative momentum operators canonically conjugate to the relative position operators @xmath91 , @xmath88 and @xmath89 . that is , we are looking for operators @xmath92 , and @xmath93 satisfying the canonical commutation relations @xmath94=i\delta_{\alpha\beta}$ ] . a lengthy but straightforward calculation leads to the following set of operators : @xmath95 next , let us re - express the relative shift operator @xmath96 in terms of these canonically conjugate momentum operators : @xmath97 where we have used the fact that @xmath98 . from the above expression , it becomes clear that @xmath86 acts _ non - locally _ on the hilbert space spanned by the relative co - ordinates @xmath60 and @xmath68 , shifting both co - ordinates . previously we argued that the inaccessibility of particle 3 by particle 1 would allow us to trace over @xmath68 . however , this argument does not hold , because the relative shift operator @xmath99 also affects @xmath68 . we can now finally understand from particle 1 s perspective how it can extract the phase from the entangled state @xmath100 , as @xmath101 and therefore @xmath102 in agreement with . this solves the paradox . from particle 1 s perspective , it seemed impossible to retrieve the phase without interacting with particle 3 , due to the entanglement present . however , here we see that an operation that particle 1 can carry out on particle 2 alone ( essentially a measurement of its relative velocity ) , also indirectly affects the relative position of particle 3 , allowing the phase to be extracted even when particle 3 is physically inaccessible . the final piece of the puzzle is to understand physically the meaning of the apparent non - locality of the shift operator ; more precisely why it acts not only on the relative co - ordinate @xmath60 but also on @xmath68 . the key observation is that particle 1 has a finite mass . thus when it shifts particle 2 it also necessarily shifts itself , since the centre of mass of particle 1 and 2 must remain fixed . this shift in particle 1 s position changes the relative position of particle 3 versus particle 1 . in other words , when particle 1 performs an interference experiment on particle 2 , the relative distance between particles 1 and 3 is also necessarily changed . finally , it is worth noting that in the limit where particle 1 becomes very massive , and thus experiences no back - action , the non - locality of the hilbert space disappears , i.e. the observables @xmath103 and @xmath104 become once again canonically conjugate , as seen from eq . . the above illustrates that the hilbert space has a rich structure . in the two particle case the hilbert space possesses the normal bipartite structure that we expect , namely @xmath105 , where @xmath106 describes the centre of mass degrees of freedom and @xmath107 describes the degrees of freedom of particle 2 relative to particle 1 . in particular , the eigenstates of relative position and of relative momentum belong to the relative hilbert space , @xmath108 , @xmath109 , , and our statements can be taken to refer to these . ] and each can be written as a superposition of the other , e.g. @xmath110 , where @xmath111 . one is tempted to think that when we add a third particle and go to relative co - ordinates all we have to do is simply add on the hilbert space for the relative degrees of freedom for this third particle , i.e. that @xmath112 . however , this can not be the case indeed from eq . we already know that the operator @xmath87 does not commute with @xmath89 , hence they can not live in two different hilbert spaces . the hilbert spaces @xmath107 and @xmath113 therefore can have no meaning . in other words it is now no longer possible to have a hilbert space such as @xmath107 which contains both eigenstates of the relative position and momentum of particle 2 relative to particle 1 ; this also implies that the decomposition @xmath110 is no longer possible . in fact there exists two different decompositions of the hilbert space , @xmath114 and @xmath115 . in the first decomposition , the spaces are spanned by the eigenstates of relative position , @xmath88 and @xmath89 and of their conjugate momenta , @xmath116 and @xmath93 . in the second decomposition the spaces are spanned by the eigenstates of relative momenta , @xmath87 and @xmath117 and their conjugate position , which we shall denote @xmath118 and @xmath119 . following from the above , it is insightful to reverse the problem , that is to start from the relative momentum operators and find explicitly their conjugate relative positions . let us first complete the set of relative momentum operators : @xmath120 where @xmath121 is the total momentum , and @xmath87 and @xmath117 are the relative momenta of particles 2 and 3 versus particle 1 . now we look for the relative position operators canonically conjugate to these relative momentum operators ( i.e. satisfying @xmath122=i\delta_{\alpha\beta}$ ] ) . again , a lengthy but straightforward calculation leads to @xmath123 with @xmath124 . re - writing the state in terms of these new @xmath125 co - ordinates , we find that @xmath126 where @xmath127 . note that with these co - ordinates , the state is fully separable . in this basis , the shift operator acts ` locally ' on the @xmath118 coordinate : @xmath128 and hence @xmath129 as expected in conclusion , we have seen that the relative position and relative momentum operators , which are the physically relevant operators from the perspective of an internal observer , are not canonically conjugate . this has the consequence that some operators ( such as shifting particle 2 relative to particle 1 ) have an apparently ` non - local ' effect . consequently , one can not trace over the relative positions of inaccessible particles to give a description of the remaining accessible degrees of freedom . note that in the classical case , although there is no entanglement , it is still true that the usual relative positions and momenta are not canonically conjugate . there are two crucial distinctions however : firstly , a quantum measurement of @xmath87 must disturb @xmath89 due to their non - commutativity , whilst classically a measurement can always be taken to produce an arbitrarily small disturbance . secondly , the classical relative positions and momenta of particle 2 and 3 can all be independently specified , whilst quantum mechanically there is no state in which @xmath87 and @xmath89 are both simultaneously well defined . hence in quantum theory , the relative state is inherently non - local , with particle 2 and particle 3 intertwined . in all the cases we have considered , we have begun by defining the state in an external frame , and then moved to relative co - ordinates . when we refer to the centre of mass coordinate , we are really describing its position in this external frame . but does our analysis depend on which external frame we initially choose ? do we require an external frame of reference at all ? if so , what properties should it have ? these issues are of particular importance in section ii , where the experimental results depend critically on whether the relative coordinate is entangled with the centre of mass coordinate or not . the first thing to note is that if the centre of mass is in a superposition of different places in relation to one frame of reference it is also in a superposition relative to any other frame of reference that is classically related to the first frame , i.e. which has a well defined displacement and speed relative to this frame a galilean transformation . so certainly we do not single out one particular frame amongst all the frames related by a galilean transformation . however , in quantum mechanics , one might consider alternative ` non - classical ' transformations of the external reference frame such as moving to a frame in a superposition of different locations relative to the original frame @xcite . looking at figure ( 1b ) it is possible to imagine a frame of reference in which the interferometer is not in a superposition but located at the origin just go to a frame of reference which is in a superposition versus the original one and correlated with the location of the interferometer . in this frame of reference , the position of the centre of mass will be in the same place in each branch of the superposition , and hence it will factorise . does it mean that we are now able to measure interference ? of course , whether or not we are able to see interference is just a question of internal measurements and should be independent of the external reference frame , so there is an inconsistency . does this therefore mean that such reference frames which are in a superposition are not permitted ? is it the case that there exists an absolute _ set _ of reference frames which are related by galilean transformations but all other frames which are in superposition relative to them do not exist ? we seem to have run into a serious problem . these problems have several aspects , and below we will address them in turn . the first issue is whether we can determine the existence or properties of the external frame from inside the system alone . i.e. without interacting directly with the external frame . unsurprisingly , the answer to this question is no . as in classical mechanics , we are at liberty to describe any physical situation only in terms of relative co - ordinates . for each of the physical situations we consider , the correct relative description is obtained by tracing out the centre of mass degree of freedom , yielding a state of only the relative degrees of freedom . the hamiltonian can also always be decomposed into two commuting terms , one of which is a function of the total momentum alone , and the other is a function of only the relative positions and momenta . keeping only the second term will correctly describe the evolution of all the relative degrees of freedom without reference to the centre of mass . however , our results highlight two curious aspects of this approach : ( i ) that some situations which can be described by pure states in an external frame must be described using mixed states in terms of relative co - ordinates @xcite . ( ii ) that the concept of locality is much harder to pin down in terms of relative co - ordinates , as the relative position and momentum of one particle do not commute with those of the other particles . hence we can not simply trace out the relative co - ordinates of particles we are not interested in , and operations by the observer on one particle will affect the relative co - ordinates of the others . another interesting consideration is that of transforming between different external reference frames , rather than eliminating the external frame entirely . there are two ways in which we can think of an external reference frame , either ( i ) as an abstract labelling of space - time , or ( ii ) as a physical laboratory containing rulers and clocks . in case ( i ) transforming between reference frames is a purely mathematical transformation , a simple relabelling of the co - ordinates . in case ( ii ) the issue of transformation is more subtle . there are actually two subcases . the first way ( iia ) is to consider one single physical frame and going to a different frame being a purely mathematical transformation on the co - ordinates indicated on this physical frame . for example , the physical frame may be a ruler , and we can go to a displaced frame by simply relabelling the marks of the ruler . the second way ( iib ) is to have two actual physical systems . for example , we could take two rulers , one displaced relative to the other . classically these different approaches are completely equivalent to one another . quantum mechanically however , the situation is more complicated . let us illustrate this in the example of section ii , where we analysed a particle s interaction with an interferometer in two different situations . in the second case ( depicted in fig . ( 1b ) ) , the interferometer was prepared in a superposition of locations in the external reference frame , whilst the particle had a well - defined location . consider now moving to a new reference frame in which the interferometer is localised near the origin . let us see what happens if we attempt to change co - ordinates via a simple relabellings as in ( i ) or ( iia ) . the description of the state in the new reference frame can not be given by a pure state , because the only pure state in which the particle can be found with equal probability at @xmath1 and @xmath0 and the interferometer always found at the origin is of the form up to some well defined relative phase between @xmath130 and @xmath131 . this state however leads to different physical predictions than those made in the original frame , namely that there is interference in the interferometer . alternatively , we might seek to describe the state in the new reference frame using a mixed state , but then the transformation between reference frames must be irreversible , and we would not be able to transform back to the first reference frame and recover the original description . all of these problems stem from the fact that it makes no sense to relabel in superposition . superposition and entanglement are properties of the states of physical objects it is the physical object whose states can be in superposition or entangled ; there is no notion of abstract mathematical relabellings being in superposition . given the above difficulties , we now turn to the case ( iib ) , in which different _ external _ frames are considered as distinct physical laboratories . we take these laboratories to be non - interacting rigid quantum objects of very large mass , such that the positions and momenta of objects in the system relative to the laboratory obey essentially the usual canonical commutation relations ( consider ( [ q3p2 ] ) in the limit as @xmath132 ) . we can now approach the paradox discussed at the beginning of this section . consider the situation depicted in figure ( 1b ) but now modified to include two physical frames , the original frame @xmath133 and a second frame @xmath134 , in which the interferometer is located at the origin . in order for the interferometer to be located at the origin in @xmath134 it must be the case that it is actually entangled with the position of the interferometer . as we will show below , given that we now consider explicitly two frames of reference , the predictions made from each of them are in fact consistent , and there is no longer a paradox . as seen from frame @xmath133 , the state of the particle , interferometer and @xmath134 is @xmath135 note that this state is now different from ( [ eqn : setup_b ] ) , which did not include the frame @xmath134 . rewriting this in terms of the relative co - ordinates @xmath136 and @xmath137 of the particle and interferometer relative to the @xmath134 , we obtain @xmath138 tracing out the centre of mass , we see that the description of the state in @xmath134 is the mixed state @xmath139 , which will not yield interference . hence in this case the description relative to the original external frame @xmath133 and the new frame @xmath134 both give the same physical predictions . let us now modify the set - up in figure ( 1a ) in a similar manner , introducing two frames , the original frame @xmath133 and a new frame @xmath134 in which the particle is located at the origin . this modified set - up is now completely equivalent to the modified set - up of figure ( 1b ) with the role of the two frames interchanged . as such there can not be interference in the modified set - up ( 1a ) . this may seem surprising , since in the original set - up there was interference . indeed , the state now , instead of being ( [ eqn : setup_a ] ) is given by @xmath140 as seen from @xmath133 . if the frame @xmath134 is much more massive than the interferometer , this state can be expressed in relative co - ordinates as @xmath141 by tracing out the centre of mass , we obtain a mixed state from the perspective of the interferometer , hence interference will be lost . ) , as the position of the lab acts as a ` which - way ' detector for the photon . ] introducing the second physical frame @xmath134 into the state has therefore _ changed _ the physical predictions ( destroying the interference ) . this may seem surprising , since all the measurements we perform involve only the particle and the interferometer and we do not touch the frame @xmath134 during the experiment . why does then the mere existence of the frame @xmath134 change the physics ? this stems from the fact that we wanted to introduce the frame @xmath134 in such a way that it is entangled with our system . in the classical case we can introduce physical frames at will without any need to interact with the system and hence without changing its behaviour . however , in order to prepare an entangled state , there must be an interaction between the system and the frame . this is when the system is affected . finally , we can consider introducing a quantum frame that is not entangled with the system , but is in a superposition of locations . after tracing out the centre of mass , the situation observed from the perspective of such a frame will be the same as if it were prepared in a mixture of locations . this will not change the physical predictions relating to internal properties of the system ( such as interference in our interferometry experiments ) , but it will nevertheless constitute an irreversible transformation of the original state of the system , mapping pure states to mixed states and losing information . we investigated the physics of quantum reference frames by studying simple situations , involving only a few particles , whereby we promoted one of these particles to the role of a quantum observer . we then asked the question of how this quantum observer would describe the rest of the system . we first considered the case in which one of the particles is in superposition . we argued that the centre of mass , in particular whether it is entangled to the relative co - ordinates or not , is a central issue . second , we considered the situation in which two particles of the system are entangled such that the centre of mass and relative position are unentangled . we then introduced a third particle which led us to realise that the hilbert space of the quantum observer is inherently non - local . finally we discussed the issue of an absolute quantum frame of reference and showed that in quantum theory , introducing external reference frames can affect the behaviour of the system being studied in particular , when the additional reference frame and the system are taken to be entangled . in the future it would be interesting to explore further the consequences of the non - local structure of the hilbert space of quantum observers . furthermore it would be important to go beyond the one - dimensional case as new physical quantities arise in this setting , for example in two - dimensions and above we must also consider angular momentum . we acknowledge financial support from cnpq / brazil , the uk epsrc , the projects qip - irc and qap . ajs acknowledges support from the royal society . 99 y. aharonov and l. susskind , phys . rev . * 155 * , 1428 ( 1967 ) . y. aharonov and l. susskind , phys . rev . * 158 * 1237 ( 1967 ) y. aharonov and g. carmi , found . phys . * 3 * , 493 ( 1973 ) . y. aharonov and g. carmi , found . physics * 4 * , 75 ( 1974 ) . y. aharonov and t. kaufherr , phys . d * 30 * , 368 ( 1984 ) . d. poulin and j. yard , new j. phys . * 9 * , 156 ( 2007 ) . boileau , l. sheridan , m. laforest and s.d . bartlett , j. math . phys . * 49 * , 032105 ( 2008 ) . f. girelli and d. poulin , phys . d * 77 * , 104012 ( 2008 ) . g. gour and r. spekkens , new j. phys . * 10 * , 033023 ( 2008 ) . g. gour , i. marvian and r. spekkens , phys . a * 80 * , 012307 ( 2009 ) . j. oppenheim and b. reznik , arxiv:0902.2361 [ hep - th ] s.d . barlett , t. rudolph , and r. spekkens , rev . phys . * 79 * , 555 ( 2007 ) . s. massar and s. popescu , phys . lett . * 74 * , 1259 ( 1995 ) . n. gisin and s. popescu , phys . lett . * 83 * , 432 ( 1999 ) . a. peres and p. f. scudo , phys . lett . * 86 * , 4160 ( 2001 ) . e. bagan , m. baig , a. brey , r. munoz - tapia and r. tarrach , phys . a * 63 * , 052309 ( 2001 ) . bartlett , t. rudolph , r. spekkens and p.s . turner , new j. phys . * 11 * , 063013 ( 2009 ) . ioannou , and m. mosca , arxiv:0903.5156 . f. costa , n. harrigan , t. rudolph , and c. brukner , new j. phys . * 11 * , 123007 ( 2009 ) . liang , n. harrigan , s.d . bartlett , and t. rudolph , phys . lett . * 104 * , 050401 ( 2010 ) . y. aharonov and d. rohlich , _ quantum paradoxes quantum theory for the perplexed _ , wiley - vch verlag gmbh & co. kgaa , weinheim ( 2005 ) . y. aharonov , h. pendleton , and a. petersen , int . * 2 * , 213 ( 1969 ) . a. kiteav , d. mayers and j. preskil , phys . a * 69 * , 052326 ( 2004 ) s.d . bartlett , t. rudolph and r. spekkens , int . . inf . * 4 * , 17 ( 2006 ) . in this appendix , we further discuss the subtle issues of entanglement due to finite wavepacket width that were introduced in section [ sec:2_particle ] . we note in the main text that a product state of two wavepackets in the external frame of reference will generally become entangled when expressed in terms of relative co - ordinates . for clarity , we have made an approximation in which this entanglement of microscopic fluctuations is ignored , such that a product state in the external reference frame remains a product state when transformed into relative co - ordinates . we therefore write @xmath142 throughout . note that although fluctuations in the centre of mass and relative coordinate only extend over a small physical distance , this sense of approximation does not correspond to equivalence between the states for all measurements ( as the trace distance would ) . in particular , fine - grained measurements on the scale of a wavepacket might be able to distinguish @xmath12 from any particular product state @xmath143 . however , such measurements are not relevant to our arguments . furthermore , for some special choices of initial wavepacket in the two particle case , our approximation holds exactly . for example consider a product state of two gaussian wavepackets in the external frame : @xmath144 i.e. particle 1 is centred around the position @xmath14 and has uncertainty @xmath145 and particle 2 is centred around @xmath15 with uncertainty @xmath146 . moving to relative co - ordinates , using equations ( [ eqn : x_cm])-([eqn : x_r ] ) from the main text , we find the state transforms into @xmath147 where @xmath148 hence in general , the relative and centre of mass coordinate will be entangled . however , if we choose @xmath149 we obtain @xmath150 and recover a product state . by taking the initial wavepackets to be gaussians satisfying @xmath149 and the transformed wavepackets to be gaussians of widths @xmath151 and @xmath152 , we could therefore eliminate our approximation and have @xmath153 . however , we emphasise that our central observations extend beyond this special case . all we really require are that wavepackets are sufficiently small , and that all wavepackets for the same particle in an initial state have the same form ( i.e. they are translated copies of each other ) . with three particles , the centre of mass will again decouple from the relative co - ordinates when the initial wavepackets are gaussians satisfying @xmath154 . however , in this case there will remain some residual entanglement between the relative co - ordinates @xmath60 and @xmath68 in the transformed state . if we wished to include this entanglement explicitly , we could rewrite ( [ psi ] ) as @xmath155 where @xmath156 represents an entangled wavepacket in the relative co - ordinates , centred on @xmath157 and @xmath158 . the reduced state of @xmath159 given by ( [ red_state ] ) would then be modified to @xmath160 where @xmath161 are mixed wavepackets centred on @xmath162 , and we have used the fact that @xmath163 . note that the phase is still unobservable in this state , and hence the conclusions are identical to those in the approximated case .
|
we investigate the physics of quantum reference frames . specifically , we study several simple scenarios involving a small number of quantum particles , whereby we promote one of these particles to the role of a quantum observer and ask what is the description of the rest of the system , as seen by this observer ?
we highlight the interesting aspects of such questions by presenting a number of apparent paradoxes . by unravelling these paradoxes
we get a better understanding of the physics of quantum reference frames .
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.